https://dispersivewiki.org/DispersiveWiki/api.php?action=feedcontributions&user=Pblue&feedformat=atom DispersiveWiki - User contributions [en] 2020-01-28T07:24:26Z User contributions MediaWiki 1.33.1 https://dispersivewiki.org/DispersiveWiki/index.php?title=Maxwell-Klein-Gordon_equations&diff=7160 Maxwell-Klein-Gordon equations 2010-07-16T16:00:21Z <p>Pblue: </p> <hr /> <div>The '''Maxwell-Klein-Gordon equation''' is the special case of the [[YMH|Yang-Mills-Higgs equation]] when the Lie group G is just the circle U(1), and there is no potential energy term &lt;math&gt;V(\phi)&lt;/math&gt;. Thus A is now purely imaginary, and &lt;math&gt;\phi&lt;/math&gt; is complex.<br /> <br /> ==Overview==<br /> The Maxwell-Klein-Gordon equation is the [[Maxwell equation]] coupled with a massless [[Klein-Gordon equation]] (i.e. a [[free wave equation]]). If the scalar field &lt;math&gt;\phi&lt;/math&gt; is set to 0, the equation collapses to the linear [[Maxwell equations]], which are basically a vector-valued variant of the free wave equation.<br /> <br /> As with Yang-Mills, the three standard gauges are the [[Lorenz gauge]], the [[Coulomb gauge]], and the [[temporal gauge]]. The [http://en.wikipedia.org/wiki/Lorenz_gauge Lorenz gauge] is most natural from a co-ordinate free viewpoint, but is difficult to work with technically. In principle the temporal gauge is the easiest to work with, being local in space, but in practice the Coulomb gauge is preferred because the null form structure of Maxwell-Klein-Gordon is most apparent in this gauge.<br /> <br /> In the Coulomb gauge, MKG has the schematic form<br /> <br /> &lt;center&gt;&lt;math&gt;\Delta A_0 = O( \phi \phi_t ) + O( \Phi^3 )&lt;/math&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;math&gt; \Box A = \nabla^{-1} Q( \phi , \phi )&lt;/math&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;math&gt; \Box \phi = Q( \nabla^{-1} A, \phi ) + O( (A_0)_t \phi ) + O( A_0 \phi_t ) + O( \Phi^3 )&lt;/math&gt;&lt;/center&gt;<br /> <br /> where &lt;math&gt;O(\Phi^3)&lt;/math&gt; denotes terms that are cubic in &lt;math&gt;(A_0, A, \phi)&lt;/math&gt;. Unfortunately, the equation for the A_0 portion of the Coloumb gauge is elliptic, which generates some low frequency issues. However, if we ignore the A_0 terms and the cubic terms then we reduce to the model equation<br /> <br /> &lt;center&gt;&lt;math&gt;\Box A = \nabla^{-1} Q( \phi , \phi ) &lt;/math&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;math&gt;\Box \phi = Q( \nabla^{-1} A, \phi )&lt;/math&gt;&lt;/center&gt;<br /> <br /> which is slightly better than the corresponding model for Yang-Mills.<br /> <br /> MKG has the advantage over YM that the Coulomb gauge is easily constructed globally using Riesz transforms, so there are less technical issues involved with this gauge.<br /> <br /> ==Maxwell-Klein-Gordon on R==<br /> <br /> * Scaling is s_c = -1/2.<br /> * LWP can be shown in the temporal gauge for s&gt;1/2 by energy estimates. For s&lt;1/2 one begins to have difficulty interpreting the solution even in the distributional sense, but this might be avoidable, perhaps by a good choice of gauge. (The Coulomb gauge seems to have some technical difficulties however).<br /> * GWP is easy to show in the temporal gauge for s \geq 1 by energy methods and Hamiltonian conservation. Presumably one can improve the s \geq 1 constraint substantially.<br /> <br /> ==Maxwell-Klein-Gordon on R&lt;sup&gt;2&lt;/sup&gt;==<br /> <br /> * Scaling is s_c = 0.<br /> * Heuristically, one expects X^{s,\delta} methods to give LWP for s &gt; 1/4, but we do not know if this has been done rigorously.<br /> ** Strichartz estimates give s &gt; 1/2 ([[PoSi1993]]), while energy methods give s&gt;1.<br /> * GWP is known for smooth data in the temporal gauge ([[Mc1980]]).<br /> ** This should extend to s \geq 1 and probably below, but we do not know if this is in the literature.<br /> <br /> ==Maxwell-Klein-Gordon on R&lt;sup&gt;3&lt;/sup&gt;==<br /> <br /> * Scaling is s_c = 1/2.<br /> * LWP for s&gt;1/2 in the Coulomb Gauge [[MaStz-p]]<br /> ** For the model equation, LWP fails for s &lt; 3/4 [[MaStz-p]]. Thus the MKG result exploits additional structure in the MKG equation which is not present in the model equation.<br /> ** For s&gt;3/4 this was proven in the Coloumb gauge in [[Cu1999]].<br /> ** For s\geq1 this was proven in the Coulomb and Temporal gauges in [[KlMa1994]].<br /> ** For s&gt;1 this follows (in any of the three gauges) from Strichartz estimates [[PoSi1993]]<br /> ** For s&gt;3/2 this follows (in any of the three gauges) from energy estimates.<br /> ** There is a tentative conjecture that one in fact has ill-posedness in the energy class for the Lorenz gauge.<br /> ** The endpoint s=1/2 looks extremely difficult, even for the model equation. Perhaps things would be easier if one only had to deal with the null form &lt;math&gt;\nabla^{-1} Q( \phi , \phi )&lt;/math&gt;, as this is slightly smoother than &lt;math&gt; Q( \nabla^{-1}A, \phi )&lt;/math&gt;.<br /> * GWP for s&gt;7/8 in the Coloumb gauge [[KeTa-p]].<br /> ** For s\geq 1 this was proven in [[KlMa1997]].<br /> ** For smooth data this was proven in [[EaMc1982]].<br /> * For physical applications it is of interest to study MKG when the scalar field &lt;math&gt;\phi&lt;/math&gt; propagates with a strictly slower velocity than the electromagnetic field A. In this case one cannot exploit the null form estimates; nevertheless, the estimates are more favourable, mainly because the two light cones are now transverse. Indeed, one has GWP for s\geq1 in all three standard gauges [[Tg2000]]. The local and global theory for this equation may well be improvable.<br /> * In the nonrelativistic limit this equation converges to a Maxwell-Poisson system [[MasNa2003]]<br /> <br /> [[Category:Equations]]<br /> [[Category:Wave]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Maxwell-Klein-Gordon_equations&diff=7159 Maxwell-Klein-Gordon equations 2010-07-16T15:49:01Z <p>Pblue: </p> <hr /> <div>The '''Maxwell-Klein-Gordon equation''' is the special case of the [[YMH|Yang-Mills-Higgs equation]] when the Lie group G is just the circle U(1), and there is no potential energy term &lt;math&gt;V(\phi)&lt;/math&gt;. Thus A is now purely imaginary, and &lt;math&gt;\phi&lt;/math&gt; is complex.