https://dispersivewiki.org/DispersiveWiki/api.php?action=feedcontributions&user=Stanford88&feedformat=atomDispersiveWiki - User contributions [en]2024-03-28T18:45:47ZUser contributionsMediaWiki 1.39.3https://dispersivewiki.org/DispersiveWiki/index.php?title=Bibliography&diff=7013Bibliography2009-07-18T01:19:18Z<p>Stanford88: </p>
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<div>__NOTOC__<br />
<br />
=== Important note ===<br />
<br />
This bibliography was ported over from the previous incarnation of these pages. It should now be<br />
considered obsolete. Instead, we are now [[Help:References|giving each reference its own page]].<br />
<br />
The naming system for these articles was somewhat implicit and ''ad hoc''; we are currently forming [[Help:Bibliographic naming conventions|more explicit standards]] for this.<br />
<br />
<br />
=== References ===<br />
<br />
<center>A</center><br />
<br />
<br />
=====[[Aa2002]]=====<br />
M. Aassilla, ''Nonexistence de solutions globales de certains equations d'ondes nonlineaires'', C. R. Acad. Sci. Paris. '''334''' (2002), 961-966.<br />
<br />
=====[[AbdBnFelSau1989]]=====<br />
L. Abdelouhab, J. Bona, M. Felland, J. Saut, ''Nonlocal models for nonlinear dispersive waves'', Phys. D '''40''' (1989), 360-392.<br />
<br />
=====[[AbKauNeSe1974]]=====<br />
M. Ablowitz, D. Kaup, A. Newell, H. Segur, ''The inverse scattering transform-Fourier analysis for nonlinear problems'', Studies in Appl. Math. '''53''' (1974), no. 4, 249-315.<br />
<br />
=====[[AbFs1983]]=====<br />
M. Ablowitz, A. Fokas, ''The inverse scattering transform for the Benjamin-Ono equation, a pivot for multidimensional problems'', Stud. Appl. Math. '''68''' (1983), 1-10.<br />
<br />
=====[[AbHab1975]]=====<br />
M. Ablowitz, R. Haberman, ''Nonlinear evolution equations in two and three dimensions'', Phys. Rev. Lett. '''35''' (1975), 1185-1188.<br />
<br />
=====[[AbMa1981]]=====<br />
M. Ablowitz, Y. Ma, The periodic nonlinear Schrodinger equation, Stud. Appl. Math. 65 (1981), 113-158.<br />
<br />
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M. Ablowitz, H. Segur, ''Asymptotic solutions of the Korteweg de Vries equation'', Stud. Appl. Math. '''57''' (1977), 13-44.<br />
<br />
=====[[AdhAds1988]]=====<br />
H. Added, S. Added, ''Equations of Langmuir turbulence and non-linear Schrodinger equation: Smoothness and approximations'', JFA '''79''' (1988), 183-210.<br />
<br />
=====[[AgYok1998]]=====<br />
R. Agemi, K. Yokoyama, The null condition and global existence of solutions to systems of wave equations with different speeds, Adv. Nonlinear PDE and Stochastics (1998), 43-86.<br />
<br />
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J.C. Alexander, R.L. Pego, R.L. Sachs, ''On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation'', Phys. Lett. A. '''226''' (1997), 187-192.<br />
<br />
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S. Alinhac, ''Blowup for nonlinear hyperbolic equations'', Boston<nowiki>: Birkhauser, 1995, Progress in </nowiki>Nonlinear DE and their Applications, 17.<br />
<br />
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S. Alinhac, ''Blow up of small data solutions for a class of quasilinear wave equations in two space dimensions I'', Annals of Mathematics 149 (1999), 97-127.<br />
<br />
=====[[Al1999b]]=====<br />
S. Alinhac, ''Blow up of small data solutions for a class of quasilinear wave equations in two space dimensions II,'' Acta Math. 182 (1999), 1-23.<br />
<br />
=====[[Al2000]]=====<br />
S. Alinhac, Rank 2 singular solutions for quasilinear wave equations, IMRN 18 (2000), 955-984.<br />
<br />
=====[[Al2001]]=====<br />
S. Alinhac, ''The null condition for quasilinear wave equations in two space dimensions I'', Invent. Math. '''145''' (2001), 597-618.<br />
<br />
=====[[Al2001b]]=====<br />
S. Alinhac, ''The null condition for quasilinear wave equations in two space dimensions II, ''AJM '''123''' (2001), 1071-1101.<br />
<br />
=====[[Al2003]]=====<br />
S. Alinhac, An example of blowup at infinity for a quasilinear wave equation, Asterisque 284 (2003), 1-91.<br />
<br />
=====[[AcIb2000]]=====<br />
S. Anco, J. Isenberg, ''Global existence for wave maps with torsion'', CPDE '''25''' (2000), 1669-1702.<br />
<br />
=====[[AnMc2003]]=====<br />
L. Andersson, V. Moncrief, ''[http://www.arxiv.org/abs/gr-qc/0110111 Elliptic-Hyperbolic systems and the Einstein equations]'', Ann. Henri Poincaré '''4''' (2003), 1-34<br />
<br />
=====[[Ant2003]]=====<br />
C. Antonini, Lower bounds for the L^2 minimal periodic blowup solutions of critical nonlinear Schrodinger equation, DIE 15 (2002), 749-768.<br />
<br />
=====[[AntMe2001]]=====<br />
C. Antonini, F. Merle, ''Optimal bounds on positive blowup solutions for a semilinear wave equation'', IMRN '''21''' (2001), 1143-1167.<br />
<br />
=====[[ArYa2000]]=====<br />
G. Artbazar, K. Yajima, ''The L^p continuity of wave operators for one dimensional Schrodinger operators'', J. Math. Sci. Univ. Tokyo '''7''' (2000), 221-240.<br />
<br />
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J. M. Ash, J. Cohen, G. Wang, On strongly interacting internal solitary waves, J. Fourier Anal. Appl., '''2''' (1996), 507-517.<br />
<br />
<center>B</center><br />
<br />
<br />
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A. Bachelot, ''Probleme de Cauchy pour des systemes hyperboliques semilineaires'', Ann. IHP (Anal. nonlin), '''1''' (1984), 453-478.<br />
<br />
=====[[BchNic1993]]=====<br />
A. Bachelot, J-P. Nicolas, Equation non-lineaire de Klein-Gordon dans des metriquesde type Schwarzchild, C.R. Acad. Sci. Paris. 316 (1993), 1047-1050<br />
<br />
=====[[BlChd1978]]=====<br />
J. B. Baillon, J. M. Chadam, ''The Cauchy problem for the coupled Schrodinger-Klein-Gordon equations'', in "Contemporary Developments in Continuum Mechanics and Partial Differential Equations", G. M. de La Penha and L. A. Medeiros eds., North Holland, Amsterdam 1978<br />
<br />
=====[[BaeSgZz1990]]=====<br />
J. Baez, I. Sega, Z. Zhou, ''The global Goursat problem and scattering for nonlinear wave equations'', J. Funct. Anal. '''93 '''(1990), 239-269.<br />
<br />
=====[[BaCh1999]]=====<br />
H. Bahouri, J-Y. Chemin, ''Équations d'ondes quasilinéaires et les inegalites de Strichartz'', Amer. J. Math.'''121''' (1999), 1337-1377.<br />
<br />
=====[[BaCh1999b]]=====<br />
H. Bahouri, J-Y. Chemin, ''Équations d'ondes quasilinéaires et effet dispersif'', IMRN '''21''' (1999), 1141-1178.<br />
<br />
=====[[BaCh2002]]=====<br />
H. Bahouri, J-Y. Chemin, ''[http://arXiv.org/abs/math.AP/0304390 Quasilinear Wave equations and Microlocal Analysis]'', ICM 2002, Vol III, 141-154.<br />
<br />
=====[[BaCh 2003]]=====<br />
H. Bahouri, J-Y. Chemin, ''Microlocal analysis, bilinear estimates, and cubic quasilinear wave equation'', Asterique (2003), 93-141<br />
<br />
=====[[BaGd1997]]=====<br />
H. Bahouri and P.Gerard, ''High frequency approximation of solutions to critical nonlinear wave equations'', Prepublications 97-34, Universite de Paris-Sud, Mai 1997. Appeared in: Amer. J. Math '''121''' (1999), 131-175.<br />
<br />
=====[[BaGdXu2000]]=====<br />
H. Bahouri, P. Gerard, C. Xu, ''Espaces de Besov et estimations de Strichartz generalisees sur le groupe de Heisenberg'', Journal d'Analyse Mathematique '''82''' (2000), 93-118.<br />
<br />
=====[[BaSa1998]]=====<br />
H. Bahouri and J.Shatah, ''Decay estimates for the critical semilinear wave equation'', Ann. Inst. H. Poincare Anal. Non Lineaire '''15''' (1998), no. 6, 783-789.<br />
<br />
=====[[Ban2004a]]=====<br />
V. Banica, ''On the nonlinear Schrödinger dynamics on S^2'', J. Math. Pures Appl. (9)'''83''' (2004), 77-98. <br />
<br />
=====[[Ban2004b]]=====<br />
V. Banica, ''Dispersion and Strichartz inequalities for Schrödinger equations with singular coefficients.'', SIAM J. Math. Anal. '''35'''(2004), 868-883. <br />
<br />
=====[[Ban2004c]]=====<br />
V. Banica, ''Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain. '',Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) '''3'''(2004), 139-170. <br />
<br />
=====[[Bb1984]]=====<br />
J.E. Barab, ''Nonexistence of asymptotically free solutions for a nonlinear Schrodinger equation'', J. Math. Phys. '''25''' (1984), 3270-3273.<br />
<br />
=====[[Be1983]]=====<br />
M. Beals, ''Self-Spreading and strength of Singularities for solutions to semilinear wave equations'', Annals of Math '''118''' (1983), 187-214<br />
<br />
=====[[BeBz1996]]=====<br />
M. Beals, M. Bezard, ''Low regularity local solutions for field equations'', Comm. Partial Differential Equations'''21''' (1996), 79-124.<br />
<br />
=====[[BecMauSb-p]]=====<br />
P. Bechouce, N. Mauser, S. Selberg, ''[http://arXiv.org/abs/math.AP/0202201 Nonrelativistic limit of Klein-Gordon-Maxwell to Schrodinger Poisson]'', preprint<br />
<br />
=====[[BecMauSb-p2]]=====<br />
P. Bechouce, N. Mauser, S. Selberg, ''[http://arXiv.org/abs/math.AP/0303079 On the asymptotic analysis of the Dirac-Maxwell system in the nonrelativistic limit]'', preprint<br />
<br />
=====[[BkOgPo1996]]=====<br />
D. Bekiranov, T. Ogawa, G. Ponce, ''On the well-posedness of Benney's interaction equation of short and long waves'', Adv. Diff. Eq. '''1''' (1996), 919-937.<br />
<br />
=====[[BkOgPo1997]]=====<br />
D. Bekiranov, T. Ogawa, G. Ponce, ''Weak solvability and well-posedness of the coupled Schrodinger Korteweg-de Vries equation in the capillary-gravity interaction waves'', Proc. Amer. Math. Soc. '''125''' (1997), 2907-2919.<br />
<br />
=====[[BkOgPo1998]]=====<br />
D. Bekiranov, T. Ogawa, G. Ponce, ''Interaction equations for short and long dispersive waves'', J. Funct. Anal. '''158''' (1998), 357-388.<br />
<br />
=====[[BcKkZz1999]]=====<br />
E. Belchev, M. Kepka, Z. Zhou, ''Global existence of solutions to nonlinear wave equations'', CPDE '''24 '''(1999), 2297-2331.<br />
<br />
=====[[BcKkZz2002]]=====<br />
E. Belchev, M. Kepka, Z. Zhou, ''Finite time blowup of solutions to semilinear wave equations'', JFA '''190 '''(2002), 233-254.<br />
<br />
=====[[Bj1967]]=====<br />
T. Benjamin, ''Internal waves of permanent form in fluids of great depth'', J. Fluid Mech. '''29''' (1967), 559-562.<br />
<br />
=====[[Bj1972]]=====<br />
T. Benjamin, ''The stability of solitary waves'', Proc. R. Soc. Lond. A. '''328''' (1972), 153-183.<br />
<br />
=====[[BjBnMah1972]]=====<br />
T. Benjamin, J. Bona, J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil. Trans. R. Soc. London A 272 (1972), 47-78.<br />
<br />
=====[[BenKl1992]]=====<br />
M. Ben-Artzi, S. Klainerman, ''Decay and regularity for the Schrodinger equation'', J. Anal. Math. '''58''' (1992), 25-37.<br />
<br />
=====[[BenSau1999]]=====<br />
M. Ben-Artzi, J.C. Saut, ''Uniform decay estimates for a class of oscillatory integrals and applications'', Diff. Int. Eq. '''12''' (1999), 137-145.<br />
<br />
=====[[BenKocSau2003]]=====<br />
M. Ben-Artzi, H. Koch, J.C. Saut, Dispersion estimates for third order equations in two dimensions, CPDE 28 (2003), 1943-1974.<br />
<br />
=====[[ByNe1967]]=====<br />
D.J. Benney, A.C. Newell, ''The propagation of nonlinear wave envelopes'', J. Math. and Phys. '''46''' (1967), 133-139.<br />
<br />
=====[[BerCa1981]]=====<br />
H. Berestycki, T. Cazenave, ''Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires'', C. R. Acad. Sc. Paris, t. 293 (1981), 489-492.<br />
<br />
=====[[BgCcMc1995]]=====<br />
B.K. Berger, P. T. Chrusciel, V. Moncrief, ''On "asymptotically flat" spacetimes with G_2 invariant Cauchy surfaces'', Ann. Physics '''237''' (1995), 322-354.<br />
<br />
=====[[BiLi-p]]=====<br />
<br />
H. Biagioni, F. Linares, ''Ill-posedness for the Derivative Schrodinger and Generalized Benjamin-Ono equations'', preprint.<br />
<br />
=====[[BirPoVe-p]]=====<br />
B. Birnir, G. Ponce, L. Vega, On the ill-posedness for the initial value problem for the modified Korteweg-de Vries equation, preprint.<br />
<br />
=====[[BirKnPoSvVe1996]]=====<br />
B. Birnir, C. Kenig, G. Ponce, N. Svanstedt, L. Vega, ''On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrödinger equations''. J. London Math. Soc. (2) '''53''' (1996), 551-559.<br />
<br />
=====[[Biz2000]]=====<br />
P. Bizon, ''Equivariant self-similar wave maps from Minkowski spacetime into 3-sphere.'' Comm. Math. Phys. '''215''' (2000), 45-56.<br />
<br />
=====[[Biz-p]]=====<br />
P. Bizon, ''[http://arXiv.org/abs/math-ph/0206004 Formation of singularities in Yang-Mills equations]'', preprint<br />
<br />
=====[[BizCjTb2001]]=====<br />
P. Bizon, T. Chmaj, Z. Tabor, ''[http://arxiv.org/abs/math-ph/0011005 Formation of singularities for equivariant 2+1 dimensional wave maps into two-sphere]'', Nonlinearity '''14''' (2001), no. 5, 1041-1053.<br />
<br />
=====[[BizOvSi-p]]=====<br />
P. Bizon, Y. Ovchinnikov, I. Sigal, ''[http://arxiv.org/abs/math.AP/0307026 Collapse of an Instanton]'', preprint.<br />
<br />
=====[[BizTb2001]]=====<br />
P. Bizon, Z. Tabor, ''[http://arxiv.org/abs/math-ph/0105016 On blowup for Yang-Mills fields]'', Phys. Rev. D (3) '''64''' (2001), 121701.<br />
<br />
=====[[BluSf2003]]=====<br />
P. Blue, A. Soffer, Semilinear wave equations on the Schwarzschild manifold I.Local decay estimates, Adv. Diff Eq. 8 (2003), 595-614.<br />
<br />
=====[[Bn1975]]=====<br />
J. Bona, ''On the stability theory of solitary waves'', Proc. R. Soc. Lond. A. '''344''' (1975), 363-374.<br />
<br />
=====[[BnDgKar1986]]=====<br />
J. Bona, V.A. Dougalis, O.A. Karakashian, ''Fully discrete Galerkin methods for the Korteweg-de Vries equation'', Comput. Math. Appl. Ser. A '''12''' (1986), 859-884.<br />
<br />
=====[[BnLiu2002]]=====<br />
J. Bona, Y. Liu, ''Instability of solitary-wave solutions of the Kadomtsev-Petviashvili equation in three dimensions'', ADE '''7''' (2002), 1-33.<br />
<br />
=====[[BnPoSauTm1992]]=====<br />
J. Bona, G. Ponce, J-C. Saut, M. Tom, ''A model system for strong interaction between internal solitary waves'', CMP '''143''' (1992), 287-313.<br />
<br />
=====[[BnSco1976]]=====<br />
J. Bona, R. Scott, ''Solutions of the Korteweg-de Vries equation, in fractional order Sobolev spaces'', Duke Math J. '''43''' (1976), 87-99.<br />
<br />
=====[[BnSmr1975]]=====<br />
J. Bona, R. Smith, ''The initial-value problem for the Korteweg-de Vries equation'', Philos. Trans. Royal Soc. London Series A 278 (1975), 555-601.<br />
<br />
=====[[BnSouSr1987]]=====<br />
J. Bona, P.E. Sougandis, W. Strauss, ''Stability and instability of solitary waves of Korteweg-de Vries type'', Proc. R. Soc. Lond. '''411''' (1987), 395-412.<br />
<br />
=====[[BnSuZh-p]]=====<br />
J. Bona, S. Sun, B-Y. Zhang, ''A non-homogeneous boundary-value problem for the Korteweg-de Vries Equation in a Quarter Plane'', preprint<br />
<br />
=====[[BnWi1983]]=====<br />
J. Bona, R. Winther, ''The Korteweg-de Vries equation, posed in a quarter plane'', SIAM J. Math. Anal. '''14''' (1983), 1056-1106.<br />
<br />
=====[[BnWe-p]]=====<br />
J. Bona, F. B. Weissler, ''Similarity solutions of the generalized Korteweg-de Vries equations'', preprint.<br />
<br />
=====[[Bd1993]]=====<br />
A. de Bouard, Analytic solutions to nonelliptic nonlinear Schrodinger equations, JDE 104 (1993), 196-213.<br />
<br />
=====[[BdDbTs1999]]=====<br />
A. de Bouard, A. Debussche, Y. Tsutsumi, ''[http://www.ams.org/mathscinet?screen=leavingmsn&publ=Academic%20Press&url=http://www.idealibrary.com/cgi%2Dbin/links/citation/0022%2D1236/169/532&back_url=msnmain%3Fpg3=ICN%26s3=Tsutsumi%26co3=AND%26pg4=TI%26s4=%26co4=AND%26pg5=CC%26s5=%26co5=A%20 White noise driven Korteweg-de Vries equation.]'' J. Funct. Anal. '''169''' (1999), no. 2, 532-558<br />
<br />
=====[[BdSau1997]]=====<br />
A. de Bouard, J.C. Saut, ''Solitary waves of generalized Kadomtsev-Petviashvili equations'', Annales IHP, Analyse non-lineaire '''14''' (1997) 211-236.<br />
<br />
=====[[BdHaSau1997]]=====<br />
A. de Bouard, N. Hayashi, J.C. Saut, Global existence of small solutions to a nonlinear relativistic Schrodinger equation, CMP 189 (1997), 73-105.<br />
<br />
=====[[BdMt2004]]=====<br />
A. de Bouard, Y. Martel, Nonexistence of L^2-compact solutions of the Kadomptsev-Petviashvili II equation, Math. Ann. 328 (2004), 525\u2014554.<br />
<br />
=====[[BctGolPal-p]]=====<br />
F. Bouchut, F. Golse, C. Pallard, ''[http://front.math.ucdavis.edu/math.AP/0301175 On classical solutions to the 3D relativistic Vlasov-Maxwell system: Glassey-Strauss's theorem revisited]'', preprint.<br />
<br />
=====[[Bo1992]]=====<br />
J. Bourgain, ''A remark on Schrodinger operators'', Israel J. Math. '''77''' (1992), 1-16.<br />
<br />
=====[[Bo1993]]=====<br />
J. Bourgain, ''Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I. Schrodinger equations,'' [http://link.springer.de/link/service/journals/00039/index.htm Geom. Funct. Anal.] '''3''' (1993) 107-156.<br />