<br /> <br /> ==Overview==<br /> The Maxwell-Klein-Gordon equation is the [[Maxwell equation]] coupled with a massless [[Klein-Gordon equation]] (i.e. a [[free wave equation]]). If the scalar field &lt;math&gt;\phi&lt;/math&gt; is set to 0, the equation collapses to the linear [[Maxwell equations]], which are basically a vector-valued variant of the free wave equation.<br /> <br /> As with Yang-Mills, the three standard gauges are the [[Lorenz gauge]], the [[Coulomb gauge]], and the [[temporal gauge]]. The [http://en.wikipedia.org/wiki/Lorenz_gauge Lorenz gauge] is most natural from a co-ordinate free viewpoint, but is difficult to work with technically. In principle the temporal gauge is the easiest to work with, being local in space, but in practice the Coulomb gauge is preferred because the null form structure of Maxwell-Klein-Gordon is most apparent in this gauge.<br /> <br /> In the Coulomb gauge, MKG has the schematic form<br /> <br /> &lt;center&gt;&lt;math&gt;\Delta A_0 = O( \phi \phi_t ) + O( \Phi^3 )&lt;/math&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;math&gt; \Box A = \nabla^{-1} Q( \phi , \phi )&lt;/math&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;math&gt; \Box \phi = Q( \nabla^{-1} A, \phi ) + O( (A_0)_t \phi ) + O( A_0 \phi_t ) + O( \Phi^3 )&lt;/math&gt;&lt;/center&gt;<br /> <br /> where &lt;math&gt;O(\Phi^3)&lt;/math&gt; denotes terms that are cubic in &lt;math&gt;(A_0, A, \phi)&lt;/math&gt;. Unfortunately, the equation for the A_0 portion of the Coloumb gauge is elliptic, which generates some low frequency issues. However, if we ignore the A_0 terms and the cubic terms then we reduce to the model equation<br /> <br /> &lt;center&gt;&lt;math&gt;\Box A = \nabla^{-1} Q( \phi , \phi ) &lt;/math&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;math&gt;\Box \phi = Q( \nabla^{-1} A, \phi )&lt;/math&gt;&lt;/center&gt;<br /> <br /> which is slightly better than the corresponding model for Yang-Mills.<br /> <br /> MKG has the advantage over YM that the Coulomb gauge is easily constructed globally using Riesz transforms, so there are less technical issues involved with this gauge.<br /> <br /> ==Maxwell-Klein-Gordon on R==<br /> <br /> * Scaling is s_c = -1/2.<br /> * LWP can be shown in the temporal gauge for s&gt;1/2 by energy estimates. For s&lt;1/2 one begins to have difficulty interpreting the solution even in the distributional sense, but this might be avoidable, perhaps by a good choice of gauge. (The Coulomb gauge seems to have some technical difficulties however).<br /> * GWP is easy to show in the temporal gauge for s \geq 1 by energy methods and Hamiltonian conservation. Presumably one can improve the s \geq 1 constraint substantially.<br /> <br /> ==Maxwell-Klein-Gordon on R&lt;sup&gt;2&lt;/sup&gt;==<br /> <br /> * Scaling is s_c = 0.<br /> * Heuristically, one expects X^{s,\delta} methods to give LWP for s &gt; 1/4, but we do not know if this has been done rigorously.<br /> ** Strichartz estimates give s &gt; 1/2 ([[PoSi1993]]), while energy methods give s&gt;1.<br /> * GWP is known for smooth data in the temporal gauge ([[Mc1980]]).<br /> ** This should extend to s \geq 1 and probably below, but we do not know if this is in the literature.<br /> <br /> ==Maxwell-Klein-Gordon on R&lt;sup&gt;3&lt;/sup&gt;==<br /> <br /> * Scaling is s_c = 1/2.<br /> * LWP for s&gt;1/2 in the Coulomb Gauge [[MaStz-p]]<br /> ** For the model equation, certain estimates which would be useful for proving LWP fail for s &lt; 2/3 [[MaStz-p]]. Thus the MKG result exploits additional structure in the MKG equation which is not present in the model equation.<br /> ** For s&gt;3/4 this was proven in the Coloumb gauge in [[Cu1999]].<br /> ** For s\geq1 this was proven in the Coulomb and Temporal gauges in [[KlMa1994]].<br /> ** For s&gt;1 this follows (in any of the three gauges) from Strichartz estimates [[PoSi1993]]<br /> ** For s&gt;3/2 this follows (in any of the three gauges) from energy estimates.<br /> ** There is a tentative conjecture that one in fact has ill-posedness in the energy class for the Lorenz gauge.<br /> ** The endpoint s=1/2 looks extremely difficult, even for the model equation. Perhaps things would be easier if one only had to deal with the null form &lt;math&gt;\nabla^{-1} Q( \phi , \phi )&lt;/math&gt;, as this is slightly smoother than &lt;math&gt; Q( \nabla^{-1}A, \phi )&lt;/math&gt;.<br /> * GWP for s&gt;7/8 in the Coloumb gauge [[KeTa-p]].<br /> ** For s\geq 1 this was proven in [[KlMa1997]].<br /> ** For smooth data this was proven in [[EaMc1982]].<br /> * For physical applications it is of interest to study MKG when the scalar field &lt;math&gt;\phi&lt;/math&gt; propagates with a strictly slower velocity than the electromagnetic field A. In this case one cannot exploit the null form estimates; nevertheless, the estimates are more favourable, mainly because the two light cones are now transverse. Indeed, one has GWP for s\geq1 in all three standard gauges [[Tg2000]]. The local and global theory for this equation may well be improvable.<br /> * In the nonrelativistic limit this equation converges to a Maxwell-Poisson system [[MasNa2003]]<br /> <br /> [[Category:Equations]]<br /> [[Category:Wave]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Null_structure&diff=7147 Null structure 2010-06-02T17:19:49Z <p>Pblue: </p> <hr /> <div>A semilinear equation is said to have ''null structure'' if the [[resonant]] component of the nonlinearity vanishes; in other words, plane waves obeying the [[dispersion relation]] cannot interact via the nonlinearity to generate forcing terms which also obey the dispersion relation. This generally requires the nonlinearity to be a linear combination of a certain special set of nonlinear forms, known as '''null forms'''. The term is primarily used for [[wave equations|non-linear wave equations]], but also applies to a number of other nonlinear dispersive equations.<br /> <br /> Quasilinear equations can also exhibit null structure, although the precise definition of this concept in this case is still not fully understood. For instance, the [[Einstein equations]] do not exhibit classical null structure, because the (resonant) self-interaction of plane waves is non-trivial, but nevertheless enjoys a kind of &quot;nilpotent&quot; null structure which can achieve a similar effect as classical null structure.