<br />
=====[[Bo1993b]]=====<br />
J. Bourgain, ''Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II. The periodic KdV equation,'' [http://link.springer.de/link/service/journals/00039/index.htm Geom. Funct. Anal.] '''3''' (1993) 209-262.<br />
<br />
=====[[Bo1993c]]=====<br />
J. Bourgain, ''On the Cauchy problem for the Kadomtsev-Petviashvili equation'', [http://link.springer.de/link/service/journals/00039/index.htm Geom. Funct. Anal.] '''3''' (1993) 315-341.<br />
<br />
=====[[Bo1993d]]=====<br />
J. Bourgain, ''Exponential sums and nonlinear Schrodinger equations'', [http://link.springer.de/link/service/journals/00039/index.htm Geom. Funct. Anal.] '''3''' (1993) 157-178.<br />
<br />
=====[[Bo1994]]=====<br />
J. Bourgain, ''On the Cauchy and invariant measure problem for the periodic Zakharov system'', Duke Math. J. '''76''' (1994), 175-202.<br />
<br />
=====[[Bo1994b]]=====<br />
J. Bourgain, ''Harmonic analysis and nonlinear partial differential equations'', Proceedings of the ICM 1994, 31-44, Birkhauser Basel, 1995.<br />
<br />
=====[[Bo1994c]]=====<br />
J. Bourgain, ''Periodic nonlinear Schrodinger equation and invariant measures'', CMP '''166''' (1994), 1-26.<br />
<br />
=====[[Bo1995]]=====<br />
J. Bourgain, ''Estimates for cone multipliers'', Operator Theory: Advances and Applications, '''77''' (1995), 41-60.<br />
<br />
=====[[Bo1995b]]=====<br />
J. Bourgain, ''Aspects of longtime behaviour of solutions of nonlinear Hamiltonian evolution equations'', GAFA ''5'' (1995), 105-140.<br />
<br />
=====[[Bo1995c]]=====<br />
J. Bourgain, ''Periodic nonlinear Schrodinger equations and invariant measures'', CMP '''166''' (1995), 1-26.<br />
<br />
=====[[Bo1996]]=====<br />
J. Bourgain, ''On well-posedness of the Zakharov system'', [http://www.imrn.org Int. Math. Research Notices] '''11''' (1996), 515-546.<br />
<br />
=====[[Bo1996b]]=====<br />
J. Bourgain, ''Invariant measures for the 2d-defocussing nonlinear Schrodinger equation'', Comm. Math. Phys. '''176''' (1996), 421-445.<br />
<br />
=====[[Bo1996c]]=====<br />
J. Bourgain, ''On the growth in time of higher order Sobolev norms of smooth solutions of Hamiltonian PDE'', IMRN '''6''' (1996), 277-304.<br />
* [[kdv_measures.pdf Bo1997]] J. Bourgain, ''[kdv_measures.pdf Periodic Korteweg de Vries equation with measures as initial data]'', Selecta Math. (N.S.) '''3''' (1997), 115-159.<br />
<br />
=====[[Bo1997b]]=====<br />
J. Bourgain, ''On the compactness of support of solutions of dispersive equations'', [http://www.imrn.org Int. Math. Research Notices] '''9''' (1997), 437-447.<br />
<br />
=====[[Bo1998]]=====<br />
J. Bourgain, ''Refinements of Strichartz' inequality and applications to 2D-NLS with critical non-linearity'', [http://www.imrn.org Int. Math. Research Notices], '''5''' (1998), 253-283.<br />
<br />
=====[[Bo1998b]]=====<br />
J. Bourgain, ''Scattering in the energy space and below for 3D NLS'', J. Anal. Math. '''75''' (1998), 267-297.<br />
<br />
<br />
=====[[Bo1999]]===== J. Bourgain, ''[http://www.math.ucla.edu/~tao/bourgain/borg_nls.dvi New global well-posedness results for non-linear Schrodinger equations]'', AMS 1999.<br />
<br />
=====[[Bo1999b]]=====<br />
J. Bourgain, ''Global well-posedness of defocussing 3D critical NLS in the radial case'', J. Amer. Math. Soc. '''12''' (1999), 145-171.<br />
<br />
=====[[Bo-p]]=====<br />
J. Bourgain, Remarks on stability and diffusion in high-dimensional Hamiltonian systems and PDE, preprint.<br />
<br />
=====[[Bo-p2]]=====<br />
J. Bourgain, A remark on normal forms and the I-method for periodic NLS, preprint.<br />
<br />
=====[[Bo-p3]]=====<br />
J. Bourgain, On the Cauchy problem for periodic KdV type equations, preprint.<br />
<br />
=====[[BoCo1996]]=====<br />
J. Bourgain, J. Colliander, ''On the well-posedness of the Zakharov system'', IMRN '''11''' (1996), 515-546.<br />
<br />
=====[[BoWg1997]]=====<br />
J. Bourgain, W. Wang, ''Construction of blowup solutions for the nonlinear Schrodinger equation with critical non-linearity,'' Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) '''25''' (1997), no. 1-2, 197-215.<br />
<br />
=====[[Bou1996]]=====<br />
N. Bournaveas, ''Local existence for the Maxwell-Dirac equations in three space dimensions''. Comm. Partial Differential Equations '''21''' (1996) 693-720.<br />
<br />
=====[[Bou1999]]=====<br />
N. Bournaveas, ''Local existence of energy class solutions for the Dirac-Klein-Gordon equations''. Comm. Partial Differential Equations '''24''' (1999), 1167-1193.<br />
<br />
=====[[Bou2000]]=====<br />
N. Bournaveas, ''A new proof of global existence for the Dirac Klein-Gordon equations in one space dimension''. J. Funct. Anal. '''173''' (2000), 203-213.<br />
<br />
=====[[Bou2001]]=====<br />
N. Bournaveas, ''Low regularity solutions of the Dirac Klein-Gordon equations in two space dimensions''. Comm. Partial Differential Equations '''26''' (2001), 1345-1366.<br />
<br />
=====[[BdmFsShp-p]]=====<br />
A. Boutet de Monvel, A. Fokas, ''The mKdV equation on a half-line'', preprint.<br />
<br />
=====[[BdmShp-p]]=====<br />
A. Boutet de Monvel, D. Shepelsky, ''[http://arxiv.org/abs/math.AP/0307194 The mKdV equation on an interval]'', preprint.<br />
<br />
=====[[BrlForMai1994]]=====<br />
P. Breitenlohner, P. Forgacs, D. Maison, ''Static spherically symmetric solutions of the Einstein-Yang-Mills equations. ''Comm. Math. Phys. '''163''' (1994), no. 1, 141-172.<br />
<br />
=====[[Br1975]]=====<br />
P. Brenner, ''On L^p - L^{p'} estimates for the wave equation'', Math Z. '''145''' (1975) 251-254.<br />
<br />
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[[Category:Bibliography]]</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=Bibliography&diff=7012Bibliography2009-07-18T01:14:55Z<p>Stanford88: /* Ban2004a */</p>
<hr />
<div>__NOTOC__<br />
<br />
=== Important note ===<br />
<br />
This bibliography was ported over from the previous incarnation of these pages. It should now be<br />
considered obsolete. Instead, we are now [[Help:References|giving each reference its own page]].<br />
<br />
The naming system for these articles was somewhat implicit and ''ad hoc''; we are currently forming [[Help:Bibliographic naming conventions|more explicit standards]] for this.<br />
<br />
<br />
=== References ===<br />
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{{Article <br />
| author = V. Banica <br />
| title = Dispersion and Strichartz inequalities for Schrödinger equations with singular coefficients. <br />
| journal = SIAM J. Math. Anal.<br />
| jvol = 35<br />
| jpage = 868-883 <br />
| year = 2003 <br />
| arxivid = <br />
| mathsciid = MR2049025 <br />
}}<br />
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K. Yajima, G. Zhang, Smoothing property for Schrodinger equations with potential superquadratic at infinity, CMP 221 (2001), 573-590.<br />
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[[Category:Bibliography]]</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=Ban-p3&diff=7011Ban-p32009-07-18T01:08:21Z<p>Stanford88: Ban-p3 moved to Ban2004c: not preprint any longer</p>
<hr />
<div>#REDIRECT [[Ban2004c]]</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=Ban2004c&diff=7010Ban2004c2009-07-18T01:08:21Z<p>Stanford88: Ban-p3 moved to Ban2004c: not preprint any longer</p>
<hr />
<div>{{Article <br />
| author = V. Banica <br />
| title = Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain. <br />
| journal = Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)<br />
| jvol = 3<br />
| jpage = 139-170 <br />
| year = 2004<br />
| arxivid = math/0401129 <br />
| mathsciid = MR2064970 <br />
}}</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=Bibliography&diff=7009Bibliography2009-07-18T01:07:09Z<p>Stanford88: /* Ban-p2 */</p>
<hr />
<div>__NOTOC__<br />
<br />
=== Important note ===<br />
<br />
This bibliography was ported over from the previous incarnation of these pages. It should now be<br />
considered obsolete. Instead, we are now [[Help:References|giving each reference its own page]].<br />
<br />
The naming system for these articles was somewhat implicit and ''ad hoc''; we are currently forming [[Help:Bibliographic naming conventions|more explicit standards]] for this.<br />
<br />
<br />
=== References ===<br />
<br />
<center>A</center><br />
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H. Bahouri, J-Y. Chemin, ''Microlocal analysis, bilinear estimates, and cubic quasilinear wave equation'', Asterique (2003), 93-141<br />
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H. Bahouri and P.Gerard, ''High frequency approximation of solutions to critical nonlinear wave equations'', Prepublications 97-34, Universite de Paris-Sud, Mai 1997. Appeared in: Amer. J. Math '''121''' (1999), 131-175.<br />
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H. Bahouri, P. Gerard, C. Xu, ''Espaces de Besov et estimations de Strichartz generalisees sur le groupe de Heisenberg'', Journal d'Analyse Mathematique '''82''' (2000), 93-118.<br />
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H. Bahouri and J.Shatah, ''Decay estimates for the critical semilinear wave equation'', Ann. Inst. H. Poincare Anal. Non Lineaire '''15''' (1998), no. 6, 783-789.<br />
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=====[[Ban2004a]]=====<br />
{{Article <br />
| author = V. Banica <br />
| title = On the nonlinear Schrödinger dynamics on S^2<br />
| journal = J. Math. Pures Appl. (9)<br />
| jvol = 83<br />
| jpage = 77-98 <br />
| year = 2004 <br />
| arxivid = <br />
| mathsciid = MR2032582<br />
}}<br />
<br />
=====[[Ban2004b]]=====<br />
{{Article <br />
| author = V. Banica <br />
| title = Dispersion and Strichartz inequalities for Schrödinger equations with singular coefficients. <br />
| journal = SIAM J. Math. Anal.<br />
| jvol = 35<br />
| jpage = 868-883 <br />
| year = 2003 <br />
| arxivid = <br />
| mathsciid = MR2049025 <br />
}}<br />
<br />
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V. Banica, ''[http://www.arxiv.org/abs/math.AP/0401129 Remarks on the blowup for the nonlinear Schrodinger equation with critical mass on a plane domain]'', preprint.<br />
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J.E. Barab, ''Nonexistence of asymptotically free solutions for a nonlinear Schrodinger equation'', J. Math. Phys. '''25''' (1984), 3270-3273.<br />
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M. Beals, ''Self-Spreading and strength of Singularities for solutions to semilinear wave equations'', Annals of Math '''118''' (1983), 187-214<br />
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M. Beals, M. Bezard, ''Low regularity local solutions for field equations'', Comm. Partial Differential Equations'''21''' (1996), 79-124.<br />
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P. Bechouce, N. Mauser, S. Selberg, ''[http://arXiv.org/abs/math.AP/0202201 Nonrelativistic limit of Klein-Gordon-Maxwell to Schrodinger Poisson]'', preprint<br />
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P. Bechouce, N. Mauser, S. Selberg, ''[http://arXiv.org/abs/math.AP/0303079 On the asymptotic analysis of the Dirac-Maxwell system in the nonrelativistic limit]'', preprint<br />
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D. Bekiranov, T. Ogawa, G. Ponce, ''Weak solvability and well-posedness of the coupled Schrodinger Korteweg-de Vries equation in the capillary-gravity interaction waves'', Proc. Amer. Math. Soc. '''125''' (1997), 2907-2919.<br />
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D. Bekiranov, T. Ogawa, G. Ponce, ''Interaction equations for short and long dispersive waves'', J. Funct. Anal. '''158''' (1998), 357-388.<br />
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E. Belchev, M. Kepka, Z. Zhou, ''Global existence of solutions to nonlinear wave equations'', CPDE '''24 '''(1999), 2297-2331.<br />
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E. Belchev, M. Kepka, Z. Zhou, ''Finite time blowup of solutions to semilinear wave equations'', JFA '''190 '''(2002), 233-254.<br />
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T. Benjamin, ''Internal waves of permanent form in fluids of great depth'', J. Fluid Mech. '''29''' (1967), 559-562.<br />
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T. Benjamin, ''The stability of solitary waves'', Proc. R. Soc. Lond. A. '''328''' (1972), 153-183.<br />
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T. Benjamin, J. Bona, J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil. Trans. R. Soc. London A 272 (1972), 47-78.<br />
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M. Ben-Artzi, H. Koch, J.C. Saut, Dispersion estimates for third order equations in two dimensions, CPDE 28 (2003), 1943-1974.<br />
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D.J. Benney, A.C. Newell, ''The propagation of nonlinear wave envelopes'', J. Math. and Phys. '''46''' (1967), 133-139.<br />
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H. Biagioni, F. Linares, ''Ill-posedness for the Derivative Schrodinger and Generalized Benjamin-Ono equations'', preprint.<br />
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P. Bizon, ''Equivariant self-similar wave maps from Minkowski spacetime into 3-sphere.'' Comm. Math. Phys. '''215''' (2000), 45-56.<br />
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N. Zabusky, M. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15 (1965), 240-243.<br />
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V.E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP 35 (1972), 908-914.<br />
<br />
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V.E. Zakharov, Weakly nonlinear waves on the surface of an ideal finite depth fluid, Amer. Math. Soc. Transl. 182 (1998), 167-197.<br />
<br />
=====[[ZkMan1976]]=====<br />
V.E. Zakharov, S.V. Manakov, Asymptotic behavior of non-linear wave systems integrated by the inverse scattering method, Soviet Physics JETP 44 (1976), 106-112.<br />
<br />
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V.E. Zakharov, A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media, Sov. Phys. JETP 34 (1972), 62-69.<br />
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=====[[ZkSha1973]]=====<br />
V.E. Zakharov, A.B. Shabat, Interaction between solitons in a stable medium, Sov. Phys. JETP 37 (1973), 834-828.<br />
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V.E. Zakharov, E.I. Schulman, Degenerated dispersion laws, motion invariant and kinetic equations, Physica 1D (1980), 185-250.<br />
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V.E. Zakharov, L.A. Takhtahan, Equivalence of a nonlinear Schrodinger equation and a Heisenberg ferromagnetic equation, Teor. Math. Phys. 8 (1979), 17-23.<br />
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B. Zhang, Unique continuation for the KdV equation, SIAM J. Math. Anal. 23 (1992), 55-71.<br />
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B. Zhang, A remark on the Cauchy problem for the Korteweg-de Vries equation on a periodic domain, DIE 8 (1995), 1191-1204.<br />
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B. Zhang, Unique continuation properties of the nonlinear Schrodinger equation, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 191-205.<br />
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X. Zhou, Inverse scattering transform for the time-dependent Schrodinger equation with application to the KP-I equations, CMP 128 (1990), 551-564.<br />
<br />
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Y. Zhou, Cauchy problem for semilinear wave equations in four space dimensions with small initial data, JPDE 8 (1995), 135-144.<br />
<br />
=====[[Zh1997]]=====<br />
Y. Zhou, Local existence with minimal regularity for nonlinear wave equations, Amer. J. Math, 119 (1997), 671-703.<br />
<br />
=====[[Zh1997b]]=====<br />
Y. Zhou, Uniqueness of weak solution of the KdV equation, IMRN 6 (1997), 271-283.<br />
<br />
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Y. Zhou, Uniqueness of weak solutions of 1+1 dimensional wave maps, Math. Z. 232 (1999), 707-719.<br />
<br />
=====[[Zh2000]]=====<br />
Y. Zhou, Uniqueness of generalized solutions to nonlinear wave equations, AJM 122 (2000), 939-965.<br />
<br />
=====[[ZhGouTan1991]]=====<br />
Y. Zhou, B. Gou, S. Tan, Existence and uniqueness of smooth solution for system of ferromagnetic chain, Science in China Ser. A. 34 (1991), 257.<br />
<br />
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L. Zielinski, Asymptotic completeness for multiparticle dispersive charge transfer models, JFA 150 (1997), 453-470.<br />
<br />
[[Category:Bibliography]]</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=Ban-p2&diff=7008Ban-p22009-07-18T01:06:03Z<p>Stanford88: Ban-p2 moved to Ban2004b: not preprint any more</p>
<hr />
<div>#REDIRECT [[Ban2004b]]</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=Ban2004b&diff=7007Ban2004b2009-07-18T01:06:03Z<p>Stanford88: Ban-p2 moved to Ban2004b: not preprint any more</p>
<hr />
<div>{{Article <br />
| author = V. Banica <br />
| title = Dispersion and Strichartz inequalities for Schrödinger equations with singular coefficients. <br />
| journal = SIAM J. Math. Anal.<br />
| jvol = 35<br />
| jpage = 868-883 <br />
| year = 2003 <br />
| arxivid = <br />
| mathsciid = MR2049025 <br />
}}</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=Cubic_NLS_on_2d_manifolds&diff=7006Cubic NLS on 2d manifolds2009-07-18T01:04:22Z<p>Stanford88: /* Cubic NLS on the sphere S^2 */</p>
<hr />
<div>In this page we discuss the [[cubic NLS]] on various two-dimensional domains (other than [[cubic NLS on R2|on R^2]]). in all cases the [[critical]] regularity<br />
is <math>s_c = 0</math>, thus this is a [[mass-critical NLS]].<br />
<br />
==Cubic NLS on the torus T^2==<br />
<br />
* One has LWP for <math>s>0\,</math> [[Bo1993]].<br />
* In the defocussing case one has GWP for <math>s>1\,</math> in by Hamiltonian conservation.<br />
** One can improve this to <math>s > 2/3\,</math> by the I-method by De Silva, Pavlovic, Staffilani, and Tzirakis (and also in an unpublished work of Bourgain).<br />
* In the focusing case one has blowup for data close to the ground state, with a blowup rate of <math>(T^* -t )^{-1}\,</math> [[BuGdTz-p]]<br />
* The <math>H^k\,</math> norm grows like <math>O(t^{2(k-1)+})\,</math> as long as the <math>H^1\,</math> norm stays bounded. <br />
<br />
==Cubic NLS on the cylinder <math>R \times T</math>==<br />
<br />
* One has LWP for <math>s>0\,</math> [[TkTz-p2]].<br />
<br />
==Cubic NLS on the sphere S^2==<br />
<br />
* Uniform local well-posedness fails for <math>3/20 < s < 1/4\,</math> [[BuGdTz2002]], [[Ban2004a]], but holds for <math>s>1/4\,</math> [[BuGdTz-p7]].<br />
** For <math>s >1/2\,</math> this is in [[BuGdTz-p3]].<br />
** These results for the sphere can mostly be generalized to other Zoll manifolds.<br />
<br />
==Cubic NLS on bounded domains==<br />
<br />
See [[BuGdTz-p]]. Sample results: blowup solutions exist close to the ground state, with a blowup rate of <math>(T-t)^{-1}\,</math>. If the domain is a disk then uniform LWP fails for <math>1/5 < s < 1/3\,</math>, while for a square one has LWP for all <math>s>0\,.</math> In general domains one has LWP for <math>s>2.</math>.<br />
<br />
[[Category:Schrodinger]]<br />
[[Category:Equations]]</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=Bibliography&diff=7005Bibliography2009-07-18T01:02:39Z<p>Stanford88: /* Ban-p */</p>
<hr />
<div>__NOTOC__<br />
<br />
=== Important note ===<br />
<br />
This bibliography was ported over from the previous incarnation of these pages. It should now be<br />
considered obsolete. Instead, we are now [[Help:References|giving each reference its own page]].<br />
<br />
The naming system for these articles was somewhat implicit and ''ad hoc''; we are currently forming [[Help:Bibliographic naming conventions|more explicit standards]] for this.<br />
<br />
<br />
=== References ===<br />
<br />
<center>A</center><br />
<br />
<br />
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M. Aassilla, ''Nonexistence de solutions globales de certains equations d'ondes nonlineaires'', C. R. Acad. Sci. Paris. '''334''' (2002), 961-966.<br />
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L. Abdelouhab, J. Bona, M. Felland, J. Saut, ''Nonlocal models for nonlinear dispersive waves'', Phys. D '''40''' (1989), 360-392.<br />
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M. Ablowitz, D. Kaup, A. Newell, H. Segur, ''The inverse scattering transform-Fourier analysis for nonlinear problems'', Studies in Appl. Math. '''53''' (1974), no. 4, 249-315.<br />
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M. Ablowitz, A. Fokas, ''The inverse scattering transform for the Benjamin-Ono equation, a pivot for multidimensional problems'', Stud. Appl. Math. '''68''' (1983), 1-10.<br />
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M. Ablowitz, R. Haberman, ''Nonlinear evolution equations in two and three dimensions'', Phys. Rev. Lett. '''35''' (1975), 1185-1188.<br />
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M. Ablowitz, Y. Ma, The periodic nonlinear Schrodinger equation, Stud. Appl. Math. 65 (1981), 113-158.<br />
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M. Ablowitz, H. Segur, ''Asymptotic solutions of the Korteweg de Vries equation'', Stud. Appl. Math. '''57''' (1977), 13-44.<br />
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H. Added, S. Added, ''Equations of Langmuir turbulence and non-linear Schrodinger equation: Smoothness and approximations'', JFA '''79''' (1988), 183-210.<br />
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R. Agemi, K. Yokoyama, The null condition and global existence of solutions to systems of wave equations with different speeds, Adv. Nonlinear PDE and Stochastics (1998), 43-86.<br />
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<br />
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S. Alinhac, ''Blow up of small data solutions for a class of quasilinear wave equations in two space dimensions II,'' Acta Math. 182 (1999), 1-23.<br />
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S. Alinhac, Rank 2 singular solutions for quasilinear wave equations, IMRN 18 (2000), 955-984.<br />
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S. Alinhac, ''The null condition for quasilinear wave equations in two space dimensions I'', Invent. Math. '''145''' (2001), 597-618.<br />
<br />
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S. Alinhac, ''The null condition for quasilinear wave equations in two space dimensions II, ''AJM '''123''' (2001), 1071-1101.<br />
<br />
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S. Alinhac, An example of blowup at infinity for a quasilinear wave equation, Asterisque 284 (2003), 1-91.<br />
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S. Anco, J. Isenberg, ''Global existence for wave maps with torsion'', CPDE '''25''' (2000), 1669-1702.<br />
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L. Andersson, V. Moncrief, ''[http://www.arxiv.org/abs/gr-qc/0110111 Elliptic-Hyperbolic systems and the Einstein equations]'', Ann. Henri Poincaré '''4''' (2003), 1-34<br />
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C. Antonini, Lower bounds for the L^2 minimal periodic blowup solutions of critical nonlinear Schrodinger equation, DIE 15 (2002), 749-768.<br />
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C. Antonini, F. Merle, ''Optimal bounds on positive blowup solutions for a semilinear wave equation'', IMRN '''21''' (2001), 1143-1167.<br />
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G. Artbazar, K. Yajima, ''The L^p continuity of wave operators for one dimensional Schrodinger operators'', J. Math. Sci. Univ. Tokyo '''7''' (2000), 221-240.<br />
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J. M. Ash, J. Cohen, G. Wang, On strongly interacting internal solitary waves, J. Fourier Anal. Appl., '''2''' (1996), 507-517.<br />
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<center>B</center><br />
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A. Bachelot, ''Probleme de Cauchy pour des systemes hyperboliques semilineaires'', Ann. IHP (Anal. nonlin), '''1''' (1984), 453-478.<br />
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A. Bachelot, J-P. Nicolas, Equation non-lineaire de Klein-Gordon dans des metriquesde type Schwarzchild, C.R. Acad. Sci. Paris. 316 (1993), 1047-1050<br />
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J. B. Baillon, J. M. Chadam, ''The Cauchy problem for the coupled Schrodinger-Klein-Gordon equations'', in "Contemporary Developments in Continuum Mechanics and Partial Differential Equations", G. M. de La Penha and L. A. Medeiros eds., North Holland, Amsterdam 1978<br />
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J. Baez, I. Sega, Z. Zhou, ''The global Goursat problem and scattering for nonlinear wave equations'', J. Funct. Anal. '''93 '''(1990), 239-269.<br />
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H. Bahouri, J-Y. Chemin, ''Équations d'ondes quasilinéaires et les inegalites de Strichartz'', Amer. J. Math.'''121''' (1999), 1337-1377.<br />
<br />
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H. Bahouri, J-Y. Chemin, ''Équations d'ondes quasilinéaires et effet dispersif'', IMRN '''21''' (1999), 1141-1178.<br />
<br />
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H. Bahouri, J-Y. Chemin, ''[http://arXiv.org/abs/math.AP/0304390 Quasilinear Wave equations and Microlocal Analysis]'', ICM 2002, Vol III, 141-154.<br />
<br />
=====[[BaCh 2003]]=====<br />
H. Bahouri, J-Y. Chemin, ''Microlocal analysis, bilinear estimates, and cubic quasilinear wave equation'', Asterique (2003), 93-141<br />
<br />
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H. Bahouri and P.Gerard, ''High frequency approximation of solutions to critical nonlinear wave equations'', Prepublications 97-34, Universite de Paris-Sud, Mai 1997. Appeared in: Amer. J. Math '''121''' (1999), 131-175.<br />
<br />
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H. Bahouri, P. Gerard, C. Xu, ''Espaces de Besov et estimations de Strichartz generalisees sur le groupe de Heisenberg'', Journal d'Analyse Mathematique '''82''' (2000), 93-118.<br />
<br />
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H. Bahouri and J.Shatah, ''Decay estimates for the critical semilinear wave equation'', Ann. Inst. H. Poincare Anal. Non Lineaire '''15''' (1998), no. 6, 783-789.<br />
<br />
=====[[Ban2004a]]=====<br />
{{Article <br />
| author = V. Banica <br />
| title = On the nonlinear Schrödinger dynamics on S^2<br />
| journal = J. Math. Pures Appl. (9)<br />
| jvol = 83<br />
| jpage = 77-98 <br />
| year = 2004 <br />
| arxivid = <br />
| mathsciid = MR2032582<br />
}}<br />
<br />
=====[[Ban-p2]]=====<br />
V. Banica, Dispersion and Strichartz inequalities for Schrodinger equations with singular coefficients, preprint.<br />
<br />
=====[[Ban-p3]]=====<br />
V. Banica, ''[http://www.arxiv.org/abs/math.AP/0401129 Remarks on the blowup for the nonlinear Schrodinger equation with critical mass on a plane domain]'', preprint.<br />
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J.E. Barab, ''Nonexistence of asymptotically free solutions for a nonlinear Schrodinger equation'', J. Math. Phys. '''25''' (1984), 3270-3273.<br />
<br />
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M. Beals, ''Self-Spreading and strength of Singularities for solutions to semilinear wave equations'', Annals of Math '''118''' (1983), 187-214<br />
<br />
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M. Beals, M. Bezard, ''Low regularity local solutions for field equations'', Comm. Partial Differential Equations'''21''' (1996), 79-124.<br />
<br />
=====[[BecMauSb-p]]=====<br />
P. Bechouce, N. Mauser, S. Selberg, ''[http://arXiv.org/abs/math.AP/0202201 Nonrelativistic limit of Klein-Gordon-Maxwell to Schrodinger Poisson]'', preprint<br />
<br />
=====[[BecMauSb-p2]]=====<br />
P. Bechouce, N. Mauser, S. Selberg, ''[http://arXiv.org/abs/math.AP/0303079 On the asymptotic analysis of the Dirac-Maxwell system in the nonrelativistic limit]'', preprint<br />
<br />
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D. Bekiranov, T. Ogawa, G. Ponce, ''On the well-posedness of Benney's interaction equation of short and long waves'', Adv. Diff. Eq. '''1''' (1996), 919-937.<br />
<br />
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D. Bekiranov, T. Ogawa, G. Ponce, ''Weak solvability and well-posedness of the coupled Schrodinger Korteweg-de Vries equation in the capillary-gravity interaction waves'', Proc. Amer. Math. Soc. '''125''' (1997), 2907-2919.<br />
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D. Bekiranov, T. Ogawa, G. Ponce, ''Interaction equations for short and long dispersive waves'', J. Funct. Anal. '''158''' (1998), 357-388.<br />
<br />
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E. Belchev, M. Kepka, Z. Zhou, ''Global existence of solutions to nonlinear wave equations'', CPDE '''24 '''(1999), 2297-2331.<br />
<br />
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E. Belchev, M. Kepka, Z. Zhou, ''Finite time blowup of solutions to semilinear wave equations'', JFA '''190 '''(2002), 233-254.<br />
<br />
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T. Benjamin, ''Internal waves of permanent form in fluids of great depth'', J. Fluid Mech. '''29''' (1967), 559-562.<br />
<br />
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T. Benjamin, ''The stability of solitary waves'', Proc. R. Soc. Lond. A. '''328''' (1972), 153-183.<br />
<br />
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T. Benjamin, J. Bona, J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil. Trans. R. Soc. London A 272 (1972), 47-78.<br />
<br />
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M. Ben-Artzi, S. Klainerman, ''Decay and regularity for the Schrodinger equation'', J. Anal. Math. '''58''' (1992), 25-37.<br />
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M. Ben-Artzi, J.C. Saut, ''Uniform decay estimates for a class of oscillatory integrals and applications'', Diff. Int. Eq. '''12''' (1999), 137-145.<br />
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M. Ben-Artzi, H. Koch, J.C. Saut, Dispersion estimates for third order equations in two dimensions, CPDE 28 (2003), 1943-1974.<br />
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D.J. Benney, A.C. Newell, ''The propagation of nonlinear wave envelopes'', J. Math. and Phys. '''46''' (1967), 133-139.<br />
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H. Berestycki, T. Cazenave, ''Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires'', C. R. Acad. Sc. Paris, t. 293 (1981), 489-492.<br />
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B.K. Berger, P. T. Chrusciel, V. Moncrief, ''On "asymptotically flat" spacetimes with G_2 invariant Cauchy surfaces'', Ann. Physics '''237''' (1995), 322-354.<br />
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<br />
H. Biagioni, F. Linares, ''Ill-posedness for the Derivative Schrodinger and Generalized Benjamin-Ono equations'', preprint.<br />
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B. Birnir, G. Ponce, L. Vega, On the ill-posedness for the initial value problem for the modified Korteweg-de Vries equation, preprint.<br />
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B. Birnir, C. Kenig, G. Ponce, N. Svanstedt, L. Vega, ''On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrödinger equations''. J. London Math. Soc. (2) '''53''' (1996), 551-559.<br />
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P. Bizon, ''Equivariant self-similar wave maps from Minkowski spacetime into 3-sphere.'' Comm. Math. Phys. '''215''' (2000), 45-56.<br />
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=====[[Biz-p]]=====<br />
P. Bizon, ''[http://arXiv.org/abs/math-ph/0206004 Formation of singularities in Yang-Mills equations]'', preprint<br />
<br />
=====[[BizCjTb2001]]=====<br />
P. Bizon, T. Chmaj, Z. Tabor, ''[http://arxiv.org/abs/math-ph/0011005 Formation of singularities for equivariant 2+1 dimensional wave maps into two-sphere]'', Nonlinearity '''14''' (2001), no. 5, 1041-1053.<br />
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=====[[BizOvSi-p]]=====<br />
P. Bizon, Y. Ovchinnikov, I. Sigal, ''[http://arxiv.org/abs/math.AP/0307026 Collapse of an Instanton]'', preprint.<br />
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P. Bizon, Z. Tabor, ''[http://arxiv.org/abs/math-ph/0105016 On blowup for Yang-Mills fields]'', Phys. Rev. D (3) '''64''' (2001), 121701.<br />
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P. Blue, A. Soffer, Semilinear wave equations on the Schwarzschild manifold I.Local decay estimates, Adv. Diff Eq. 8 (2003), 595-614.<br />
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J. Bona, ''On the stability theory of solitary waves'', Proc. R. Soc. Lond. A. '''344''' (1975), 363-374.<br />
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J. Bona, V.A. Dougalis, O.A. Karakashian, ''Fully discrete Galerkin methods for the Korteweg-de Vries equation'', Comput. Math. Appl. Ser. A '''12''' (1986), 859-884.<br />
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J. Bona, Y. Liu, ''Instability of solitary-wave solutions of the Kadomtsev-Petviashvili equation in three dimensions'', ADE '''7''' (2002), 1-33.<br />