<br /> <br /> == Null forms in wave equations ==<br /> <br /> The non-linear expressions which occur in [[wave equations|non-linear wave equations]] often have null form structure. Roughly speaking, this means that travelling waves &lt;math&gt;exp(i (k.x +- |k|t))&lt;/math&gt; do not self-interact, or only self-interact very weakly. When one has a null form present, the local and global well-posedness theory often improves substantially. There are several reasons for this. One is that null forms behave better under conformal compactification. Another is that null forms often have a nice representation in terms of conformal Killing vector fields. Finally, bilinear null forms enjoy much better estimates than other bilinear forms, as the interactions of parallel frequencies (which would normally be the worst case) is now zero.<br /> <br /> The standard bilinear null forms are<br /> <br /> &lt;center&gt;&lt;math&gt;Q_0(\phi,\psi) := \partial^\alpha \phi \partial_\alpha \psi = - \phi_t \psi_t + \nabla \phi \cdot \nabla \psi&lt;/math&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;math&gt;Q_{0i}(\phi,\psi) := \phi_t \psi_i - \phi_i \psi_t&lt;/math&gt;&lt;/center&gt;<br /> &lt;center&gt;&lt;math&gt;Q_{ij}(\phi,\psi) := \phi_i \psi_j - \phi_j \psi_i.&lt;/math&gt;&lt;/center&gt;<br /> The only bilinear forms in first derivatives of scalar solutions to the wave equation which have null structure are linear combinations of the above forms. However, more null forms are possible if one has trilinear or higher nonlinearities, if higher derivatives of solutions are permitted, or if one is considering vector or tensor equations (such as [[Maxwell equations|Maxwell]] or [[Dirac equations|Dirac]] type equations) which obey additional compatibility conditions.<br /> <br /> The presence of null structure seems to be related to the covariance of the underlying equation or Lagrangian, although the exact connection is not well understood.<br /> <br /> == Equations with null structure ==<br /> <br /> * [[Dirac-Klein-Gordon equation]]<br /> * [[Einstein equation]]<br /> * [[Maxwell-Dirac equation]]<br /> * [[Maxwell-Klein-Gordon system]]<br /> * [[Korteweg-de Vries equation]] (and more general [[KdV-type equations]])<br /> * [[nonlinear Dirac equation]]<br /> * [[Schrodinger maps]]<br /> * [[Wave maps]]<br /> * [[Yang-Mills equations]]<br /> <br /> [[Category:wave]]<br /> [[Category:concept]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Wave_estimates&diff=7068 Wave estimates 2009-12-11T16:37:23Z <p>Pblue: </p> <hr /> <div>Solutions to the linear wave equation and its perturbations are either estimated in mixed space-time norms &lt;math&gt;L^q_t L^r_x&lt;/math&gt;, or in &lt;math&gt;X^{s,b}_{}&lt;/math&gt; spaces, defined by<br /> <br /> &lt;center&gt;&lt;math&gt;\| u \|_{X^{s,b}} = \| &lt;\xi&gt;^s &lt;|\xi| - |\tau|&gt;^b \hat{u} ( \tau, \xi )\|_2 &lt;/math&gt;&lt;/center&gt;<br /> <br /> Linear space-time estimates are known as [[Strichartz estimates]]. They are especially useful for the [[NLW|semilinear NLW without derivatives]], and also have applications to other non-linearities, although the results obtained are often non-optimal (Strichartz estimates do not exploit any null structure of the equation). The &lt;math&gt;X^{s,b}_{}&lt;/math&gt; spaces are used primarily for [[bilinear wave estimates|bilinear estimates]], although more recently [[multilinear wave estimates|multilinear estimates have begun to appear]]. These spaces first appear in one-dimension in [[RaRe1982]] and in higher dimensions in [[Be1983]] in the context of [[propagation of singularities]]; they were used implicitly for LWP in [[KlMa1993]], while the Schrodinger and KdV analogues were developed in [[Bo1993]], [[Bo1993b]].<br /> <br /> == Specific wave estimates ==<br /> <br /> * [[Linear wave estimates]] ([[Strichartz estimates|Strichartz]], etc.)<br /> * [[Bilinear wave estimates]]<br /> * [[Multilinear wave estimates]]<br /> <br /> [[Category:wave]]<br /> [[Category:Estimates]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Conferences&diff=5627 Conferences 2009-02-12T17:55:25Z <p>Pblue: /* Analysis &amp; PDE conferences */</p> <hr /> <div>== Analysis &amp; PDE conferences ==<br /> <br /> * Feb 19-20 2009, U. Maryland: [http://www.norbertwiener.umd.edu/FFT/FFT09/ February Fourier talks] <br /> * Mar 9-13, 2009 CRM, Barcleona [http://www.crm.cat/harmonicpde/ Harmonic Analysis and PDE: Fluid Mechanics and Kato's Problem]<br /> * Mar 30-Apr 4 2009, U. Edinburgh: [http://www.maths.ed.ac.uk/~wright/Ionescu/ LMS Invited Series: Black holes in vacuum: examples and uniqueness properties, Main Speaker Alexandru Ionescu (Madison, WI)]<br /> * Apr 16-18 2009, U. Arkansas: [http://comp.uark.edu/~lcapogna/SLS2009/Welcome.html &quot;New Results on the Discrepancy Function, and Related Results&quot;, University of Arkansas Spring Lecture Series, Main Speaker Michael Lacey (Georgia Tech)] <br /> * Apr 23-25 2009, New Mexico: [http://www.math.unm.edu/conferences/12thAnalysis/index.html Twelfth New Mexico Analysis Seminar] <br /> * Apr 27-30 2009, Paris: [http://www.u-cergy.fr/rech/pages/duyckaerts/trimestreEn Nonlinear waves and dispersion trimester. Conference 1.]<br /> * May 4-8, CRM, Barcleona [http://www.crm.cat/multilinearharmonic/ Multilinear Harmonic Analysis and Weights]<br /> * May 18-30 2009, Italy: [http://www.dma.unina.it/hamiltonianPDE Course in Hamiltonian PDE]<br /> * May 26-29 2009, Purdue U.: [http://www.math.purdue.edu/~danielli/symposium09/Home.html 4th Symposium on Analysis and PDEs]<br /> * June 3-12, CRM, Barcelona [http://www.crm.cat/acmappings/ Quasiconformal Mappings, PDE and Geometric measure Theory]<br /> * June 15-18, CRM, Barcelona [http://www.crm.cat/charmonic/ Harmonic Analysis, Geometric Measure]<br /> * June 22-26 2009, Paris: [http://www.u-cergy.fr/rech/pages/duyckaerts/trimestreEn Nonlinear waves and dispersion trimester. Conference 2.]