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J. Bona, G. Ponce, J-C. Saut, M. Tom, ''A model system for strong interaction between internal solitary waves'', CMP '''143''' (1992), 287-313.<br />
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J. Bona, R. Scott, ''Solutions of the Korteweg-de Vries equation, in fractional order Sobolev spaces'', Duke Math J. '''43''' (1976), 87-99.<br />
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J. Bona, R. Smith, ''The initial-value problem for the Korteweg-de Vries equation'', Philos. Trans. Royal Soc. London Series A 278 (1975), 555-601.<br />
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J. Bona, P.E. Sougandis, W. Strauss, ''Stability and instability of solitary waves of Korteweg-de Vries type'', Proc. R. Soc. Lond. '''411''' (1987), 395-412.<br />
<br />
=====[[BnSuZh-p]]=====<br />
J. Bona, S. Sun, B-Y. Zhang, ''A non-homogeneous boundary-value problem for the Korteweg-de Vries Equation in a Quarter Plane'', preprint<br />
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J. Bona, R. Winther, ''The Korteweg-de Vries equation, posed in a quarter plane'', SIAM J. Math. Anal. '''14''' (1983), 1056-1106.<br />
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J. Bona, F. B. Weissler, ''Similarity solutions of the generalized Korteweg-de Vries equations'', preprint.<br />
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A. de Bouard, Analytic solutions to nonelliptic nonlinear Schrodinger equations, JDE 104 (1993), 196-213.<br />
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A. de Bouard, A. Debussche, Y. Tsutsumi, ''[http://www.ams.org/mathscinet?screen=leavingmsn&publ=Academic%20Press&url=http://www.idealibrary.com/cgi%2Dbin/links/citation/0022%2D1236/169/532&back_url=msnmain%3Fpg3=ICN%26s3=Tsutsumi%26co3=AND%26pg4=TI%26s4=%26co4=AND%26pg5=CC%26s5=%26co5=A%20 White noise driven Korteweg-de Vries equation.]'' J. Funct. Anal. '''169''' (1999), no. 2, 532-558<br />
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A. de Bouard, J.C. Saut, ''Solitary waves of generalized Kadomtsev-Petviashvili equations'', Annales IHP, Analyse non-lineaire '''14''' (1997) 211-236.<br />
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A. de Bouard, N. Hayashi, J.C. Saut, Global existence of small solutions to a nonlinear relativistic Schrodinger equation, CMP 189 (1997), 73-105.<br />
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A. de Bouard, Y. Martel, Nonexistence of L^2-compact solutions of the Kadomptsev-Petviashvili II equation, Math. Ann. 328 (2004), 525\u2014554.<br />
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J. Ginibre, T. Ozawa, G. Velo, ''On the existence of the wave operators for a class of nonlinear Schrodinger equations'', Ann. Inst. H. Poincare Pys. Theor. '''60''' (1994), 211-239.<br />
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J. Ginibre, G. Velo, ''On a class of nonlinear Schrodinger equations'', J. Funct. Anal. '''32''' (1979), 1-71.<br />
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J. Ginibre, G. Velo, ''On a class of nonlinear Schrodinger equations with nonlocal interactions'', Math Z. '''170''' (1980), 109-136.<br />
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J. Ginibre, G. Velo, ''The Cauchy problem for the O(N), CP(N-1), and GC(N,P) models'', Ann. Physics, '''142''' (1982), 393-415.<br />
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J. Ginibre, G. Velo, ''Scattering Theory in the Energy Space for a Class of Nonlinear Schrodinger Equations'', J. Math. Pure Appl., '''64''' (1985), 363-401.<br />
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J. Ginibre and G. Velo, ''Time decay of finite energy solutions of the non linear Klein-Gordon and Schr\"odinger equations'', Ann. Inst. Henri. Poincar\'e, '''43''' (1985), 399-442.<br />
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J. Ginibre, G. Velo, ''The global Cauchy problem for the non linear Klein-Gordon equation'', Math. Z. '''189''' (1985), 487-505.<br />
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[[Category:Bibliography]]</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=Ban-p&diff=7004Ban-p2009-07-18T00:59:56Z<p>Stanford88: Ban-p moved to Ban2004a: not preprint anymore</p>
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<div>#REDIRECT [[Ban2004a]]</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=Ban2004a&diff=7003Ban2004a2009-07-18T00:59:56Z<p>Stanford88: Ban-p moved to Ban2004a: not preprint anymore</p>
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<div>{{Article <br />
| author = V. Banica <br />
| title = On the nonlinear Schrödinger dynamics on S^2<br />
| journal = J. Math. Pures Appl. (9)<br />
| jvol = 83<br />
| jpage = 77-98 <br />
| year = 2004 <br />
| arxivid = <br />
| mathsciid = MR2032582<br />
}}</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=Bibliography&diff=7002Bibliography2009-07-18T00:56:07Z<p>Stanford88: /* AnMc-p */</p>
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<div>__NOTOC__<br />
<br />
=== Important note ===<br />
<br />
This bibliography was ported over from the previous incarnation of these pages. It should now be<br />
considered obsolete. Instead, we are now [[Help:References|giving each reference its own page]].<br />
<br />
The naming system for these articles was somewhat implicit and ''ad hoc''; we are currently forming [[Help:Bibliographic naming conventions|more explicit standards]] for this.<br />
<br />
<br />
=== References ===<br />
<br />
<center>A</center><br />
<br />
<br />
=====[[Aa2002]]=====<br />
M. Aassilla, ''Nonexistence de solutions globales de certains equations d'ondes nonlineaires'', C. R. Acad. Sci. Paris. '''334''' (2002), 961-966.<br />
<br />
=====[[AbdBnFelSau1989]]=====<br />
L. Abdelouhab, J. Bona, M. Felland, J. Saut, ''Nonlocal models for nonlinear dispersive waves'', Phys. D '''40''' (1989), 360-392.<br />
<br />
=====[[AbKauNeSe1974]]=====<br />
M. Ablowitz, D. Kaup, A. Newell, H. Segur, ''The inverse scattering transform-Fourier analysis for nonlinear problems'', Studies in Appl. Math. '''53''' (1974), no. 4, 249-315.<br />
<br />
=====[[AbFs1983]]=====<br />
M. Ablowitz, A. Fokas, ''The inverse scattering transform for the Benjamin-Ono equation, a pivot for multidimensional problems'', Stud. Appl. Math. '''68''' (1983), 1-10.<br />
<br />
=====[[AbHab1975]]=====<br />
M. Ablowitz, R. Haberman, ''Nonlinear evolution equations in two and three dimensions'', Phys. Rev. Lett. '''35''' (1975), 1185-1188.<br />
<br />
=====[[AbMa1981]]=====<br />
M. Ablowitz, Y. Ma, The periodic nonlinear Schrodinger equation, Stud. Appl. Math. 65 (1981), 113-158.<br />
<br />
=====[[AbSe1977]]=====<br />
M. Ablowitz, H. Segur, ''Asymptotic solutions of the Korteweg de Vries equation'', Stud. Appl. Math. '''57''' (1977), 13-44.<br />
<br />
=====[[AdhAds1988]]=====<br />
H. Added, S. Added, ''Equations of Langmuir turbulence and non-linear Schrodinger equation: Smoothness and approximations'', JFA '''79''' (1988), 183-210.<br />
<br />
=====[[AgYok1998]]=====<br />
R. Agemi, K. Yokoyama, The null condition and global existence of solutions to systems of wave equations with different speeds, Adv. Nonlinear PDE and Stochastics (1998), 43-86.<br />
<br />
=====[[AxPgSac1997]]=====<br />
J.C. Alexander, R.L. Pego, R.L. Sachs, ''On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation'', Phys. Lett. A. '''226''' (1997), 187-192.<br />
<br />
=====[[Al1995]]=====<br />
S. Alinhac, ''Blowup for nonlinear hyperbolic equations'', Boston<nowiki>: Birkhauser, 1995, Progress in </nowiki>Nonlinear DE and their Applications, 17.<br />
<br />
=====[[Al1999]]=====<br />
S. Alinhac, ''Blow up of small data solutions for a class of quasilinear wave equations in two space dimensions I'', Annals of Mathematics 149 (1999), 97-127.<br />
<br />
=====[[Al1999b]]=====<br />
S. Alinhac, ''Blow up of small data solutions for a class of quasilinear wave equations in two space dimensions II,'' Acta Math. 182 (1999), 1-23.<br />
<br />
=====[[Al2000]]=====<br />
S. Alinhac, Rank 2 singular solutions for quasilinear wave equations, IMRN 18 (2000), 955-984.<br />
<br />
=====[[Al2001]]=====<br />
S. Alinhac, ''The null condition for quasilinear wave equations in two space dimensions I'', Invent. Math. '''145''' (2001), 597-618.<br />
<br />
=====[[Al2001b]]=====<br />
S. Alinhac, ''The null condition for quasilinear wave equations in two space dimensions II, ''AJM '''123''' (2001), 1071-1101.<br />
<br />
=====[[Al2003]]=====<br />
S. Alinhac, An example of blowup at infinity for a quasilinear wave equation, Asterisque 284 (2003), 1-91.<br />
<br />
=====[[AcIb2000]]=====<br />
S. Anco, J. Isenberg, ''Global existence for wave maps with torsion'', CPDE '''25''' (2000), 1669-1702.<br />
<br />
=====[[AnMc2003]]=====<br />
L. Andersson, V. Moncrief, ''[http://www.arxiv.org/abs/gr-qc/0110111 Elliptic-Hyperbolic systems and the Einstein equations]'', Ann. Henri Poincaré '''4''' (2003), 1-34<br />
<br />
=====[[Ant2003]]=====<br />
C. Antonini, Lower bounds for the L^2 minimal periodic blowup solutions of critical nonlinear Schrodinger equation, DIE 15 (2002), 749-768.<br />
<br />
=====[[AntMe2001]]=====<br />
C. Antonini, F. Merle, ''Optimal bounds on positive blowup solutions for a semilinear wave equation'', IMRN '''21''' (2001), 1143-1167.<br />
<br />
=====[[ArYa2000]]=====<br />
G. Artbazar, K. Yajima, ''The L^p continuity of wave operators for one dimensional Schrodinger operators'', J. Math. Sci. Univ. Tokyo '''7''' (2000), 221-240.<br />
<br />
=====[[AsCoeWgg1996]]=====<br />
J. M. Ash, J. Cohen, G. Wang, On strongly interacting internal solitary waves, J. Fourier Anal. Appl., '''2''' (1996), 507-517.<br />
<br />
<center>B</center><br />
<br />
<br />
=====[[Bch1984]]=====<br />
A. Bachelot, ''Probleme de Cauchy pour des systemes hyperboliques semilineaires'', Ann. IHP (Anal. nonlin), '''1''' (1984), 453-478.<br />
<br />
=====[[BchNic1993]]=====<br />
A. Bachelot, J-P. Nicolas, Equation non-lineaire de Klein-Gordon dans des metriquesde type Schwarzchild, C.R. Acad. Sci. Paris. 316 (1993), 1047-1050<br />
<br />
=====[[BlChd1978]]=====<br />
J. B. Baillon, J. M. Chadam, ''The Cauchy problem for the coupled Schrodinger-Klein-Gordon equations'', in "Contemporary Developments in Continuum Mechanics and Partial Differential Equations", G. M. de La Penha and L. A. Medeiros eds., North Holland, Amsterdam 1978<br />
<br />
=====[[BaeSgZz1990]]=====<br />
J. Baez, I. Sega, Z. Zhou, ''The global Goursat problem and scattering for nonlinear wave equations'', J. Funct. Anal. '''93 '''(1990), 239-269.<br />
<br />
=====[[BaCh1999]]=====<br />
H. Bahouri, J-Y. Chemin, ''Équations d'ondes quasilinéaires et les inegalites de Strichartz'', Amer. J. Math.'''121''' (1999), 1337-1377.<br />
<br />
=====[[BaCh1999b]]=====<br />
H. Bahouri, J-Y. Chemin, ''Équations d'ondes quasilinéaires et effet dispersif'', IMRN '''21''' (1999), 1141-1178.<br />
<br />
=====[[BaCh2002]]=====<br />
H. Bahouri, J-Y. Chemin, ''[http://arXiv.org/abs/math.AP/0304390 Quasilinear Wave equations and Microlocal Analysis]'', ICM 2002, Vol III, 141-154.<br />
<br />
=====[[BaCh 2003]]=====<br />
H. Bahouri, J-Y. Chemin, ''Microlocal analysis, bilinear estimates, and cubic quasilinear wave equation'', Asterique (2003), 93-141<br />
<br />
=====[[BaGd1997]]=====<br />
H. Bahouri and P.Gerard, ''High frequency approximation of solutions to critical nonlinear wave equations'', Prepublications 97-34, Universite de Paris-Sud, Mai 1997. Appeared in: Amer. J. Math '''121''' (1999), 131-175.<br />
<br />
=====[[BaGdXu2000]]=====<br />
H. Bahouri, P. Gerard, C. Xu, ''Espaces de Besov et estimations de Strichartz generalisees sur le groupe de Heisenberg'', Journal d'Analyse Mathematique '''82''' (2000), 93-118.<br />
<br />
=====[[BaSa1998]]=====<br />
H. Bahouri and J.Shatah, ''Decay estimates for the critical semilinear wave equation'', Ann. Inst. H. Poincare Anal. Non Lineaire '''15''' (1998), no. 6, 783-789.<br />
<br />
=====[[Ban-p]]=====<br />
V. Banica, On the nonlinear Schrodinger dynamics on S^2, preprint.<br />
<br />
=====[[Ban-p2]]=====<br />
V. Banica, Dispersion and Strichartz inequalities for Schrodinger equations with singular coefficients, preprint.<br />
<br />
=====[[Ban-p3]]=====<br />
V. Banica, ''[http://www.arxiv.org/abs/math.AP/0401129 Remarks on the blowup for the nonlinear Schrodinger equation with critical mass on a plane domain]'', preprint.<br />
<br />
=====[[Bb1984]]=====<br />
J.E. Barab, ''Nonexistence of asymptotically free solutions for a nonlinear Schrodinger equation'', J. Math. Phys. '''25''' (1984), 3270-3273.<br />
<br />
=====[[Be1983]]=====<br />
M. Beals, ''Self-Spreading and strength of Singularities for solutions to semilinear wave equations'', Annals of Math '''118''' (1983), 187-214<br />
<br />
=====[[BeBz1996]]=====<br />
M. Beals, M. Bezard, ''Low regularity local solutions for field equations'', Comm. Partial Differential Equations'''21''' (1996), 79-124.<br />
<br />
=====[[BecMauSb-p]]=====<br />
P. Bechouce, N. Mauser, S. Selberg, ''[http://arXiv.org/abs/math.AP/0202201 Nonrelativistic limit of Klein-Gordon-Maxwell to Schrodinger Poisson]'', preprint<br />
<br />
=====[[BecMauSb-p2]]=====<br />
P. Bechouce, N. Mauser, S. Selberg, ''[http://arXiv.org/abs/math.AP/0303079 On the asymptotic analysis of the Dirac-Maxwell system in the nonrelativistic limit]'', preprint<br />
<br />
=====[[BkOgPo1996]]=====<br />
D. Bekiranov, T. Ogawa, G. Ponce, ''On the well-posedness of Benney's interaction equation of short and long waves'', Adv. Diff. Eq. '''1''' (1996), 919-937.<br />
<br />
=====[[BkOgPo1997]]=====<br />
D. Bekiranov, T. Ogawa, G. Ponce, ''Weak solvability and well-posedness of the coupled Schrodinger Korteweg-de Vries equation in the capillary-gravity interaction waves'', Proc. Amer. Math. Soc. '''125''' (1997), 2907-2919.<br />
<br />
=====[[BkOgPo1998]]=====<br />
D. Bekiranov, T. Ogawa, G. Ponce, ''Interaction equations for short and long dispersive waves'', J. Funct. Anal. '''158''' (1998), 357-388.<br />
<br />
=====[[BcKkZz1999]]=====<br />
E. Belchev, M. Kepka, Z. Zhou, ''Global existence of solutions to nonlinear wave equations'', CPDE '''24 '''(1999), 2297-2331.<br />
<br />
=====[[BcKkZz2002]]=====<br />
E. Belchev, M. Kepka, Z. Zhou, ''Finite time blowup of solutions to semilinear wave equations'', JFA '''190 '''(2002), 233-254.<br />
<br />
=====[[Bj1967]]=====<br />
T. Benjamin, ''Internal waves of permanent form in fluids of great depth'', J. Fluid Mech. '''29''' (1967), 559-562.<br />
<br />
=====[[Bj1972]]=====<br />
T. Benjamin, ''The stability of solitary waves'', Proc. R. Soc. Lond. A. '''328''' (1972), 153-183.<br />
<br />
=====[[BjBnMah1972]]=====<br />
T. Benjamin, J. Bona, J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil. Trans. R. Soc. London A 272 (1972), 47-78.<br />
<br />
=====[[BenKl1992]]=====<br />
M. Ben-Artzi, S. Klainerman, ''Decay and regularity for the Schrodinger equation'', J. Anal. Math. '''58''' (1992), 25-37.<br />
<br />
=====[[BenSau1999]]=====<br />
M. Ben-Artzi, J.C. Saut, ''Uniform decay estimates for a class of oscillatory integrals and applications'', Diff. Int. Eq. '''12''' (1999), 137-145.<br />
<br />
=====[[BenKocSau2003]]=====<br />
M. Ben-Artzi, H. Koch, J.C. Saut, Dispersion estimates for third order equations in two dimensions, CPDE 28 (2003), 1943-1974.<br />
<br />
=====[[ByNe1967]]=====<br />
D.J. Benney, A.C. Newell, ''The propagation of nonlinear wave envelopes'', J. Math. and Phys. '''46''' (1967), 133-139.<br />
<br />
=====[[BerCa1981]]=====<br />
H. Berestycki, T. Cazenave, ''Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires'', C. R. Acad. Sc. Paris, t. 293 (1981), 489-492.<br />
<br />
=====[[BgCcMc1995]]=====<br />
B.K. Berger, P. T. Chrusciel, V. Moncrief, ''On "asymptotically flat" spacetimes with G_2 invariant Cauchy surfaces'', Ann. Physics '''237''' (1995), 322-354.<br />
<br />
=====[[BiLi-p]]=====<br />
<br />
H. Biagioni, F. Linares, ''Ill-posedness for the Derivative Schrodinger and Generalized Benjamin-Ono equations'', preprint.<br />
<br />
=====[[BirPoVe-p]]=====<br />
B. Birnir, G. Ponce, L. Vega, On the ill-posedness for the initial value problem for the modified Korteweg-de Vries equation, preprint.<br />
<br />
=====[[BirKnPoSvVe1996]]=====<br />
B. Birnir, C. Kenig, G. Ponce, N. Svanstedt, L. Vega, ''On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrödinger equations''. J. London Math. Soc. (2) '''53''' (1996), 551-559.<br />
<br />
=====[[Biz2000]]=====<br />
P. Bizon, ''Equivariant self-similar wave maps from Minkowski spacetime into 3-sphere.'' Comm. Math. Phys. '''215''' (2000), 45-56.<br />
<br />
=====[[Biz-p]]=====<br />
P. Bizon, ''[http://arXiv.org/abs/math-ph/0206004 Formation of singularities in Yang-Mills equations]'', preprint<br />
<br />
=====[[BizCjTb2001]]=====<br />
P. Bizon, T. Chmaj, Z. Tabor, ''[http://arxiv.org/abs/math-ph/0011005 Formation of singularities for equivariant 2+1 dimensional wave maps into two-sphere]'', Nonlinearity '''14''' (2001), no. 5, 1041-1053.<br />
<br />
=====[[BizOvSi-p]]=====<br />
P. Bizon, Y. Ovchinnikov, I. Sigal, ''[http://arxiv.org/abs/math.AP/0307026 Collapse of an Instanton]'', preprint.<br />
<br />
=====[[BizTb2001]]=====<br />
P. Bizon, Z. Tabor, ''[http://arxiv.org/abs/math-ph/0105016 On blowup for Yang-Mills fields]'', Phys. Rev. D (3) '''64''' (2001), 121701.<br />
<br />
=====[[BluSf2003]]=====<br />
P. Blue, A. Soffer, Semilinear wave equations on the Schwarzschild manifold I.Local decay estimates, Adv. Diff Eq. 8 (2003), 595-614.<br />
<br />
=====[[Bn1975]]=====<br />
J. Bona, ''On the stability theory of solitary waves'', Proc. R. Soc. Lond. A. '''344''' (1975), 363-374.<br />
<br />
=====[[BnDgKar1986]]=====<br />
J. Bona, V.A. Dougalis, O.A. Karakashian, ''Fully discrete Galerkin methods for the Korteweg-de Vries equation'', Comput. Math. Appl. Ser. A '''12''' (1986), 859-884.<br />
<br />
=====[[BnLiu2002]]=====<br />
J. Bona, Y. Liu, ''Instability of solitary-wave solutions of the Kadomtsev-Petviashvili equation in three dimensions'', ADE '''7''' (2002), 1-33.<br />
<br />
=====[[BnPoSauTm1992]]=====<br />
J. Bona, G. Ponce, J-C. Saut, M. Tom, ''A model system for strong interaction between internal solitary waves'', CMP '''143''' (1992), 287-313.<br />
<br />
=====[[BnSco1976]]=====<br />
J. Bona, R. Scott, ''Solutions of the Korteweg-de Vries equation, in fractional order Sobolev spaces'', Duke Math J. '''43''' (1976), 87-99.<br />
<br />
=====[[BnSmr1975]]=====<br />
J. Bona, R. Smith, ''The initial-value problem for the Korteweg-de Vries equation'', Philos. Trans. Royal Soc. London Series A 278 (1975), 555-601.<br />
<br />
=====[[BnSouSr1987]]=====<br />
J. Bona, P.E. Sougandis, W. Strauss, ''Stability and instability of solitary waves of Korteweg-de Vries type'', Proc. R. Soc. Lond. '''411''' (1987), 395-412.<br />
<br />
=====[[BnSuZh-p]]=====<br />
J. Bona, S. Sun, B-Y. Zhang, ''A non-homogeneous boundary-value problem for the Korteweg-de Vries Equation in a Quarter Plane'', preprint<br />
<br />
=====[[BnWi1983]]=====<br />
J. Bona, R. Winther, ''The Korteweg-de Vries equation, posed in a quarter plane'', SIAM J. Math. Anal. '''14''' (1983), 1056-1106.<br />
<br />
=====[[BnWe-p]]=====<br />
J. Bona, F. B. Weissler, ''Similarity solutions of the generalized Korteweg-de Vries equations'', preprint.<br />
<br />
=====[[Bd1993]]=====<br />
A. de Bouard, Analytic solutions to nonelliptic nonlinear Schrodinger equations, JDE 104 (1993), 196-213.<br />
<br />
=====[[BdDbTs1999]]=====<br />
A. de Bouard, A. Debussche, Y. Tsutsumi, ''[http://www.ams.org/mathscinet?screen=leavingmsn&publ=Academic%20Press&url=http://www.idealibrary.com/cgi%2Dbin/links/citation/0022%2D1236/169/532&back_url=msnmain%3Fpg3=ICN%26s3=Tsutsumi%26co3=AND%26pg4=TI%26s4=%26co4=AND%26pg5=CC%26s5=%26co5=A%20 White noise driven Korteweg-de Vries equation.]'' J. Funct. Anal. '''169''' (1999), no. 2, 532-558<br />
<br />
=====[[BdSau1997]]=====<br />
A. de Bouard, J.C. Saut, ''Solitary waves of generalized Kadomtsev-Petviashvili equations'', Annales IHP, Analyse non-lineaire '''14''' (1997) 211-236.<br />
<br />
=====[[BdHaSau1997]]=====<br />
A. de Bouard, N. Hayashi, J.C. Saut, Global existence of small solutions to a nonlinear relativistic Schrodinger equation, CMP 189 (1997), 73-105.<br />
<br />
=====[[BdMt2004]]=====<br />
A. de Bouard, Y. Martel, Nonexistence of L^2-compact solutions of the Kadomptsev-Petviashvili II equation, Math. Ann. 328 (2004), 525\u2014554.<br />
<br />
=====[[BctGolPal-p]]=====<br />
F. Bouchut, F. Golse, C. Pallard, ''[http://front.math.ucdavis.edu/math.AP/0301175 On classical solutions to the 3D relativistic Vlasov-Maxwell system: Glassey-Strauss's theorem revisited]'', preprint.<br />
<br />
=====[[Bo1992]]=====<br />
J. Bourgain, ''A remark on Schrodinger operators'', Israel J. Math. '''77''' (1992), 1-16.<br />
<br />
=====[[Bo1993]]=====<br />
J. Bourgain, ''Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I. Schrodinger equations,'' [http://link.springer.de/link/service/journals/00039/index.htm Geom. Funct. Anal.] '''3''' (1993) 107-156.<br />
<br />
=====[[Bo1993b]]=====<br />
J. Bourgain, ''Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II. The periodic KdV equation,'' [http://link.springer.de/link/service/journals/00039/index.htm Geom. Funct. Anal.] '''3''' (1993) 209-262.<br />
<br />
=====[[Bo1993c]]=====<br />
J. Bourgain, ''On the Cauchy problem for the Kadomtsev-Petviashvili equation'', [http://link.springer.de/link/service/journals/00039/index.htm Geom. Funct. Anal.] '''3''' (1993) 315-341.<br />
<br />
=====[[Bo1993d]]=====<br />
J. Bourgain, ''Exponential sums and nonlinear Schrodinger equations'', [http://link.springer.de/link/service/journals/00039/index.htm Geom. Funct. Anal.] '''3''' (1993) 157-178.<br />
<br />
=====[[Bo1994]]=====<br />
J. Bourgain, ''On the Cauchy and invariant measure problem for the periodic Zakharov system'', Duke Math. J. '''76''' (1994), 175-202.<br />
<br />
=====[[Bo1994b]]=====<br />
J. Bourgain, ''Harmonic analysis and nonlinear partial differential equations'', Proceedings of the ICM 1994, 31-44, Birkhauser Basel, 1995.<br />
<br />
=====[[Bo1994c]]=====<br />
J. Bourgain, ''Periodic nonlinear Schrodinger equation and invariant measures'', CMP '''166''' (1994), 1-26.<br />
<br />
=====[[Bo1995]]=====<br />
J. Bourgain, ''Estimates for cone multipliers'', Operator Theory: Advances and Applications, '''77''' (1995), 41-60.<br />
<br />
=====[[Bo1995b]]=====<br />
J. Bourgain, ''Aspects of longtime behaviour of solutions of nonlinear Hamiltonian evolution equations'', GAFA ''5'' (1995), 105-140.<br />
<br />
=====[[Bo1995c]]=====<br />
J. Bourgain, ''Periodic nonlinear Schrodinger equations and invariant measures'', CMP '''166''' (1995), 1-26.<br />
<br />
=====[[Bo1996]]=====<br />
J. Bourgain, ''On well-posedness of the Zakharov system'', [http://www.imrn.org Int. Math. Research Notices] '''11''' (1996), 515-546.<br />
<br />
=====[[Bo1996b]]=====<br />
J. Bourgain, ''Invariant measures for the 2d-defocussing nonlinear Schrodinger equation'', Comm. Math. Phys. '''176''' (1996), 421-445.<br />
<br />
=====[[Bo1996c]]=====<br />
J. Bourgain, ''On the growth in time of higher order Sobolev norms of smooth solutions of Hamiltonian PDE'', IMRN '''6''' (1996), 277-304.<br />
* [[kdv_measures.pdf Bo1997]] J. Bourgain, ''[kdv_measures.pdf Periodic Korteweg de Vries equation with measures as initial data]'', Selecta Math. (N.S.) '''3''' (1997), 115-159.<br />
<br />
=====[[Bo1997b]]=====<br />
J. Bourgain, ''On the compactness of support of solutions of dispersive equations'', [http://www.imrn.org Int. Math. Research Notices] '''9''' (1997), 437-447.<br />
<br />
=====[[Bo1998]]=====<br />
J. Bourgain, ''Refinements of Strichartz' inequality and applications to 2D-NLS with critical non-linearity'', [http://www.imrn.org Int. Math. Research Notices], '''5''' (1998), 253-283.<br />
<br />
=====[[Bo1998b]]=====<br />
J. Bourgain, ''Scattering in the energy space and below for 3D NLS'', J. Anal. Math. '''75''' (1998), 267-297.<br />
<br />
<br />
=====[[Bo1999]]===== J. Bourgain, ''[http://www.math.ucla.edu/~tao/bourgain/borg_nls.dvi New global well-posedness results for non-linear Schrodinger equations]'', AMS 1999.<br />
<br />
=====[[Bo1999b]]=====<br />
J. Bourgain, ''Global well-posedness of defocussing 3D critical NLS in the radial case'', J. Amer. Math. Soc. '''12''' (1999), 145-171.<br />
<br />
=====[[Bo-p]]=====<br />
J. Bourgain, Remarks on stability and diffusion in high-dimensional Hamiltonian systems and PDE, preprint.<br />
<br />
=====[[Bo-p2]]=====<br />
J. Bourgain, A remark on normal forms and the I-method for periodic NLS, preprint.<br />
<br />
=====[[Bo-p3]]=====<br />
J. Bourgain, On the Cauchy problem for periodic KdV type equations, preprint.<br />
<br />
=====[[BoCo1996]]=====<br />
J. Bourgain, J. Colliander, ''On the well-posedness of the Zakharov system'', IMRN '''11''' (1996), 515-546.<br />
<br />
=====[[BoWg1997]]=====<br />
J. Bourgain, W. Wang, ''Construction of blowup solutions for the nonlinear Schrodinger equation with critical non-linearity,'' Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) '''25''' (1997), no. 1-2, 197-215.<br />
<br />
=====[[Bou1996]]=====<br />
N. Bournaveas, ''Local existence for the Maxwell-Dirac equations in three space dimensions''. Comm. Partial Differential Equations '''21''' (1996) 693-720.<br />
<br />
=====[[Bou1999]]=====<br />
N. Bournaveas, ''Local existence of energy class solutions for the Dirac-Klein-Gordon equations''. Comm. Partial Differential Equations '''24''' (1999), 1167-1193.<br />
<br />
=====[[Bou2000]]=====<br />
N. Bournaveas, ''A new proof of global existence for the Dirac Klein-Gordon equations in one space dimension''. J. Funct. Anal. '''173''' (2000), 203-213.<br />
<br />
=====[[Bou2001]]=====<br />
N. Bournaveas, ''Low regularity solutions of the Dirac Klein-Gordon equations in two space dimensions''. Comm. Partial Differential Equations '''26''' (2001), 1345-1366.<br />
<br />
=====[[BdmFsShp-p]]=====<br />
A. Boutet de Monvel, A. Fokas, ''The mKdV equation on a half-line'', preprint.<br />
<br />
=====[[BdmShp-p]]=====<br />
A. Boutet de Monvel, D. Shepelsky, ''[http://arxiv.org/abs/math.AP/0307194 The mKdV equation on an interval]'', preprint.<br />
<br />
=====[[BrlForMai1994]]=====<br />
P. Breitenlohner, P. Forgacs, D. Maison, ''Static spherically symmetric solutions of the Einstein-Yang-Mills equations. ''Comm. Math. Phys. '''163''' (1994), no. 1, 141-172.<br />
<br />
=====[[Br1975]]=====<br />
P. Brenner, ''On L^p - L^{p'} estimates for the wave equation'', Math Z. '''145''' (1975) 251-254.<br />
<br />
=====[[Br1984]]=====<br />
P. Brenner, ''On spacetime means and everywhere defined scattering operators'', Math. Z. '''186''' (1984), 383-391.<br />
<br />
=====[[Br1985]]=====<br />
P. Brenner, ''On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations'', JDE '''56''' (1985), 310-344.<br />
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[[Category:Bibliography]]</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=Bibliography&diff=7001Bibliography2009-07-18T00:53:52Z<p>Stanford88: /* AnMc-p */</p>
<hr />
<div>__NOTOC__<br />
<br />
=== Important note ===<br />
<br />
This bibliography was ported over from the previous incarnation of these pages. It should now be<br />
considered obsolete. Instead, we are now [[Help:References|giving each reference its own page]].<br />
<br />
The naming system for these articles was somewhat implicit and ''ad hoc''; we are currently forming [[Help:Bibliographic naming conventions|more explicit standards]] for this.<br />
<br />
<br />
=== References ===<br />
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R. Weder, The W_{k,p} continuity of the Schrodinger wave operators on the line, CMP 208 (1999), 507-520.<br />
<br />
=====[[Wed1999b]]=====<br />
R. Weder, Inverse scattering for the nonlinear Schrodinger equation, CPDE 22 (1997), 2089-2103.<br />
<br />
=====[[Wed2000]]=====<br />
R. Weder, L^p - L^p' estimates for the Schrodinger equation on the line and inverse scattering for the nonlinear Schrodinger equation with a potential, JFA 170 (2000), 37-68.<br />
<br />
=====[[Wed2000b]]=====<br />
R. Weder, Center manifold for nonintegrable nonlinear Schrodinger equations on the line, CMP 215 (2000), 343-356.<br />
<br />
=====[[Wed2001]]=====<br />
R. Weder, Inverse scattering for the nonlinear Schrodinger equation: reconstruction of the potential and the non-linearity, Math. Methods Appl. Sci. 24 (2001), 245-254.<br />
<br />
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R. Weder, Inverse scattering for the nonlinear Schrodinger equation II. reconstruction of the potential and the non-linearity in the multidimensional case, Proc. Amer. Math. Soc. 129 (2001), 3627-3645.<br />
<br />
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R. Weder, Direct and inverse scattering for the nonlinear Schrodinger equation with potential, VI Seminar on Free Boundary Value Problems and their Applications, part 1 (Rosario 1998), 13-20, Mat Ser. A. Conf. Semin. Trab. Math. 3, Univ. Austral, Rosario, 2001.<br />
<br />
=====[[Wed2001d]]=====<br />
R. Weder, Inverse scattering on the line for the nonlinear Klein-Gordon equation with a potential. J. Math. Anal. Appl. 252 (2000), 102-123.<br />
<br />
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R. Weder, Multidimensional inverse scattering for the nonlinear Klein-Gordon equation with a potential, JDE 184 (2002), 62-77.<br />
<br />
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R. Weder, The forced non-linear Schrodinger equation with a potential on the half-line, preprint.<br />
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R. Weder, The L^p - L^p' estimate for the Schrodinger equation on the half-line, preprint.<br />
<br />
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R. Weder, Scattering for the forced nonlinear Schrodinger equation with a potential on the half-line, preprint.<br />
<br />
=====[[Ws1983]]=====<br />
M. Weinstein, Nonlinear Schrodinger equations and sharp interpolation estimates, CMP 87 (1983), 567-576.<br />
<br />
=====[[Ws1985]]=====<br />
M. Weinstein, Modulational stability of ground states of nonlinear Schrodinger equations, SIAM J. Math. Anal. 16 (1985), 472-491.<br />
<br />
=====[[Ws1986]]=====<br />
M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive equations, CPAM 39 (1986), 51-68.<br />
<br />
=====[[Ws1986b]]=====<br />
M. Weinstein, On the structure and formation of singularities in solutions to nonlinear dispersive equations, CPDE 11 (1986), 545-565.<br />
<br />
=====[[WsXi1996]]=====<br />
M. Weinstein, J. Xin, Dynamic stability of vortex solutions of Ginzberg-Landau and nonlinear Schrodinger equations, CMP 180 (1996), 389-428.<br />
<br />
=====[[Wc1987]]=====<br />
M.V. Wickerhauser, Inverse scattering for the heat operators and evolutions in 2+1 variables, CMP 108 (1987) 67-89.<br />
<br />
=====[[Wx1959]]=====<br />
C. Wilcox, The initial boundary-value problem for the wave equation in an exterior domain with spherical boundary, Not. Amer. Math. Soc. 6 (1959).<br />
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T. Wolff, A sharp bilinear cone estimate, to appear, Ann. Math.<br />
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S Wollman, An existence and uniqueness theorem for the Vlasov-Maxwell system, CPAM 37 (1984), 457-462.<br />
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T. Woods, 1+1 Lorentz wave maps, preprint.<br />
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=====[[Wu1991]]=====<br />
U. Wuller, Geometric methods in scattering theory of the charge transfer model, Duke Math J. 62 (1991), 273-313.<br />
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K. Yagi, Normal forms and nonlinear Klein-Gordon equations in one space dimension, Master Thesis, Waseda University, 1994.<br />
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K. Yajima, Nonrelavistic limit of the Dirac theory, scattering theory, J. Fac. Sci. Univ. Tokyo. Sect. IA Math. 23 (1976), 517-523.<br />
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K. Yajima, A multichannel scattering theory for some time dependent Hamiltonians, charge transfer problem, CMP 75 (1980), 153-178.<br />
<br />
=====[[Ya1987]]=====<br />
K. Yajima, Existence of solutions for Schrodinger evolution equations, Commun. Math. Phys. 110 (1987), 415-426.<br />
<br />
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K. Yajima, The W^{k,p} continuity of wave operators for Schrodinger operators, J. Math. Soc. Japan 47 (1995), 551-581.<br />
<br />
=====[[Ya1999]]=====<br />
K. Yajima, The L^p continuity of wave operators for two-dimensional Schrodinger operators, CMP 208 (1999), 125-152.<br />
<br />
=====[[YaZgg2001]]=====<br />
K. Yajima, G. Zhang, Smoothing property for Schrodinger equations with potential superquadratic at infinity, CMP 221 (2001), 573-590.<br />
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K. Yokoyama, Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions, J. Math. Soc. Japan 52 (2000), 609-632.<br />
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B. Yordanov, Blow-up for the one-dimensional Klein-Gordon equation with a cubic non-linearity, preprint.<br />
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B. Yordanov, Q. Zhang, Finite time blowup for wave equations with a potential, preprint.<br />