<br /> * Jun 30-Jul 4 2009, Lille: [http://math.univ-lille1.fr/~ramare/AAA/annonce.html Activites Additives et Analytiques ]<br /> * Aug 25-29 2009, Stockholm: [http://www.math.kth.se/ag08/ Geometry &amp; Analysis]<br /> * Nov 7-8 2009, Kansas: [http://www.math.ku.edu/conferences/prairie/prairie08/ Prairie Analysis]<br /> <br /> Other conference links:<br /> <br /> * [http://www.ams.org/meetings/ AMS meetings]<br /> * [http://www.ams.org/mathcal/ AMS conference calendar]<br /> * [http://www.math.ucla.edu/~tao/harmonic/conferences.html An old list of conferences in Analysis &amp; PDE]<br /> <br /> == Programs in analysis &amp; PDE ==<br /> <br /> * Jul-Aug 2009, BIRS: [http://www.math.ubc.ca/~gustaf/SummerSchool09/index.html Summer school in PDE]<br /> * Apr 1-Jun 6 2009, U. Napoli: [http://www.dma.unina.it/hamiltonianPDE Dynamical systems and Hamiltonian PDEs] <br /> * Apr 20-Jul 10 2009, IHP: [http://www.ihp.jussieu.fr/ceb/Trimestres/T09-3/index.html Dispersive equations and nonlinear waves and dispersion]<br /> <br /> == Research groups in analysis &amp; PDE ==<br /> <br /> * [http://www.math.chalmers.se/Math/Research/HarmonicAnalysis/ HAPDE group at Chalmers University of Technology]<br /> * [http://tyche.mat.univie.ac.at/ NuHAG group at University of Vienna]<br /> * [http://www.math.mcgill.ca/jakobson/analysish/analysis.html Analysis in Quebec] <br /> * [http://www.math.uiowa.edu/~jorgen/waveletFRG.html Wavelets, Frames, Operator Theory FRG]<br /> * [http://www.math.ucla.edu/%7Eanalysis Analysis group at UCLA]<br /> <br /> == Allied fields ==<br /> <br /> * [http://www.wavelet.org/ The Wavelet Digest]<br /> * [http://www.cosy.sbg.ac.at/~uhl/wav.html Salzburg wavelet directory]<br /> * [http://people.bath.ac.uk/masfeb/harmonic.html Harmonic Maps bibliography] <br /> * [http://math.nist.gov/opsf/ SIAM: Orthogonal Polynomials and Special Functions]<br /> * [http://finmath.com/ Financial Mathematics] <br /> * [http://relativity.livingreviews.org/ Living Reviews in Relativity] <br /> * [http://www.mathematik.uni-erlangen.de/at-net/ Approximation theory network] <br /> * [http://www.stolaf.edu/people/analysis/ Real Analysis exchange]<br /> * [http://www.math.okstate.edu/~alspach/banach/ Banach space Bulletin Board]<br /> * [http://www.math.tamu.edu/research/workshops/linanalysis/problems.html Open problems in Linear Analysis &amp; Probability] <br /> * [http://www.math.kyoto-u.ac.jp/complex/index_E.html Complex Analysis in Japan]<br /> * [http://groups.yahoo.com/group/harmonicanalysis/ Harmonic Analysis mailing list]<br /> <br /> == Other links ==<br /> <br /> * [[Courses and lecture notes]] in analysis and PDE<br /> * [http://tosio.math.toronto.edu/wiki/index.php/Main_Page DispersiveWiki]<br /> * [http://www.math.ucla.edu/~tao/harmonic An old list of links in harmonic analysis and PDE]<br /> <br /> == Old conferences ==<br /> <br /> * Feb 7-8 2009, UCLA: [http://www.math.ucla.edu/~tao/scapde.html SCAPDE] <br /> <br /> Contributions, corrections, and updates are of course are very welcome.</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Semilinear&diff=5254 Semilinear 2007-12-12T15:41:44Z <p>Pblue: Clarified antecedent of &quot;these equations&quot;</p> <hr /> <div>{{stub}}<br /> <br /> A '''semilinear''' equation is a PDE of the form<br /> &lt;center&gt;&lt;math&gt;L u = F(u)&lt;/math&gt;&lt;/center&gt;<br /> where ''L'' is a linear operator and ''F'' is a nonlinear operator which does not involve any derivatives of ''u''.<br /> <br /> A '''semilinear-with-derivatives''' equation is a PDE of the form<br /> &lt;center&gt;&lt;math&gt;L u = F(u, Du, \ldots, D^k u)&lt;/math&gt;&lt;/center&gt;<br /> where ''L'' is a linear operator, ''F'' is a nonlinear function of the first few derivatives &lt;math&gt;u, Du, \ldots, D^k u&lt;/math&gt;, with<br /> ''k'' strictly less than the order of ''L''. Semilinear-with-derivatives equations are more nonlinear than semilinear equations, but are less nonlinear than [[quasilinear]] or [[fully nonlinear]] equations.<br /> <br /> [[Category:concept]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Quasilinear&diff=5253 Quasilinear 2007-12-12T15:35:43Z <p>Pblue: typo, QNLW link</p> <hr /> <div>{{stub}}<br /> <br /> A '''quasilinear equation''' is an equation of the form<br /> &lt;center&gt;&lt;math&gt;F( u, Du, \ldots, D^k u ) = 0&lt;/math&gt;&lt;/center&gt;<br /> which is linear (and nontrivial) in the top order terms &lt;math&gt;D^k u&lt;/math&gt;. Thus a quasilinear equation takes the schematic form<br /> &lt;center&gt;&lt;math&gt;F( u, Du, \ldots, D^{k-1} u) D^k u = G( u, Du, \ldots, D^{k-1} u ).&lt;/math&gt;&lt;/center&gt;<br /> By differentiating this equation up to &lt;math&gt;k-1&lt;/math&gt; times and working with the system of fields &lt;math&gt;v := (u, Du, \ldots, D^k u)&lt;/math&gt;,<br /> one can place such equations in the slightly simpler form<br /> &lt;center&gt;&lt;math&gt;F( v) D^k v = G( v, Dv, \ldots, D^{k-1} v )&lt;/math&gt;&lt;/center&gt;<br /> though this trick comes at the cost of lowering the regularity of the fields.<br /> <br /> Quasilinear equations are more nonlinear than [[semilinear]] ones, but less nonlinear than [[fully nonlinear]] equations.<br /> <br /> ==See also==<br /> [[QNLW|Quasilinear nonlinear wave (QNLW)]]<br /> <br /> [[Category:Concept]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=BaCh1999&diff=5146 BaCh1999 2007-07-06T15:53:37Z <p>Pblue: </p> <hr /> <div>H. Bahouri, J-Y. Chemin, ''Équations d'ondes quasilinéaires et les inegalites de Strichartz'', Amer. J. Math.'''121''' (1999), 1337-1377.<br /> <br /> [[Category:Bibliography]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=BaCh1999&diff=5145 BaCh1999 2007-07-06T15:50:37Z <p>Pblue: BibBot test</p> <hr /> <div></div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=BaCh1999&diff=5144 BaCh1999 2007-07-06T15:49:21Z <p>Pblue: </p> <hr /> <div>H. Bahouri, J-Y. Chemin, ''Équations d'ondes quasilinéaires et les inegalites de Strichartz'', Amer. J. Math.'''121''' (1999), 1337-1377.<br /> <br /> [[Category:Bibliography]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=BaCh1999b&diff=5143 BaCh1999b 2007-07-06T15:48:59Z <p>Pblue: </p> <hr /> <div>H. Bahouri, J-Y. Chemin, ''Équations d'ondes quasilinéaires et effet dispersif'', IMRN '''21''' (1999), 1141-1178.