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B. Yordanov, Q. Zhang, Finite time blowup for critical wave equations in high dimensions, preprint.<br />
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N. Zabusky, M. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15 (1965), 240-243.<br />
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V.E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP 35 (1972), 908-914.<br />
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V.E. Zakharov, Weakly nonlinear waves on the surface of an ideal finite depth fluid, Amer. Math. Soc. Transl. 182 (1998), 167-197.<br />
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V.E. Zakharov, S.V. Manakov, Asymptotic behavior of non-linear wave systems integrated by the inverse scattering method, Soviet Physics JETP 44 (1976), 106-112.<br />
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V.E. Zakharov, A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media, Sov. Phys. JETP 34 (1972), 62-69.<br />
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V.E. Zakharov, A.B. Shabat, Interaction between solitons in a stable medium, Sov. Phys. JETP 37 (1973), 834-828.<br />
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V.E. Zakharov, E.I. Schulman, Degenerated dispersion laws, motion invariant and kinetic equations, Physica 1D (1980), 185-250.<br />
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V.E. Zakharov, L.A. Takhtahan, Equivalence of a nonlinear Schrodinger equation and a Heisenberg ferromagnetic equation, Teor. Math. Phys. 8 (1979), 17-23.<br />
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B. Zhang, Unique continuation for the KdV equation, SIAM J. Math. Anal. 23 (1992), 55-71.<br />
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B. Zhang, A remark on the Cauchy problem for the Korteweg-de Vries equation on a periodic domain, DIE 8 (1995), 1191-1204.<br />
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B. Zhang, Unique continuation properties of the nonlinear Schrodinger equation, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 191-205.<br />
<br />
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X. Zhou, Inverse scattering transform for the time-dependent Schrodinger equation with application to the KP-I equations, CMP 128 (1990), 551-564.<br />
<br />
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Y. Zhou, Cauchy problem for semilinear wave equations in four space dimensions with small initial data, JPDE 8 (1995), 135-144.<br />
<br />
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Y. Zhou, Local existence with minimal regularity for nonlinear wave equations, Amer. J. Math, 119 (1997), 671-703.<br />
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Y. Zhou, Uniqueness of weak solutions of 1+1 dimensional wave maps, Math. Z. 232 (1999), 707-719.<br />
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Y. Zhou, Uniqueness of generalized solutions to nonlinear wave equations, AJM 122 (2000), 939-965.<br />
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[[Category:Bibliography]]</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=Einstein_equations&diff=7000Einstein equations2009-07-18T00:52:04Z<p>Stanford88: </p>
<hr />
<div>[Note: This is an immense topic, and we do not even begin to do it justice with this very brief selection of results. Further references or expansion of this article will, of course, be very much appreciated.]<br />
<br />
== Introduction ==<br />
<br />
The (vacuum) Einstein equations, for a Riemannian manifold <math>\, (M,g)\!</math>, take the form<br />
<br />
<center><math>\, R_{ \alpha \beta } = C R g_{ \alpha \beta }+\Lambda g_{ \alpha \beta } \!</math></center><br />
<br />
where <math>\, g\!</math> is the metric for a 3+1-dimensional manifold <math>\, M\!</math>, <math>\, R\!</math> is the Ricci curvature tensor, <math>\, C\!</math> is an absolute constant, and <math>\,\Lambda\!</math> the so-called cosmological constant. The Cauchy data for this problem is thus a three-dimensional Riemannian manifold together with the second fundamental form of this manifold (roughly speaking, this is like the initial position and initial velocity for the metric <math>g</math>). However, these two quantities are not completely independent; they must obey certain ''constraint equations''. These equations are now known to be well behaved for all <math>s > 3/2</math> [[Max-p]], [[Max2005]] (see also earlier work in higher regularities in [[RenFri2000]], [[Ren2002]]).<br />
<br />
Because of the diffeomorphism invariance of the Einstein equations, these equations are not hyperbolic as stated. However, this can be remedied by choosing an appropriate choice of co-ordinate system (which is the analog of a [[gauge transformation]] in gauge theory). One popular choice is ''harmonic co-ordinates'' or ''wave co-ordinates'', where the co-ordinate functions <math> x^a </math> are assumed to obey the wave equation <math>\Box_g x^a = 0</math> with respect to the metric <math>g</math>. In this case the Einstein equations take a form which (in gross caricature) looks something like<br />
<br />
<center><math>\Box_g g = \Gamma (g) Q(dg, dg) + </math>lower order terms</center><br />
<br />
where <math>Q</math> is some quadratic form of the first two derivatives. In other words, it becomes a [[QNLW|quasilinear wave equation]]. One would then specify initial data on the initial surface <math>x = 0</math>; the co-ordinate <math>x</math> plays the role of time, locally at least.<br />
<br />
* The [[critical]] regularity is <math>s_c = 3/2</math>. Thus energy is super-critical, which seems to make a large data global theory extremely difficult.<br />
* LWP is known in <math>H^s</math> for <math>s > 5/2</math> by energy estimates (see [[HuKaMar1977]], [[AnMc2003]]; for smooth data <math>s > 4</math> this is in [[Cq1952]]) - given that the initial data obeys the constraint equations, of course.<br />
** This result can be improved to <math>s>2</math> by the [[QNLW|recent quasilinear theory]] (see in particular [[KlRo-p3]], [[KlRo-p4]], [[KlRo-p5]]).<br />
** Some results have been recently achieved in the direction of the LWP for <math>s=2</math> ([[KlRo-p6]], [[KlRo-p7]], [[KlRo-p8]]). This problem is known in the litterature as the <math>L^2</math> conjecture.<br />
** For smooth data, one has a (possibly geodesically incomplete) maximal Cauchy development ([[CqGc1969]]).<br />
* GWP for small smooth asymptotically flat data was shown in [[CdKl1993]] (see also [[CdKl1990]]). In other words, [[Minkowski space]] is stable.<br />
** Another proof using the double null foliation is in [[KlNi2003]], [[KlNi-p]]<br />
** Another proof of this fact (using the Lorenz gauge, and assuming Schwarzschild metric outside of a compact set) is in [[LbRo-p]] (see also [[LbRo2003]] for a treatment of the asymptotic dynamics)<br />
** Singularities must form if there is a trapped surface ([[Pn1965]], [[Haw66]]).<br />
* Many special solutions (Schwarzschild space, Kerr space, etc.) The stability of these spaces is a very interesting (and difficult) question.<br />
* The equations can simplify under additional symmetry assumptions. The <math>U(1)</math>-symmetric case reduces to a system of equations which closely resembles the [[wave maps on R2|two-dimensional wave maps equation]] (with the target manifold being hyperbolic space <math>H^2</math>).<br />
<br />
== Global property of solutions ==<br />
<br />
It is widely known that exact solutions of vacuum Einstein equations display a singularity. Initially, this behavior was believed to be due to the particular symmetry chosen to solve them. Roger Penrose and Stephen Hawking, with some classical papers (see [[Pn1965]] and [[Haw66]]), proved that this is a general property of solutions of Einstein equations making the role of singularities central in the understanding of these equations. We can state this result in the following way [[HwEl73]]<br />
<br />
'''Theorem:''' ''Space-time is not timelike geodesically complete if:''<br />
<br />
''(1) <math>\, R_{\alpha\beta}K^\alpha K^\beta\ge 0\!</math> for every non-space-like vector <math>\, K\!</math>;''<br />
<br />
''(2) there exists a compact spacelike 3-surface S (without edge);''<br />
<br />
''(3) the unit normals to S are everywhere converging (or everywhere diverging) on S.''<br />
<br />
Condition (1) implies that energy is always zero or positive, condition (2) and (3) say that space-time is spatially closed and expanding or contracting.<br />
<br />
About the analytical structure of the singularities, a [[BKL Conjecture|conjecture]] was put forward by V.A.Belinski, I.M. Khalatnikov and E.M. Lifshitz in [[BeKhLi1970]]. In any case, to acquire knowledge of the analytical structure of such solutions we need to solve Einstein equations in a regime of strong field (see [[FraA2006]]).<br />
<br />
== Open problems ==<br />
<br />
* [[Cosmic Censorship Hypothesis]]<br />
<br />
* [[BKL Conjecture]]<br />
<br />
== Further reading ==<br />
<br />
* For more detail, we recommend the very nice [http://relativity.livingreviews.org/Articles/lrr-2002-6/index.html survey on existence and global dynamics of the Einstein equations by Alan Rendall].<br />
<br />
* S. W. Hawking, G. F. R. Ellis, ''The large scale structure of space-time'', (Cambridge University Press, Cambridge, 1973). <br />
<br />
<br />
[[Category:Geometry]]<br />
[[Category:Wave]]<br />
[[Category:Equations]]</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=Einstein_equations&diff=6999Einstein equations2009-07-18T00:49:26Z<p>Stanford88: </p>
<hr />
<div>[Note: This is an immense topic, and we do not even begin to do it justice with this very brief selection of results. Further references or expansion of this article will, of course, be very much appreciated.]<br />
<br />
== Introduction ==<br />
<br />
The (vacuum) Einstein equations, for a Riemannian manifold <math>\, (M,g)\!</math>, take the form<br />
<br />
<center><math>\, R_{ \alpha \beta } = C R g_{ \alpha \beta }+\Lambda g_{ \alpha \beta } \!</math></center><br />
<br />
where <math>\, g\!</math> is the metric for a 3+1-dimensional manifold <math>\, M\!</math>, <math>\, R\!</math> is the Ricci curvature tensor, <math>\, C\!</math> is an absolute constant, and <math>\,\Lambda\!</math> the so-called cosmological constant. The Cauchy data for this problem is thus a three-dimensional Riemannian manifold together with the second fundamental form of this manifold (roughly speaking, this is like the initial position and initial velocity for the metric <math>g</math>). However, these two quantities are not completely independent; they must obey certain ''constraint equations''. These equations are now known to be well behaved for all <math>s > 3/2</math> [[Max-p]], [[Max2005]] (see also earlier work in higher regularities in [[RenFri2000]], [[Ren2002]]).<br />
<br />
Because of the diffeomorphism invariance of the Einstein equations, these equations are not hyperbolic as stated. However, this can be remedied by choosing an appropriate choice of co-ordinate system (which is the analog of a [[gauge transformation]] in gauge theory). One popular choice is ''harmonic co-ordinates'' or ''wave co-ordinates'', where the co-ordinate functions <math> x^a </math> are assumed to obey the wave equation <math>\Box_g x^a = 0</math> with respect to the metric <math>g</math>. In this case the Einstein equations take a form which (in gross caricature) looks something like<br />
<br />
<center><math>\Box_g g = \Gamma (g) Q(dg, dg) + </math>lower order terms</center><br />
<br />
where <math>Q</math> is some quadratic form of the first two derivatives. In other words, it becomes a [[QNLW|quasilinear wave equation]]. One would then specify initial data on the initial surface <math>x = 0</math>; the co-ordinate <math>x</math> plays the role of time, locally at least.<br />
<br />
* The [[critical]] regularity is <math>s_c = 3/2</math>. Thus energy is super-critical, which seems to make a large data global theory extremely difficult.<br />
* LWP is known in <math>H^s</math> for <math>s > 5/2</math> by energy estimates (see [[HuKaMar1977]], [[AnMc 2003]]; for smooth data <math>s > 4</math> this is in [[Cq1952]]) - given that the initial data obeys the constraint equations, of course.<br />
** This result can be improved to <math>s>2</math> by the [[QNLW|recent quasilinear theory]] (see in particular [[KlRo-p3]], [[KlRo-p4]], [[KlRo-p5]]).<br />
** Some results have been recently achieved in the direction of the LWP for <math>s=2</math> ([[KlRo-p6]], [[KlRo-p7]], [[KlRo-p8]]). This problem is known in the litterature as the <math>L^2</math> conjecture.<br />
** For smooth data, one has a (possibly geodesically incomplete) maximal Cauchy development ([[CqGc1969]]).<br />
* GWP for small smooth asymptotically flat data was shown in [[CdKl1993]] (see also [[CdKl1990]]). In other words, [[Minkowski space]] is stable.<br />
** Another proof using the double null foliation is in [[KlNi2003]], [[KlNi-p]]<br />
** Another proof of this fact (using the Lorenz gauge, and assuming Schwarzschild metric outside of a compact set) is in [[LbRo-p]] (see also [[LbRo2003]] for a treatment of the asymptotic dynamics)<br />
** Singularities must form if there is a trapped surface ([[Pn1965]], [[Haw66]]).<br />
* Many special solutions (Schwarzschild space, Kerr space, etc.) The stability of these spaces is a very interesting (and difficult) question.<br />
* The equations can simplify under additional symmetry assumptions. The <math>U(1)</math>-symmetric case reduces to a system of equations which closely resembles the [[wave maps on R2|two-dimensional wave maps equation]] (with the target manifold being hyperbolic space <math>H^2</math>).<br />
<br />
== Global property of solutions ==<br />
<br />
It is widely known that exact solutions of vacuum Einstein equations display a singularity. Initially, this behavior was believed to be due to the particular symmetry chosen to solve them. Roger Penrose and Stephen Hawking, with some classical papers (see [[Pn1965]] and [[Haw66]]), proved that this is a general property of solutions of Einstein equations making the role of singularities central in the understanding of these equations. We can state this result in the following way [[HwEl73]]<br />
<br />
'''Theorem:''' ''Space-time is not timelike geodesically complete if:''<br />
<br />
''(1) <math>\, R_{\alpha\beta}K^\alpha K^\beta\ge 0\!</math> for every non-space-like vector <math>\, K\!</math>;''<br />
<br />
''(2) there exists a compact spacelike 3-surface S (without edge);''<br />
<br />
''(3) the unit normals to S are everywhere converging (or everywhere diverging) on S.''<br />
<br />
Condition (1) implies that energy is always zero or positive, condition (2) and (3) say that space-time is spatially closed and expanding or contracting.<br />
<br />
About the analytical structure of the singularities, a [[BKL Conjecture|conjecture]] was put forward by V.A.Belinski, I.M. Khalatnikov and E.M. Lifshitz in [[BeKhLi1970]]. In any case, to acquire knowledge of the analytical structure of such solutions we need to solve Einstein equations in a regime of strong field (see [[FraA2006]]).<br />
<br />
== Open problems ==<br />
<br />
* [[Cosmic Censorship Hypothesis]]<br />
<br />
* [[BKL Conjecture]]<br />
<br />
== Further reading ==<br />
<br />
* For more detail, we recommend the very nice [http://relativity.livingreviews.org/Articles/lrr-2002-6/index.html survey on existence and global dynamics of the Einstein equations by Alan Rendall].<br />
<br />
* S. W. Hawking, G. F. R. Ellis, ''The large scale structure of space-time'', (Cambridge University Press, Cambridge, 1973). <br />
<br />
<br />
[[Category:Geometry]]<br />
[[Category:Wave]]<br />
[[Category:Equations]]</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=QNLW&diff=6998QNLW2009-07-18T00:42:18Z<p>Stanford88: </p>
<hr />
<div>In a local co-ordinate chart, '''quasilinear wave equations''' (QNLW) take the form<br />
<br />
<center><math>\partial_\alpha g^{\alpha\beta}(u) \partial_\beta u = F(u, Du)</math>.</center><br />
<br />
One could also consider equations where the metric depends on derivatives of <math>u</math>, but one can reduce to this case (giving up a derivative) by differentiating the equation. One can also reduce to the case <math>g^{00} = 1</math>, <math>g^{0i} = g^{i0} = 0</math> by a suitable change of variables. <math>F</math> is usually quadratic in the derivatives <math>Du</math>, as this formulation is then robust under many types of changes of variables.<br />
<br />
== Specific equations ==<br />
<br />
Quasilinear NLWs appear frequently in general relativity. Examples include<br />
<br />
* The [[Einstein equations]]<br />