<br /> <br /> [[Category:Bibliography]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=BaCh2002&diff=5142 BaCh2002 2007-07-06T15:48:20Z <p>Pblue: </p> <hr /> <div>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 /> [[Category:Bibliography]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=BaGd1997&diff=5141 BaGd1997 2007-07-06T15:47:50Z <p>Pblue: </p> <hr /> <div>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 /> [[Category:Bibliography]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=BaGd1997&diff=5140 BaGd1997 2007-07-06T15:40:14Z <p>Pblue: BibBot test</p> <hr /> <div></div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=BaCh2002&diff=5139 BaCh2002 2007-07-06T15:39:57Z <p>Pblue: BibBot test</p> <hr /> <div></div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=BaCh1999b&diff=5138 BaCh1999b 2007-07-06T15:39:40Z <p>Pblue: BibBot test</p> <hr /> <div></div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=BaCh1999&diff=5137 BaCh1999 2007-07-06T15:39:21Z <p>Pblue: BibBot test</p> <hr /> <div></div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=BaCh_2003&diff=5136 BaCh 2003 2007-07-06T15:39:04Z <p>Pblue: BibBot test</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> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Wave_maps&diff=5101 Wave maps 2007-06-16T16:18:28Z <p>Pblue: H^{n/2} -&gt; H^{d/2}</p> <hr /> <div>{{equation<br /> | name = Wave maps <br /> | equation = &lt;math&gt;(\phi^* \nabla)^\alpha \partial_\alpha \phi = 0&lt;/math&gt;<br /> | fields = &lt;math&gt;\phi: \R^{1+d} \to \mathfrak{g}&lt;/math&gt;<br /> | data = &lt;math&gt;\phi \in H^s \times H^{s-1}(\R^d \to TM)&lt;/math&gt;<br /> | hamiltonian = [[Hamiltonian]] ([[completely integrable]] when d=1)<br /> | linear = [[free wave equation|wave]]<br /> | nonlinear = [[semilinear|semilinear with derivatives]]<br /> | critical = &lt;math&gt;\dot H^{d/2}(\R^d)&lt;/math&gt;<br /> | criticality = energy critical for d=2<br /> | covariance = [[Lorentzian]], diffeomorphism of target<br /> | lwp = varies | gwp = varies<br /> | parent = [[DDNLW]]<br /> | special = Wave maps [[wave maps on R|on R]], [[wave maps on R2|on R^2]]<br /> | related = [[Einstein equations]]<br /> }}<br /> <br /> '''Wave maps''' are maps &lt;math&gt;\phi\,&lt;/math&gt; from &lt;math&gt;R^{d+1}&lt;/math&gt; to a Riemannian manifold &lt;math&gt;M&lt;/math&gt; which are critical points of the Lagrangian<br /> <br /> &lt;center&gt;&lt;math&gt;\int \phi_\alpha \cdot \phi^\alpha dx dt.&lt;/math&gt;&lt;/center&gt;<br /> <br /> When M is flat, wave maps just obey the [[free wave equation]] (if viewed in flat co-ordinates). More generally, they obey the equation<br /> <br /> &lt;center&gt;&lt;math&gt;\Box \phi = G ( \phi ) Q_0( \phi , \phi )&lt;/math&gt;&lt;/center&gt;<br /> <br /> where &lt;math&gt;G( \phi )&lt;/math&gt; is the second fundamental form and &lt;math&gt;Q_0\,&lt;/math&gt; is the standard [[null form]]. When the target manifold is a unit sphere, this simplifies to<br /> <br /> &lt;center&gt;&lt;math&gt;\Box \phi = - \phi Q_0( \phi , \phi ) &lt;/math&gt;&lt;/center&gt;<br /> <br /> where &lt;math&gt;\phi\,&lt;/math&gt; is viewed in Cartesian co-ordinates (and must therefore obey &lt;math&gt;| \phi |=1\,&lt;/math&gt; at all positions and times in order to stay on the sphere). The sphere case has special algebraic structure (beyond that of other symmetric spaces) while also staying compact, and so the sphere is usually considered the easiest case to study. Some additional simplifications arise if the target is a Riemann surface (because the connection group becomes &lt;math&gt;U(1)\,&lt;/math&gt;, which is abelian); thus &lt;math&gt;S^2\,&lt;/math&gt; is a particularly simple case.<br /> <br /> This equation is highly geometrical, and can be rewritten in many different ways. It is also related to the [[Einstein equations]] (if one assumes various symmetry assumptions on the metric); see e.g. [[BgCcMc1995]]).<br /> <br /> The critical regularity is &lt;math&gt;s_c = d/2\,.&lt;/math&gt; Thus the two-dimensional case is especially interesting, as the equation is then energy-critical. The sub-critical theory &lt;math&gt;s &gt; d/2\,&lt;/math&gt; is fairly well understood, but the &lt;math&gt;s_c = d/2\,&lt;/math&gt; theory is quite delicate. A big problem is that &lt;math&gt;H^{d/2}\,&lt;/math&gt; does not control &lt;math&gt;L^\infty\,&lt;/math&gt;, so one cannot localize to a small co-ordinate patch (or perform algebraic operations properly).<br /> <br /> The positive and negative curvature cases are suspected to behave differently, especially at the critical regularity. Intuitively, the negative curvature space spreads the solution out more, thus giving a better chance for LWP and GWP.More recently, distinctions have arisen between the boundedly parallelizable case (where the exists an orthonormal frame whose structure constants and derivatives are bounded), and the isometrically embeddable case.For instance, hyperbolic space is in the former category but not in the latter; smooth compact manifolds such as the sphere are in both.<br /> <br /> The general LWP/GWP theory (except for the special [[wave maps on R|one-dimensional]] and [[wave maps on R2|two-dimensional]] cases) is as follows.<br /> <br /> * For &lt;math&gt;d\geq 2&lt;/math&gt; one has LWP in &lt;math&gt;H^{d/2}\,&lt;/math&gt;, and GWP and regularity for small data, if the manifold can be isometrically embedded in Euclidean space [[Tt-p2]]<br /> ** Earlier global regularity results in &lt;math&gt;H^{d/2}\,&lt;/math&gt; are as follows.<br /> *** For a sphere in &lt;math&gt;d\ge 5\,&lt;/math&gt;, see [[Ta2001c]]; for a sphere in &lt;math&gt;d \ge 2\,&lt;/math&gt;, see [[Ta2001d]].<br /> *** The &lt;math&gt;d \ge 5\,&lt;/math&gt; has been generalized to arbitrary manifolds which are boundedly parallelizable [[KlRo-p]].<br /> *** This has been extended to &lt;math&gt;d=4\,&lt;/math&gt; by [[SaSw2001]] and [[NdStvUh2003b]]. In the constant curvature case one also has global well-posedness for small data in &lt;math&gt;H^{d/2}\,&lt;/math&gt; [[NdStvUh2003b]]. This can be extended to manifolds with bounded second fundamental form [[SaSw2001]].<br /> *** This has been extended to &lt;math&gt;d=3\,&lt;/math&gt; when the target is a Riemann surface [[Kri2003]], and to &lt;math&gt;d=2\,&lt;/math&gt; for hyperbolic space [[Kri-p]]<br /> ** For the critical Besov space &lt;math&gt;B^{d/2,1}_2\,&lt;/math&gt; this is in [[Tt1998]] when d \ge 4 and [[Tt2001b]] when &lt;math&gt;d\ge 2\,&lt;/math&gt;. (See also [[Na1999]] in the case when the wave map lies on a geodesic). For small data one also has GWP and scattering.<br /> ** In the sub-critical spaces &lt;math&gt;H^s, s &gt; d/2\,&lt;/math&gt; this was shown in [[KlMa1995b]] for the &lt;math&gt;d\ge4\,&lt;/math&gt; case and in [[KlSb1997]] for &lt;math&gt;d\ge 2\,&lt;/math&gt;.