* Equations of relativistic elasticity<br />
* Equations of relativistic hydrodynamics<br />
* The [[minimal surface equation]]<br />
<br />
The most interesting dimension is of course the physical dimension d=3.<br />
<br />
== Wellposedness theory ==<br />
<br />
Classically one has LWP for <math>H^s</math> when <math>s > d/2+1</math> ([[HuKaMar1977]]), but the [[DNLW|semilinear theory]] suggests that we should be able to improve this to <math>s > s_c = d/2</math> with a null condition, and to <math>s > d/2 + max(1/2, (d-5)/4)</math> without one (these results would be sharp even in the semilinear case). In principle Strichartz estimates should be able to push down to <math>s > d/2 + 1/2</math>, but only partial results of this type are known. Specifically:<br />
<br />
* When <math>d=2</math> one has LWP in the expected range <math>s > d/2 + 3/4</math> without a null condition ([[SmhTt-p]]).<br />
** For <math>s > d/2 + 3/4 + 1/12</math> this is in [[Tt-p5]] (using the FBI transform).<br />
** For <math>s > d/2 + 3/4 + 1/8</math> this is in [[BaCh1999]] (using FIOs) and [[Tt2000]] (using the FBI transform).<br />
* When <math>d=3,4,5</math> one has LWP for <math>s > d/2 + 1/2</math> [[SmhTt-p]] (using parametrices and the equation for the metric); in the specific case of the [[Einstein equations]] see [[KlRo-p3]], [[KlRo-p4]], [[KlRo-p5]] (using vector fields and the equation for the metric)<br />
** For <math>s > d/2 + 1/2 + 1/7</math> (approx) and <math>d=3</math> this is in [[KlRo-p2]] (vector fields and the equation for the metric)<br />
** For <math>s > d/2 + 1/2 + 1/6</math> and <math>d=3</math> this is in [[Tt-p5]] (using the FBI transform).<br />
** For <math>s > d/2 + 1/2 + 1/5</math> (approx) and <math>d=3</math> this is in [[Kl-p2]] (vector fields methods).<br />
** For <math>s > d/2 + 1/2 + 1/4</math> and <math>d \geq 3</math> this is in [[BaCh1999]] (using FIOs) and [[Tt2000]] (using the FBI transform). See also [[BaCh1999b]].<br />
<br />
A special type of QNLW is the cubic equations, where the metric g itself obeys an elliptic equation of the form Delta <math>g = |Du|^2</math>, and the non-linearity is of the form <math>Dg Du</math>. For such equations, we have LWP for <math>s > d/2 + 1/6</math> when <math>d \geq 4</math> [[BaCh 2003]], [[BaCh2002]]. This equation has some similarity with the differentiated [[wave maps|wave maps equation]] in the Coulomb gauge.<br />
<br />
For small smooth compactly supported data of size <math>\epsilon</math> and smooth non-linearities, the GWP theory for QNLW is as follows.<br />
<br />
* If the non-linearity is a null form, then one has GWP for <math>d\geq3</math>; in fact one can take the data in a weighted Sobolev space <math>H^{4,3} x H^{3,4}</math> [[Cd1986]].<br />
** Without the null structure, one has almost GWP in d=3 [[Kl1985b]], and this is sharp [[Jo1981]], [[Si1983]]<br />
*** In the semi-linear case and when the nonlinearity is quadratic in the derivatives, this is also true outside of a compact non-trapping obstacle [[KeSmhSo-p2]]. This has been generalized to the quasi-linear case in [[KeSmhSo-p3]] (and non-linear Dirichlet wave equations are also treated there, as are multiple speeds).<br />
** With a null structure and outside a star-shaped obstacle with Dirichlet conditions and <math>d=3</math>, one has GWP for small data in <math>H^{9,8} \times H^{8,9}</math> which are compatible with the boundary [KeSmhSo-p]. Earlier work in this direction is in [[Dt1990]].<br />
*** For radial data and obstacle this was obtained in [[Go1995]]; see also [[Ha1995]], [[Ha2000]].<br />
*** In the semilinear case, the non-trapping condition was removed in [[MetSo-p]], even in the multiple speed case, provided one has an exponential decay result near the obstacle (this is true, for instance, if the obstacle is a union of a finite number of sufficiently separated strictly convex bodies).<br />
** For <math>d>3</math> or for cubic nonlinearities one has GWP regardless of the null structure [[KlPo1983]], [[Sa1982]], [[Kl1985b]].<br />
*** In three dimensions with a null structure, for systems with multiple wave speeds, one has GWP [[So2001]]<br />
*** In the exterior of an nontrapping obstacle with Dirichlet conditions, with multiple speeds, one has GWP for sufficiently smooth and decaying data obeying the usual compatibility conditions at the boundary [[MetSo-p2]], if the quasilinear terms obey some symmetry conditions and the semilinear terms are quadratic in <math>Du</math><br />
***# When the obstacle is a ball this is in [[Ha1995]].<br />
***# For <math>d \geq 6</math> outside of a starshaped obstacle this is in [[ShbTs1984]], [[ShbTs1986]].<br />
<br />
[[Category:Wave]]<br />
[[Category:Equations]]</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=Bibliography&diff=6997Bibliography2009-07-17T11:10:08Z<p>Stanford88: /* BaCh 2003 */</p>
<hr />
<div>__NOTOC__<br />
<br />
=== Important note ===<br />
<br />
This bibliography was ported over from the previous incarnation of these pages. It should now be<br />
considered obsolete. Instead, we are now [[Help:References|giving each reference its own page]].<br />
<br />
The naming system for these articles was somewhat implicit and ''ad hoc''; we are currently forming [[Help:Bibliographic naming conventions|more explicit standards]] for this.<br />
<br />
<br />
=== References ===<br />
<br />
<center>A</center><br />
<br />
<br />
=====[[Aa2002]]=====<br />
M. Aassilla, ''Nonexistence de solutions globales de certains equations d'ondes nonlineaires'', C. R. Acad. Sci. Paris. '''334''' (2002), 961-966.<br />
<br />
=====[[AbdBnFelSau1989]]=====<br />
L. Abdelouhab, J. Bona, M. Felland, J. Saut, ''Nonlocal models for nonlinear dispersive waves'', Phys. D '''40''' (1989), 360-392.<br />
<br />
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M. Ablowitz, D. Kaup, A. Newell, H. Segur, ''The inverse scattering transform-Fourier analysis for nonlinear problems'', Studies in Appl. Math. '''53''' (1974), no. 4, 249-315.<br />
<br />
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M. Ablowitz, A. Fokas, ''The inverse scattering transform for the Benjamin-Ono equation, a pivot for multidimensional problems'', Stud. Appl. Math. '''68''' (1983), 1-10.<br />
<br />
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M. Ablowitz, R. Haberman, ''Nonlinear evolution equations in two and three dimensions'', Phys. Rev. Lett. '''35''' (1975), 1185-1188.<br />
<br />
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M. Ablowitz, Y. Ma, The periodic nonlinear Schrodinger equation, Stud. Appl. Math. 65 (1981), 113-158.<br />
<br />
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M. Ablowitz, H. Segur, ''Asymptotic solutions of the Korteweg de Vries equation'', Stud. Appl. Math. '''57''' (1977), 13-44.<br />
<br />
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H. Added, S. Added, ''Equations of Langmuir turbulence and non-linear Schrodinger equation: Smoothness and approximations'', JFA '''79''' (1988), 183-210.<br />
<br />
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R. Agemi, K. Yokoyama, The null condition and global existence of solutions to systems of wave equations with different speeds, Adv. Nonlinear PDE and Stochastics (1998), 43-86.<br />
<br />
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J.C. Alexander, R.L. Pego, R.L. Sachs, ''On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation'', Phys. Lett. A. '''226''' (1997), 187-192.<br />
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S. Alinhac, ''Blowup for nonlinear hyperbolic equations'', Boston<nowiki>: Birkhauser, 1995, Progress in </nowiki>Nonlinear DE and their Applications, 17.<br />
<br />
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S. Alinhac, ''Blow up of small data solutions for a class of quasilinear wave equations in two space dimensions I'', Annals of Mathematics 149 (1999), 97-127.<br />
<br />
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S. Alinhac, ''Blow up of small data solutions for a class of quasilinear wave equations in two space dimensions II,'' Acta Math. 182 (1999), 1-23.<br />
<br />
=====[[Al2000]]=====<br />
S. Alinhac, Rank 2 singular solutions for quasilinear wave equations, IMRN 18 (2000), 955-984.<br />
<br />
=====[[Al2001]]=====<br />
S. Alinhac, ''The null condition for quasilinear wave equations in two space dimensions I'', Invent. Math. '''145''' (2001), 597-618.<br />
<br />
=====[[Al2001b]]=====<br />
S. Alinhac, ''The null condition for quasilinear wave equations in two space dimensions II, ''AJM '''123''' (2001), 1071-1101.<br />
<br />
=====[[Al2003]]=====<br />
S. Alinhac, An example of blowup at infinity for a quasilinear wave equation, Asterisque 284 (2003), 1-91.<br />
<br />
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S. Anco, J. Isenberg, ''Global existence for wave maps with torsion'', CPDE '''25''' (2000), 1669-1702.<br />
<br />
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L. Andersson, V. Moncrief, ''[http://www.arxiv.org/abs/gr-qc/0110111 Elliptic-Hyperbolic systems and the Einstein equations]'', preprint.<br />
<br />
=====[[Ant2003]]=====<br />
C. Antonini, Lower bounds for the L^2 minimal periodic blowup solutions of critical nonlinear Schrodinger equation, DIE 15 (2002), 749-768.<br />
<br />
=====[[AntMe2001]]=====<br />
C. Antonini, F. Merle, ''Optimal bounds on positive blowup solutions for a semilinear wave equation'', IMRN '''21''' (2001), 1143-1167.<br />
<br />
=====[[ArYa2000]]=====<br />
G. Artbazar, K. Yajima, ''The L^p continuity of wave operators for one dimensional Schrodinger operators'', J. Math. Sci. Univ. Tokyo '''7''' (2000), 221-240.<br />
<br />
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J. M. Ash, J. Cohen, G. Wang, On strongly interacting internal solitary waves, J. Fourier Anal. Appl., '''2''' (1996), 507-517.<br />
<br />
<center>B</center><br />
<br />
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A. Bachelot, ''Probleme de Cauchy pour des systemes hyperboliques semilineaires'', Ann. IHP (Anal. nonlin), '''1''' (1984), 453-478.<br />
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A. Bachelot, J-P. Nicolas, Equation non-lineaire de Klein-Gordon dans des metriquesde type Schwarzchild, C.R. Acad. Sci. Paris. 316 (1993), 1047-1050<br />
<br />
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J. B. Baillon, J. M. Chadam, ''The Cauchy problem for the coupled Schrodinger-Klein-Gordon equations'', in "Contemporary Developments in Continuum Mechanics and Partial Differential Equations", G. M. de La Penha and L. A. Medeiros eds., North Holland, Amsterdam 1978<br />
<br />
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J. Baez, I. Sega, Z. Zhou, ''The global Goursat problem and scattering for nonlinear wave equations'', J. Funct. Anal. '''93 '''(1990), 239-269.<br />
<br />
=====[[BaCh1999]]=====<br />
H. Bahouri, J-Y. Chemin, ''Équations d'ondes quasilinéaires et les inegalites de Strichartz'', Amer. J. Math.'''121''' (1999), 1337-1377.<br />
<br />
=====[[BaCh1999b]]=====<br />
H. Bahouri, J-Y. Chemin, ''Équations d'ondes quasilinéaires et effet dispersif'', IMRN '''21''' (1999), 1141-1178.<br />
<br />
=====[[BaCh2002]]=====<br />
H. Bahouri, J-Y. Chemin, ''[http://arXiv.org/abs/math.AP/0304390 Quasilinear Wave equations and Microlocal Analysis]'', ICM 2002, Vol III, 141-154.<br />
<br />
=====[[BaCh 2003]]=====<br />
H. Bahouri, J-Y. Chemin, ''Microlocal analysis, bilinear estimates, and cubic quasilinear wave equation'', Asterique (2003), 93-141<br />
<br />
=====[[BaGd1997]]=====<br />
H. Bahouri and P.Gerard, ''High frequency approximation of solutions to critical nonlinear wave equations'', Prepublications 97-34, Universite de Paris-Sud, Mai 1997. Appeared in: Amer. J. Math '''121''' (1999), 131-175.<br />
<br />
=====[[BaGdXu2000]]=====<br />
H. Bahouri, P. Gerard, C. Xu, ''Espaces de Besov et estimations de Strichartz generalisees sur le groupe de Heisenberg'', Journal d'Analyse Mathematique '''82''' (2000), 93-118.<br />
<br />
=====[[BaSa1998]]=====<br />
H. Bahouri and J.Shatah, ''Decay estimates for the critical semilinear wave equation'', Ann. Inst. H. Poincare Anal. Non Lineaire '''15''' (1998), no. 6, 783-789.<br />
<br />
=====[[Ban-p]]=====<br />
V. Banica, On the nonlinear Schrodinger dynamics on S^2, preprint.<br />
<br />
=====[[Ban-p2]]=====<br />
V. Banica, Dispersion and Strichartz inequalities for Schrodinger equations with singular coefficients, preprint.<br />
<br />
=====[[Ban-p3]]=====<br />
V. Banica, ''[http://www.arxiv.org/abs/math.AP/0401129 Remarks on the blowup for the nonlinear Schrodinger equation with critical mass on a plane domain]'', preprint.<br />
<br />
=====[[Bb1984]]=====<br />
J.E. Barab, ''Nonexistence of asymptotically free solutions for a nonlinear Schrodinger equation'', J. Math. Phys. '''25''' (1984), 3270-3273.<br />
<br />
=====[[Be1983]]=====<br />
M. Beals, ''Self-Spreading and strength of Singularities for solutions to semilinear wave equations'', Annals of Math '''118''' (1983), 187-214<br />
<br />
=====[[BeBz1996]]=====<br />
M. Beals, M. Bezard, ''Low regularity local solutions for field equations'', Comm. Partial Differential Equations'''21''' (1996), 79-124.<br />
<br />
=====[[BecMauSb-p]]=====<br />
P. Bechouce, N. Mauser, S. Selberg, ''[http://arXiv.org/abs/math.AP/0202201 Nonrelativistic limit of Klein-Gordon-Maxwell to Schrodinger Poisson]'', preprint<br />
<br />
=====[[BecMauSb-p2]]=====<br />
P. Bechouce, N. Mauser, S. Selberg, ''[http://arXiv.org/abs/math.AP/0303079 On the asymptotic analysis of the Dirac-Maxwell system in the nonrelativistic limit]'', preprint<br />
<br />
=====[[BkOgPo1996]]=====<br />
D. Bekiranov, T. Ogawa, G. Ponce, ''On the well-posedness of Benney's interaction equation of short and long waves'', Adv. Diff. Eq. '''1''' (1996), 919-937.<br />
<br />
=====[[BkOgPo1997]]=====<br />
D. Bekiranov, T. Ogawa, G. Ponce, ''Weak solvability and well-posedness of the coupled Schrodinger Korteweg-de Vries equation in the capillary-gravity interaction waves'', Proc. Amer. Math. Soc. '''125''' (1997), 2907-2919.<br />
<br />
=====[[BkOgPo1998]]=====<br />
D. Bekiranov, T. Ogawa, G. Ponce, ''Interaction equations for short and long dispersive waves'', J. Funct. Anal. '''158''' (1998), 357-388.<br />
<br />
=====[[BcKkZz1999]]=====<br />
E. Belchev, M. Kepka, Z. Zhou, ''Global existence of solutions to nonlinear wave equations'', CPDE '''24 '''(1999), 2297-2331.<br />
<br />
=====[[BcKkZz2002]]=====<br />
E. Belchev, M. Kepka, Z. Zhou, ''Finite time blowup of solutions to semilinear wave equations'', JFA '''190 '''(2002), 233-254.<br />
<br />
=====[[Bj1967]]=====<br />
T. Benjamin, ''Internal waves of permanent form in fluids of great depth'', J. Fluid Mech. '''29''' (1967), 559-562.<br />
<br />
=====[[Bj1972]]=====<br />
T. Benjamin, ''The stability of solitary waves'', Proc. R. Soc. Lond. A. '''328''' (1972), 153-183.<br />
<br />
=====[[BjBnMah1972]]=====<br />
T. Benjamin, J. Bona, J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil. Trans. R. Soc. London A 272 (1972), 47-78.<br />
<br />
=====[[BenKl1992]]=====<br />
M. Ben-Artzi, S. Klainerman, ''Decay and regularity for the Schrodinger equation'', J. Anal. Math. '''58''' (1992), 25-37.<br />
<br />
=====[[BenSau1999]]=====<br />
M. Ben-Artzi, J.C. Saut, ''Uniform decay estimates for a class of oscillatory integrals and applications'', Diff. Int. Eq. '''12''' (1999), 137-145.<br />
<br />
=====[[BenKocSau2003]]=====<br />
M. Ben-Artzi, H. Koch, J.C. Saut, Dispersion estimates for third order equations in two dimensions, CPDE 28 (2003), 1943-1974.<br />
<br />
=====[[ByNe1967]]=====<br />
D.J. Benney, A.C. Newell, ''The propagation of nonlinear wave envelopes'', J. Math. and Phys. '''46''' (1967), 133-139.<br />
<br />
=====[[BerCa1981]]=====<br />
H. Berestycki, T. Cazenave, ''Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires'', C. R. Acad. Sc. Paris, t. 293 (1981), 489-492.<br />
<br />
=====[[BgCcMc1995]]=====<br />
B.K. Berger, P. T. Chrusciel, V. Moncrief, ''On "asymptotically flat" spacetimes with G_2 invariant Cauchy surfaces'', Ann. Physics '''237''' (1995), 322-354.<br />
<br />
=====[[BiLi-p]]=====<br />
<br />
H. Biagioni, F. Linares, ''Ill-posedness for the Derivative Schrodinger and Generalized Benjamin-Ono equations'', preprint.<br />
<br />
=====[[BirPoVe-p]]=====<br />
B. Birnir, G. Ponce, L. Vega, On the ill-posedness for the initial value problem for the modified Korteweg-de Vries equation, preprint.<br />
<br />
=====[[BirKnPoSvVe1996]]=====<br />
B. Birnir, C. Kenig, G. Ponce, N. Svanstedt, L. Vega, ''On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrödinger equations''. J. London Math. Soc. (2) '''53''' (1996), 551-559.<br />