<br /> *** For the model wave map equation this was shown for &lt;math&gt;d\ge 3\,&lt;/math&gt; in [[KlMa1997b]].<br /> ** If one replaces the critical Besov space by &lt;math&gt;H^{d/2}\,&lt;/math&gt; then one has failure of analytic or &lt;math&gt;C^2\,&lt;/math&gt; local well-posedness for &lt;math&gt;d\ge 3\,&lt;/math&gt; [DanGe-p], and one has failure of continuous local well-posedness for &lt;math&gt;d=1\,&lt;/math&gt; [[Na1999]], [[Ta2000]]<br /> ** GWP is also known for smooth data close to a geodesic [[Si1989]]. For smooth data close to a point this was in [[Cq1987]].<br /> * For &lt;math&gt;d \ge 3\,&lt;/math&gt; singularities can form from large data, even when the data is smooth and rotationally symmetric [[CaSaTv1998]]<br /> ** For &lt;math&gt;d=3\,&lt;/math&gt; this was proven in [[Sa1988]]<br /> ** For &lt;math&gt;d\ge 7\,&lt;/math&gt; one can have singularities even when the target has negative curvature [[CaSaTv1998]]<br /> ** For &lt;math&gt;d=3\,&lt;/math&gt;, numerics suggest that there is a transition between global existence for small data and blowup for large data, with the self-similar blowup solution being an intermediate attractor [[Lie-p]]<br /> <br /> === Special cases ===<br /> <br /> * [[wave maps on R|one-dimensional wave maps]]<br /> * [[wave maps on R2|two-dimensional wave maps]]<br /> <br /> === Further reading ===<br /> <br /> Surveys of wave maps can be found in [[Sw1997]], [[SaSw1998]], [[KlSb-p]].<br /> <br /> [[Category:Equations]]<br /> <br /> [[Category:Wave]]<br /> <br /> [[Category:Geometry]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=AnMc-p&diff=4544 AnMc-p 2007-02-09T22:48:39Z <p>Pblue: </p> <hr /> <div>{{Bibliography<br /> | author = Andersson, Lars and Moncrief, Vincent<br /> | title = Elliptic-hyperbolic systems and the Einstein equations<br /> | journal = Ann. Henri Poincar\'e<br /> | Fjournal = Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics<br /> | jvol = 4<br /> | year = 2003<br /> | NUMBER = 1<br /> | jpage = 1--34<br /> | ISSN = 1424-0637<br /> | MRCLASS = 58J45 (35Q75 83C05)<br /> | arxivlink = http://www.arxiv.org/abs/gr-qc/0110111<br /> | mathsciid = MR1967177<br /> | MRREVIEWER = Alan D. Rendall <br /> }}</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Template:Bibliography&diff=4543 Template:Bibliography 2007-02-09T22:37:17Z <p>Pblue: Added NUMBER field and blank defaults</p> <hr /> <div>{{{author|&lt;noinclude&gt;&lt;nowiki&gt;{{{author}}}&lt;/nowiki&gt;&lt;/noinclude&gt;}}},<br /> ''{{{title|&lt;noinclude&gt;&lt;nowiki&gt;{{{title}}}&lt;/nowiki&gt;&lt;/noinclude&gt;}}}'',<br /> {{{journal|&lt;noinclude&gt;&lt;nowiki&gt;{{{journal}}}&lt;/nowiki&gt;&lt;/noinclude&gt;}}},<br /> '''{{{jvol|&lt;noinclude&gt;&lt;nowiki&gt;{{{jvol}}}&lt;/nowiki&gt;&lt;/noinclude&gt;}}}''',<br /> ({{{year|&lt;noinclude&gt;&lt;nowiki&gt;{{{year}}}&lt;/nowiki&gt;&lt;/noinclude&gt;}}}), <br /> {{{NUMBER|&lt;noinclude&gt;&lt;nowiki&gt;{{{NUMBER}}}&lt;/nowiki&gt;&lt;/noinclude&gt;}}}, <br /> {{{jpage|&lt;noinclude&gt;&lt;nowiki&gt;{{{jpage}}}&lt;/nowiki&gt;&lt;/noinclude&gt;}}}<br /> <br /> ArXiv: [http://front.math.ucdavis.edu/{{{arxivlink}}} {{{arxivlink}}}]<br /> <br /> MathSciNet: [http://www.ams.org/mathscinet-getitem?mr={{{mathsciid}}} {{{mathsciid}}}]<br /> <br /> &lt;includeonly&gt;[[Category:Bibliography]]&lt;/includeonly&gt;<br /> &lt;noinclude&gt;<br /> == How to use this template ==<br /> <br /> Insert the text<br /> <br /> &lt;nowiki&gt;{{Bibliography &lt;/nowiki&gt;<br /> &lt;nowiki&gt; | author = &lt;list authors here&gt; &lt;/nowiki&gt;<br /> &lt;nowiki&gt; | title = &lt;list title here&gt; &lt;/nowiki&gt;<br /> &lt;nowiki&gt; | journal = &lt;list journal here, or &quot;preprint&quot;&gt; &lt;/nowiki&gt;<br /> &lt;nowiki&gt; | jvol = &lt;list journal volume here, or &quot;-&quot;&gt; &lt;/nowiki&gt;<br /> &lt;nowiki&gt; | year = &lt;list year of publication (or of preprint release)&gt; &lt;/nowiki&gt;<br /> &lt;nowiki&gt; | jpage = &lt;list journal pages, or &quot;-&quot;&gt; &lt;/nowiki&gt;<br /> &lt;nowiki&gt; | arxivlink = &lt;list arxiv number, or &quot;-&quot;&gt; &lt;/nowiki&gt;<br /> &lt;nowiki&gt; | mathsciid = &lt;list mathsci number, or &quot;-&quot;&gt; &lt;/nowiki&gt;<br /> &lt;nowiki&gt;}}&lt;/nowiki&gt;<br /> <br /> at the top of the page where a bibliography reference is desired. Linking, italicization, etc. will be done automatically, as will the adding of the page to the [[:Category:Bibliography|bibliography category]]. Note that the parameters here can themselves contain markup such as external web links or LaTeX math symbols.<br /> <br /> For an example of this template in action, see [[Ta2004]].<br /> <br /> [[Category:Template]]<br /> &lt;/noinclude&gt;</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=AxPgSac1997&diff=4542 AxPgSac1997 2007-02-09T20:47:56Z <p>Pblue: BibBot test</p> <hr /> <div>{{Bibliography<br /> | author = Alexander, J. C. and Pego, R. L. and Sachs, R. L.<br /> | title = On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation<br /> | journal = Phys. Lett. A<br /> | Fjournal = Physics Letters. A<br /> | jvol = 226<br /> | year = 1997<br /> | NUMBER = 3-4<br /> | jpage = 187--192<br /> | ISSN = 0375-9601<br /> | CODEN = PYLAAG<br /> | MRCLASS = 35Q53 (35B35)<br /> | arxivlink = <br /> | mathsciid = MR1435907<br /> }}</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=AsCoeWgg1996&diff=4541 AsCoeWgg1996 2007-02-09T20:47:40Z <p>Pblue: BibBot test</p> <hr /> <div>{{Bibliography<br /> | author = Ash, J. Marshall and Cohen, Jonathan and Wang, Gang<br /> | title = On strongly interacting internal solitary waves<br /> | journal = J. Fourier Anal. Appl.<br /> | Fjournal = The Journal of Fourier Analysis and Applications<br /> | jvol = 2<br /> | year = 1996<br /> | NUMBER = 5<br /> | jpage = 507--517<br /> | ISSN = 1069-5869<br /> | MRCLASS = 35Q53 (35Q35 76B15 76V05)<br /> | arxivlink = <br /> | mathsciid = MR1412066<br /> | MRREVIEWER = F. Pempinelli <br /> }}</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=ArYa2000&diff=4540 ArYa2000 2007-02-09T20:47:24Z <p>Pblue: BibBot test</p> <hr /> <div>{{Bibliography<br /> | author = Artbazar, Galtbayar and Yajima, Kenji<br /> | title = The $L\sp p$-continuity of wave operators for one dimensional Schr\&quot;odinger operators<br /> | journal = J. Math. Sci. Univ. Tokyo<br /> | Fjournal = The University of Tokyo. Journal of Mathematical Sciences<br /> | jvol = 7<br /> | year = 2000<br /> | NUMBER = 2<br /> | jpage = 221--240<br /> | ISSN = 1340-5705<br /> | MRCLASS = 34L25 (47A40 47E05 81U05)<br /> | arxivlink = <br /> | mathsciid = MR1768465<br /> | MRREVIEWER = Philippe Briet <br /> }}</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Ant2003&diff=4539 Ant2003 2007-02-09T20:47:02Z <p>Pblue: BibBot test</p> <hr /> <div>{{Bibliography<br /> | author = Antonini, Christophe<br /> | title = Lower bounds for the $L\sp 2$ minimal periodic blow-up solutions of critical nonlinear Schr\&quot;odinger equation<br /> | journal = Differential Integral Equations<br /> | Fjournal = Differential and Integral Equations. An International Journal for Theory \&amp; Applications<br /> | jvol = 15<br /> | year = 2002<br /> | NUMBER = 6<br /> | jpage = 749--768<br /> | ISSN = 0893-4983<br /> | MRCLASS = 35Q55 (35B10 35B45 35Q40)<br /> | arxivlink = <br /> | mathsciid = MR1893845<br /> | MRREVIEWER = Enrique Fern\'andez Cara <br /> }}</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=AnMc-p&diff=4538 AnMc-p 2007-02-09T20:46:46Z <p>Pblue: BibBot test</p> <hr /> <div>{{Bibliography<br /> | author = Andersson, Lars and Moncrief, Vincent<br /> | title = Elliptic-hyperbolic systems and the Einstein equations<br /> | journal = Ann. Henri Poincar\'e<br /> | Fjournal = Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics<br /> | jvol = 4<br /> | year = 2003<br /> | NUMBER = 1<br /> | jpage = 1--34<br /> | ISSN = 1424-0637<br /> | MRCLASS = 58J45 (35Q75 83C05)<br /> | arxivlink = <br /> | mathsciid = MR1967177<br /> | MRREVIEWER = Alan D. Rendall <br /> }}</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al2003&diff=4537 Al2003 2007-02-09T20:46:27Z <p>Pblue: BibBot test</p> <hr /> <div>{{Bibliography<br /> | author = Alinhac, Serge<br /> | title = An example of blowup at infinity for a 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 = 1--91<br /> | ISSN = 0303-1179<br /> | MRCLASS = 35L70 (35B40 35Q60 35S50)<br /> | arxivlink = <br /> | jvol = <br /> | mathsciid = MR2003417<br /> | MRREVIEWER = Varga Kalantarov <br /> }}</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al2001b&diff=4536 Al2001b 2007-02-09T20:45:30Z <p>Pblue: BibBot test</p> <hr /> <div>{{Bibliography<br /> | author = Alinhac, S.<br /> | title = The null condition for quasilinear wave equations in two space dimensions. II<br /> | journal = Amer. J. Math.<br /> | Fjournal = American Journal of Mathematics<br /> | jvol = 123<br /> | year = 2001<br /> | NUMBER = 6<br /> | jpage = 1071--1101<br /> | ISSN = 0002-9327<br /> | CODEN = AJMAAN<br /> | MRCLASS = 35L70 (35B40 35L67)<br /> | arxivlink = <br /> | mathsciid = MR1867312<br /> | MRREVIEWER = Michael Renardy <br /> }}</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al2000&diff=4535 Al2000 2007-02-09T20:44:14Z <p>Pblue: BibBot test</p> <hr /> <div>{{Bibliography<br /> | author = Alinhac, S.<br /> | title = Rank 2 singular solutions for quasilinear wave equations<br /> | journal = Internat. Math. Res. Notices<br /> | Fjournal = International Mathematics Research Notices<br /> | year = 2000<br /> | NUMBER = 18<br /> | jpage = 955--984<br /> | ISSN = 1073-7928<br /> | MRCLASS = 35L70 (35B40)<br /> | arxivlink = <br /> | jvol = <br /> | mathsciid = MR1792284<br /> | MRREVIEWER = Ralph Saxton <br /> }}</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al2000&diff=4534 Al2000 2007-02-09T20:43:49Z <p>Pblue: </p> <hr /> <div>S. Alinhac, Rank 2 singular solutions for quasilinear wave equations, IMRN 18 (2000), 955-984.<br /> <br /> [[Category:Bibliography]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al2000&diff=4533 Al2000 2007-02-09T20:32:15Z <p>Pblue: BibBot test</p> <hr /> <div>{{Bibliography<br /> | author = Alinhac, S.<br /> | title = Rank 2 singular solutions for quasilinear wave equations<br /> | journal = Internat. Math. Res. Notices<br /> | Fjournal = International Mathematics Research Notices<br /> | year = 2000<br /> | NUMBER = 18<br /> | jpage = 955--984<br /> | ISSN = 1073-7928<br /> | MRCLASS = 35L70 (35B40)<br /> | arxivlink = <br /> | mathsciid = MR1792284<br /> | MRREVIEWER = Ralph Saxton <br /> }}</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al2000&diff=4532 Al2000 2007-02-09T20:31:26Z <p>Pblue: </p> <hr /> <div>S. Alinhac, Rank 2 singular solutions for quasilinear wave equations, IMRN 18 (2000), 955-984.<br /> <br /> [[Category:Bibliography]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al2001&diff=4531 Al2001 2007-02-09T20:30:23Z <p>Pblue: BibBot test</p> <hr /> <div>{{Bibliography<br /> | author = Alinhac, S.<br /> | title = The null condition for quasilinear wave equations in two space dimensions I<br /> | journal = Invent. Math.<br /> | Fjournal = Inventiones Mathematicae<br /> | jvol = 145<br /> | year = 2001<br /> | NUMBER = 3<br /> | jpage = 597--618<br /> | ISSN = 0020-9910<br /> | CODEN = INVMBH<br /> | MRCLASS = 35L70 (35A05 35L80)<br /> | arxivlink = <br /> | mathsciid = MR1856402<br /> | MRREVIEWER = Michael Renardy <br /> }}</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al2000&diff=4530 Al2000 2007-02-09T20:30:07Z <p>Pblue: BibBot test</p> <hr /> <div>{{Bibliography<br /> | author = Alinhac, S.<br /> | title = Rank 2 singular solutions for quasilinear wave equations<br /> | journal = Internat. Math. Res. Notices<br /> | Fjournal = International Mathematics Research Notices<br /> | year = 2000<br /> | NUMBER = 18<br /> | jpage = 955--984<br /> | ISSN = 1073-7928<br /> | MRCLASS = 35L70 (35B40)<br /> | arxivlink = <br /> | mathsciid = MR1792284<br /> | MRREVIEWER = Ralph Saxton <br /> }}</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al2000&diff=4529 Al2000 2007-02-09T20:28:29Z <p>Pblue: </p> <hr /> <div>S. Alinhac, Rank 2 singular solutions for quasilinear wave equations, IMRN 18 (2000), 955-984.<br /> <br /> [[Category:Bibliography]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al2000&diff=4528 Al2000 2007-02-09T19:38:46Z <p>Pblue: BibBot test</p> <hr /> <div>{{Bibliography<br /> | author = Alinhac, S.