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=====[[Biz2000]]=====<br />
P. Bizon, ''Equivariant self-similar wave maps from Minkowski spacetime into 3-sphere.'' Comm. Math. Phys. '''215''' (2000), 45-56.<br />
<br />
=====[[Biz-p]]=====<br />
P. Bizon, ''[http://arXiv.org/abs/math-ph/0206004 Formation of singularities in Yang-Mills equations]'', preprint<br />
<br />
=====[[BizCjTb2001]]=====<br />
P. Bizon, T. Chmaj, Z. Tabor, ''[http://arxiv.org/abs/math-ph/0011005 Formation of singularities for equivariant 2+1 dimensional wave maps into two-sphere]'', Nonlinearity '''14''' (2001), no. 5, 1041-1053.<br />
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=====[[BizOvSi-p]]=====<br />
P. Bizon, Y. Ovchinnikov, I. Sigal, ''[http://arxiv.org/abs/math.AP/0307026 Collapse of an Instanton]'', preprint.<br />
<br />
=====[[BizTb2001]]=====<br />
P. Bizon, Z. Tabor, ''[http://arxiv.org/abs/math-ph/0105016 On blowup for Yang-Mills fields]'', Phys. Rev. D (3) '''64''' (2001), 121701.<br />
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=====[[BluSf2003]]=====<br />
P. Blue, A. Soffer, Semilinear wave equations on the Schwarzschild manifold I.Local decay estimates, Adv. Diff Eq. 8 (2003), 595-614.<br />
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J. Bona, ''On the stability theory of solitary waves'', Proc. R. Soc. Lond. A. '''344''' (1975), 363-374.<br />
<br />
=====[[BnDgKar1986]]=====<br />
J. Bona, V.A. Dougalis, O.A. Karakashian, ''Fully discrete Galerkin methods for the Korteweg-de Vries equation'', Comput. Math. Appl. Ser. A '''12''' (1986), 859-884.<br />
<br />
=====[[BnLiu2002]]=====<br />
J. Bona, Y. Liu, ''Instability of solitary-wave solutions of the Kadomtsev-Petviashvili equation in three dimensions'', ADE '''7''' (2002), 1-33.<br />
<br />
=====[[BnPoSauTm1992]]=====<br />
J. Bona, G. Ponce, J-C. Saut, M. Tom, ''A model system for strong interaction between internal solitary waves'', CMP '''143''' (1992), 287-313.<br />
<br />
=====[[BnSco1976]]=====<br />
J. Bona, R. Scott, ''Solutions of the Korteweg-de Vries equation, in fractional order Sobolev spaces'', Duke Math J. '''43''' (1976), 87-99.<br />
<br />
=====[[BnSmr1975]]=====<br />
J. Bona, R. Smith, ''The initial-value problem for the Korteweg-de Vries equation'', Philos. Trans. Royal Soc. London Series A 278 (1975), 555-601.<br />
<br />
=====[[BnSouSr1987]]=====<br />
J. Bona, P.E. Sougandis, W. Strauss, ''Stability and instability of solitary waves of Korteweg-de Vries type'', Proc. R. Soc. Lond. '''411''' (1987), 395-412.<br />
<br />
=====[[BnSuZh-p]]=====<br />
J. Bona, S. Sun, B-Y. Zhang, ''A non-homogeneous boundary-value problem for the Korteweg-de Vries Equation in a Quarter Plane'', preprint<br />
<br />
=====[[BnWi1983]]=====<br />
J. Bona, R. Winther, ''The Korteweg-de Vries equation, posed in a quarter plane'', SIAM J. Math. Anal. '''14''' (1983), 1056-1106.<br />
<br />
=====[[BnWe-p]]=====<br />
J. Bona, F. B. Weissler, ''Similarity solutions of the generalized Korteweg-de Vries equations'', preprint.<br />
<br />
=====[[Bd1993]]=====<br />
A. de Bouard, Analytic solutions to nonelliptic nonlinear Schrodinger equations, JDE 104 (1993), 196-213.<br />
<br />
=====[[BdDbTs1999]]=====<br />
A. de Bouard, A. Debussche, Y. Tsutsumi, ''[http://www.ams.org/mathscinet?screen=leavingmsn&publ=Academic%20Press&url=http://www.idealibrary.com/cgi%2Dbin/links/citation/0022%2D1236/169/532&back_url=msnmain%3Fpg3=ICN%26s3=Tsutsumi%26co3=AND%26pg4=TI%26s4=%26co4=AND%26pg5=CC%26s5=%26co5=A%20 White noise driven Korteweg-de Vries equation.]'' J. Funct. Anal. '''169''' (1999), no. 2, 532-558<br />
<br />
=====[[BdSau1997]]=====<br />
A. de Bouard, J.C. Saut, ''Solitary waves of generalized Kadomtsev-Petviashvili equations'', Annales IHP, Analyse non-lineaire '''14''' (1997) 211-236.<br />
<br />
=====[[BdHaSau1997]]=====<br />
A. de Bouard, N. Hayashi, J.C. Saut, Global existence of small solutions to a nonlinear relativistic Schrodinger equation, CMP 189 (1997), 73-105.<br />
<br />
=====[[BdMt2004]]=====<br />
A. de Bouard, Y. Martel, Nonexistence of L^2-compact solutions of the Kadomptsev-Petviashvili II equation, Math. Ann. 328 (2004), 525\u2014554.<br />
<br />
=====[[BctGolPal-p]]=====<br />
F. Bouchut, F. Golse, C. Pallard, ''[http://front.math.ucdavis.edu/math.AP/0301175 On classical solutions to the 3D relativistic Vlasov-Maxwell system: Glassey-Strauss's theorem revisited]'', preprint.<br />
<br />
=====[[Bo1992]]=====<br />
J. Bourgain, ''A remark on Schrodinger operators'', Israel J. Math. '''77''' (1992), 1-16.<br />
<br />
=====[[Bo1993]]=====<br />
J. Bourgain, ''Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I. Schrodinger equations,'' [http://link.springer.de/link/service/journals/00039/index.htm Geom. Funct. Anal.] '''3''' (1993) 107-156.<br />
<br />
=====[[Bo1993b]]=====<br />
J. Bourgain, ''Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II. The periodic KdV equation,'' [http://link.springer.de/link/service/journals/00039/index.htm Geom. Funct. Anal.] '''3''' (1993) 209-262.<br />
<br />
=====[[Bo1993c]]=====<br />
J. Bourgain, ''On the Cauchy problem for the Kadomtsev-Petviashvili equation'', [http://link.springer.de/link/service/journals/00039/index.htm Geom. Funct. Anal.] '''3''' (1993) 315-341.<br />
<br />
=====[[Bo1993d]]=====<br />
J. Bourgain, ''Exponential sums and nonlinear Schrodinger equations'', [http://link.springer.de/link/service/journals/00039/index.htm Geom. Funct. Anal.] '''3''' (1993) 157-178.<br />
<br />
=====[[Bo1994]]=====<br />
J. Bourgain, ''On the Cauchy and invariant measure problem for the periodic Zakharov system'', Duke Math. J. '''76''' (1994), 175-202.<br />
<br />
=====[[Bo1994b]]=====<br />
J. Bourgain, ''Harmonic analysis and nonlinear partial differential equations'', Proceedings of the ICM 1994, 31-44, Birkhauser Basel, 1995.<br />
<br />
=====[[Bo1994c]]=====<br />
J. Bourgain, ''Periodic nonlinear Schrodinger equation and invariant measures'', CMP '''166''' (1994), 1-26.<br />
<br />
=====[[Bo1995]]=====<br />
J. Bourgain, ''Estimates for cone multipliers'', Operator Theory: Advances and Applications, '''77''' (1995), 41-60.<br />
<br />
=====[[Bo1995b]]=====<br />
J. Bourgain, ''Aspects of longtime behaviour of solutions of nonlinear Hamiltonian evolution equations'', GAFA ''5'' (1995), 105-140.<br />
<br />
=====[[Bo1995c]]=====<br />
J. Bourgain, ''Periodic nonlinear Schrodinger equations and invariant measures'', CMP '''166''' (1995), 1-26.<br />
<br />
=====[[Bo1996]]=====<br />
J. Bourgain, ''On well-posedness of the Zakharov system'', [http://www.imrn.org Int. Math. Research Notices] '''11''' (1996), 515-546.<br />
<br />
=====[[Bo1996b]]=====<br />
J. Bourgain, ''Invariant measures for the 2d-defocussing nonlinear Schrodinger equation'', Comm. Math. Phys. '''176''' (1996), 421-445.<br />
<br />
=====[[Bo1996c]]=====<br />
J. Bourgain, ''On the growth in time of higher order Sobolev norms of smooth solutions of Hamiltonian PDE'', IMRN '''6''' (1996), 277-304.<br />
* [[kdv_measures.pdf Bo1997]] J. Bourgain, ''[kdv_measures.pdf Periodic Korteweg de Vries equation with measures as initial data]'', Selecta Math. (N.S.) '''3''' (1997), 115-159.<br />
<br />
=====[[Bo1997b]]=====<br />
J. Bourgain, ''On the compactness of support of solutions of dispersive equations'', [http://www.imrn.org Int. Math. Research Notices] '''9''' (1997), 437-447.<br />
<br />
=====[[Bo1998]]=====<br />
J. Bourgain, ''Refinements of Strichartz' inequality and applications to 2D-NLS with critical non-linearity'', [http://www.imrn.org Int. Math. Research Notices], '''5''' (1998), 253-283.<br />
<br />
=====[[Bo1998b]]=====<br />
J. Bourgain, ''Scattering in the energy space and below for 3D NLS'', J. Anal. Math. '''75''' (1998), 267-297.<br />
<br />
<br />
=====[[Bo1999]]===== J. Bourgain, ''[http://www.math.ucla.edu/~tao/bourgain/borg_nls.dvi New global well-posedness results for non-linear Schrodinger equations]'', AMS 1999.<br />
<br />
=====[[Bo1999b]]=====<br />
J. Bourgain, ''Global well-posedness of defocussing 3D critical NLS in the radial case'', J. Amer. Math. Soc. '''12''' (1999), 145-171.<br />
<br />
=====[[Bo-p]]=====<br />
J. Bourgain, Remarks on stability and diffusion in high-dimensional Hamiltonian systems and PDE, preprint.<br />
<br />
=====[[Bo-p2]]=====<br />
J. Bourgain, A remark on normal forms and the I-method for periodic NLS, preprint.<br />
<br />
=====[[Bo-p3]]=====<br />
J. Bourgain, On the Cauchy problem for periodic KdV type equations, preprint.<br />
<br />
=====[[BoCo1996]]=====<br />
J. Bourgain, J. Colliander, ''On the well-posedness of the Zakharov system'', IMRN '''11''' (1996), 515-546.<br />
<br />
=====[[BoWg1997]]=====<br />
J. Bourgain, W. Wang, ''Construction of blowup solutions for the nonlinear Schrodinger equation with critical non-linearity,'' Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) '''25''' (1997), no. 1-2, 197-215.<br />
<br />
=====[[Bou1996]]=====<br />
N. Bournaveas, ''Local existence for the Maxwell-Dirac equations in three space dimensions''. Comm. Partial Differential Equations '''21''' (1996) 693-720.<br />
<br />
=====[[Bou1999]]=====<br />
N. Bournaveas, ''Local existence of energy class solutions for the Dirac-Klein-Gordon equations''. Comm. Partial Differential Equations '''24''' (1999), 1167-1193.<br />
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[[Category:Bibliography]]</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=Bibliography&diff=6996Bibliography2009-07-17T11:03:15Z<p>Stanford88: /* BaCh-2003 */</p>
<hr />
<div>__NOTOC__<br />
<br />
=== Important note ===<br />
<br />
This bibliography was ported over from the previous incarnation of these pages. It should now be<br />
considered obsolete. Instead, we are now [[Help:References|giving each reference its own page]].<br />
<br />
The naming system for these articles was somewhat implicit and ''ad hoc''; we are currently forming [[Help:Bibliographic naming conventions|more explicit standards]] for this.<br />
<br />
<br />
=== References ===<br />
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[[Category:Bibliography]]</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=Bibliography&diff=6995Bibliography2009-07-17T11:02:22Z<p>Stanford88: </p>
<hr />
<div>__NOTOC__<br />
<br />
=== Important note ===<br />
<br />
This bibliography was ported over from the previous incarnation of these pages. It should now be<br />
considered obsolete. Instead, we are now [[Help:References|giving each reference its own page]].<br />
<br />
The naming system for these articles was somewhat implicit and ''ad hoc''; we are currently forming [[Help:Bibliographic naming conventions|more explicit standards]] for this.<br />
<br />
<br />
=== References ===<br />
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[[Category:Bibliography]]</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=BaCh-p&diff=6994BaCh-p2009-07-17T10:58:05Z<p>Stanford88: BaCh-p moved to BaCh 2003</p>
<hr />
<div>#REDIRECT [[BaCh 2003]]</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=BaCh_2003&diff=6993BaCh 20032009-07-17T10:58:05Z<p>Stanford88: BaCh-p moved to BaCh 2003</p>
<hr />
<div>{{Bibliography<br />
| author = Bahouri, Hajer and Chemin, Jean-Yves<br />
| title = Microlocal analysis, bilinear estimates and cubic quasilinear wave equation<br />
| NOTE = Autour de l'analyse microlocale<br />
| journal = Ast\'erisque<br />
| Fjournal = Ast\'erisque<br />
| NUMBER = 284<br />
| year = 2003<br />
| jpage = 93--141<br />
| ISSN = 0303-1179<br />
| MRCLASS = 35L70 (35A27 35B30 47G30)<br />
| arxivlink = <br />
| jvol = <br />
| mathsciid = MR2003418<br />
| MRREVIEWER = Fran\ccois Castella <br />
}}</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=Courses_and_lecture_notes&diff=6992Courses and lecture notes2009-07-17T04:39:06Z<p>Stanford88: </p>
<hr />
<div>* [http://lagrange.uark.edu/~capogna/conf2002.html Open problems in Harmonic Analysis]<br />
* [http://www.research.att.com/~agilbert/papers.html Harmonic analysis, wavelets, and applications] (Daubechies/Gilbert)<br />
* [http://www.math.ias.edu/~dfoschi/nlwe.html Lectures in Harmonic Analysis and Nonlinear Wave Equations] (Foschi/Klainerman)<br />
* [http://www.ma.utexas.edu/users/gilbert/ M391C: Wavelets: Theory and Applications] (Gilbert)<br />
* [http://www.dpmms.cam.ac.uk/~wtg10/mathsindex.html Mathematical discussions] (Gowers)<br />
* [http://www.dpmms.cam.ac.uk/~bjg23/expos.html Expositions on Kakeya and arithmetic progressions] (Green)<br />
* [http://www.dpmms.cam.ac.uk/~bjg23/rkp.html Restriction and Kakeya Phenomena] (Green)<br />
* [http://www.math.missouri.edu/~iosevich/expositorypapers.html Kakeya Lectures, and other Expository papers] (Iosevich)<br />
* [http://www.math.princeton.edu/~seri/ Notes on Nonlinear Wave Equations] (Klainerman)<br />
* [http://www.math.princeton.edu/~seri/homepage/papers/FinalPro.pdf/ The Evolution Problem in General Relativity] (Klainerman)<br />
* [http://www.math.ubc.ca/~ilaba/kakeya.html The Kakeya problem and connections to harmonic analysis] (Laba)<br />
* [http://www.math.ubc.ca/~ilaba/tiling.html Tilings of R^n and the spectral set conjecture] (Laba)<br />
* [http://www.math.ubc.ca/~ilaba/wolff/ Thomas Wolff’s Lectures on Harmonic Analysis] (Laba/Wolff)<br />
*[http://math.mit.edu/~rbm/18.156.html 18.156: Graduate Analysis Elliptic Regularity and Monopoloes] (Melrose)<br />
* [http://www-math.mit.edu/~rbm/18.157.2000.html 18.157: Introduction to microlocal analysis] (Melrose)<br />
* [http://www.math.ucla.edu/~tao/254a.1.99w Math 254A: Oscillatory integrals, Diagonal entries of matrices, Scattering for Schrodinger with potential] (Tao)<br />
* [http://www.math.ucla.edu/~tao/254b.1.99s Math 254B: Restriction theorems, Bochner-Riesz, Kakeya, etc.] (Tao)<br />
* [http://www.math.ucla.edu/~tao/254a.1.01w/ Math 254A: Harmonic Analysis in the phase plane] (Tao)<br />
* [http://www.math.ucla.edu/~tao/254a.1.03w Math 254A: Additive combinatorics] (Tao)<br />
* [http://www.math.ucsd.edu/~vanvu/262/index.html 262: Combinatorial Number Theory] (Vu)<br />
<br />
Further suggestions are welcome!<br />
<br />
<br />
<br />
Other collections of analysis notes can be found at [http://www.mth.kcl.ac.uk/MAO/Home.html Modern Analysis Online].</div>Stanford88https://dispersivewiki.org/DispersiveWiki/index.php?title=Courses_and_lecture_notes&diff=6991Courses and lecture notes2009-07-17T03:58:11Z<p>Stanford88: </p>
<hr />
<div>* [http://lagrange.uark.edu/~capogna/conf2002.html Open problems in Harmonic Analysis]<br />
* [http://www.research.att.com/~agilbert/papers.html Harmonic analysis, wavelets, and applications] (Daubechies/Gilbert)<br />
* [http://www.math.ias.edu/~dfoschi/nlwe.html Lectures in Harmonic Analysis and Nonlinear Wave Equations] (Foschi/Klainerman)<br />
* [http://www.ma.utexas.edu/users/gilbert/ M391C: Wavelets: Theory and Applications] (Gilbert)<br />
* [http://www.dpmms.cam.ac.uk/~wtg10/mathsindex.html Mathematical discussions] (Gowers)<br />
* [http://www.dpmms.cam.ac.uk/~bjg23/expos.html Expositions on Kakeya and arithmetic progressions] (Green)<br />
* [http://www.dpmms.cam.ac.uk/~bjg23/rkp.html Restriction and Kakeya Phenomena] (Green)<br />
* [http://www.math.missouri.edu/~iosevich/expositorypapers.html Kakeya Lectures, and other Expository papers] (Iosevich)<br />
* [http://www.math.princeton.edu/~seri/ Notes on Nonlinear Wave Equations] (Klainerman)<br />
* [http://www.math.princeton.edu/~seri/homepage/papers/FinalPro.pdf/ The Evolution Problem in General Relativity] (Klainerman)<br />
* [http://www.math.ubc.ca/~ilaba/kakeya.html The Kakeya problem and connections to harmonic analysis] (Laba)<br />
* [http://www.math.ubc.ca/~ilaba/tiling.html Tilings of R^n and the spectral set conjecture] (Laba)<br />
* [http://www.math.ubc.ca/~ilaba/wolff/ Thomas Wolff’s Lectures on Harmonic Analysis] (Laba/Wolff)<br />
* [http://www-math.mit.edu/~rbm/18.157.2000.html 18.157: Introduction to microlocal analysis] (Melrose)<br />
* [http://www.math.ucla.edu/~tao/254a.1.99w Math 254A: Oscillatory integrals, Diagonal entries of matrices, Scattering for Schrodinger with potential] (Tao)<br />
* [http://www.math.ucla.edu/~tao/254b.1.99s Math 254B: Restriction theorems, Bochner-Riesz, Kakeya, etc.] (Tao)<br />
* [http://www.math.ucla.edu/~tao/254a.1.01w/ Math 254A: Harmonic Analysis in the phase plane] (Tao)<br />
* [http://www.math.ucla.edu/~tao/254a.1.03w Math 254A: Additive combinatorics] (Tao)<br />
* [http://www.math.ucsd.edu/~vanvu/262/index.html 262: Combinatorial Number Theory] (Vu)<br />
<br />
Further suggestions are welcome!<br />
<br />
<br />
<br />
Other collections of analysis notes can be found at [http://www.mth.kcl.ac.uk/MAO/Home.html Modern Analysis Online].</div>Stanford88