<br /> | title = Rank 2 singular solutions for quasilinear wave equations<br /> | journal = Internat. Math. Res. Notices<br /> | Fjournal = International Mathematics Research Notices<br /> | year = 2000<br /> | NUMBER = 18<br /> | jpage = 955--984<br /> | ISSN = 1073-7928<br /> | MRCLASS = 35L70 (35B40)<br /> | arxivlink = <br /> | mathsciid = MR1792284<br /> | MRREVIEWER = Ralph Saxton <br /> }}</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al2000&diff=4488 Al2000 2007-01-19T22:25:51Z <p>Pblue: </p> <hr /> <div>S. Alinhac, Rank 2 singular solutions for quasilinear wave equations, IMRN 18 (2000), 955-984.<br /> <br /> [[Category:Bibliography]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al2000&diff=4487 Al2000 2007-01-19T22:24:44Z <p>Pblue: BibBot test</p> <hr /> <div>{{Bibliography<br /> | author = Alinhac, S.<br /> | title = Rank 2 singular solutions for quasilinear wave equations<br /> | journal = Internat. Math. Res. Notices<br /> | Fjournal = International Mathematics Research Notices<br /> | year = 2000<br /> | NUMBER = 18<br /> | jpage = 955--984<br /> | ISSN = 1073-7928<br /> | MRCLASS = 35L70 (35B40)<br /> | arxivlink = <br /> | mathsciid = MR1792284<br /> | MRREVIEWER = Ralph Saxton <br /> }}</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al1999b&diff=4486 Al1999b 2007-01-19T22:24:31Z <p>Pblue: BibBot test</p> <hr /> <div>{{Bibliography<br /> | author = Alinhac, Serge<br /> | title = Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions. II<br /> | journal = Acta Math.<br /> | Fjournal = Acta Mathematica<br /> | jvol = 182<br /> | year = 1999<br /> | NUMBER = 1<br /> | jpage = 1--23<br /> | ISSN = 0001-5962<br /> | CODEN = ACMAA8<br /> | MRCLASS = 35L70 (35B05)<br /> | arxivlink = <br /> | mathsciid = MR1687180<br /> | MRREVIEWER = Nickolai A. Larkin <br /> }}</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al2000&diff=4485 Al2000 2007-01-19T22:23:30Z <p>Pblue: </p> <hr /> <div>S. Alinhac, Rank 2 singular solutions for quasilinear wave equations, IMRN 18 (2000), 955-984.<br /> <br /> [[Category:Bibliography]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al1999b&diff=4484 Al1999b 2007-01-19T22:20:27Z <p>Pblue: </p> <hr /> <div>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 /> [[Category:Bibliography]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al2000&diff=4483 Al2000 2007-01-19T22:18:07Z <p>Pblue: BibBot test</p> <hr /> <div>{{Bibliography<br /> | author = Alinhac, S.<br /> | title = Rank 2 singular solutions for quasilinear wave equations<br /> | journal = Internat. Math. Res. Notices<br /> | Fjournal = International Mathematics Research Notices<br /> | year = 2000<br /> | NUMBER = 18<br /> | jpage = 955--984<br /> | ISSN = 1073-7928<br /> | MRCLASS = 35L70 (35B40)<br /> | arxivlink = <br /> | mathsciid = MR1792284<br /> | MRREVIEWER = Ralph Saxton <br /> }}</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al1999&diff=4482 Al1999 2007-01-19T22:16:27Z <p>Pblue: BibBot test</p> <hr /> <div>{{Bibliography<br /> | author = Alinhac, Serge<br /> | title = Blowup of small data solutions for a quasilinear wave equation in two space dimensions<br /> | journal = Ann. of Math. (2)<br /> | Fjournal = Annals of Mathematics. Second Series<br /> | jvol = 149<br /> | year = 1999<br /> | NUMBER = 1<br /> | jpage = 97--127<br /> | ISSN = 0003-486X<br /> | CODEN = ANMAAH<br /> | MRCLASS = 35L70 (35B05)<br /> | arxivlink = <br /> | mathsciid = MR1680539<br /> | MRREVIEWER = Nickolai A. Larkin <br /> }}</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al1999&diff=4361 Al1999 2006-10-02T17:44:58Z <p>Pblue: </p> <hr /> <div>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 /> [[Category:Bibliography]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al1999&diff=4360 Al1999 2006-10-02T17:32:06Z <p>Pblue: BibBot test</p> <hr /> <div><br /> @preamble{<br /> <br /> &quot;\ef\cprime{$'$} &quot;<br /> <br /> }<br /> <br /> @article {MR1680539,<br /> <br /> AUTHOR = {Alinhac, Serge},<br /> <br /> TITLE = {Blowup of small ata solutions for a quasilinear wave equation<br /> <br /> in two space imensions},<br /> <br /> JOURNAL = {Ann. of Math. (2)},<br /> <br /> FJOURNAL = {Annals of Mathematics. Secon Series},<br /> <br /> VOLUME = {149},<br /> <br /> YEAR = {1999},<br /> <br /> NUMBER = {1},<br /> <br /> PAGES = {97--127},<br /> <br /> ISSN = {0003-486X},<br /> <br /> CODEN = {ANMAAH},<br /> <br /> MRCLASS = {35L70 (35B05)},<br /> <br /> MRNUMBER = {MR1680539 (2000:35147)},<br /> <br /> MRREVIEWER = {Nickolai A. Lar{\cprime}kin},<br /> <br /> }</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al1999&diff=4359 Al1999 2006-10-02T16:57:21Z <p>Pblue: </p> <hr /> <div>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 /> [[Category:Bibliography]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al1999&diff=4358 Al1999 2006-10-02T16:56:17Z <p>Pblue: BibBot test</p> <hr /> <div>@preamble &quot;\def\cprime$'$ &quot; {{Bibliography<br /> | author = Alinhac, Serge<br /> | title = Blowup of small data solutions for a quasilinear wave equation in two space dimensions<br /> | journal = Ann. of Math. (2)<br /> | Fjournal = Annals of Mathematics. Second Series<br /> | jvol = 149<br /> | year = 1999<br /> | NUMBER = 1<br /> | jpage = 97--127<br /> | ISSN = 0003-486X<br /> | CODEN = ANMAAH<br /> | MRCLASS = 35L70 (35B05)<br /> | arxivlink = <br /> | mathsciid = MR1680539<br /> | MRREVIEWER = Nickolai A. Larkin <br /> }}</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al1999&diff=4357 Al1999 2006-10-02T16:55:24Z <p>Pblue: </p> <hr /> <div>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 /> [[Category:Bibliography]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al1995&diff=4356 Al1995 2006-10-02T16:52:03Z <p>Pblue: </p> <hr /> <div>S. Alinhac, ''Blowup for nonlinear hyperbolic equations'', Boston&lt;nowiki&gt;: Birkhauser, 1995, Progress in &lt;/nowiki&gt;Nonlinear DE and their Applications, 17.<br /> <br /> [[Category:Bibliography]]</div> Pblue https://dispersivewiki.org/DispersiveWiki/index.php?title=Al1999b&diff=4355 Al1999b 2006-10-02T16:45:20Z <p>Pblue: BibBot test</p> <hr /> <div>@preamble &quot;\def\cprime$'$ &quot; {{Bibliography<br /> | author = Alinhac, Serge<br /> | title = Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions. II<br /> | journal = Acta Math.<br /> | Fjournal = Acta Mathematica<br /> | jvol = 182<br /> | year = 1999<br /> | NUMBER = 1<br /> | jpage = 1--23<br /> | ISSN = 0001-5962<br /> | CODEN = ACMAA8<br /> | MRCLASS = 35L70 (35B05)<br /> | arxivlink = <br /> | mathsciid = MR1687180<br /> | MRREVIEWER = Nickolai A. Larkin <br /> }}</div> Pblue