https://dispersivewiki.org/DispersiveWiki/api.php?action=feedcontributions&user=Oleg+Alexandrov&feedformat=atomDispersiveWiki - User contributions [en]2020-06-05T14:39:27ZUser contributionsMediaWiki 1.33.1https://dispersivewiki.org/DispersiveWiki/index.php?title=Free_wave_equation&diff=5217Free wave equation2007-09-03T22:14:42Z<p>Oleg Alexandrov: stub to the bottom</p>
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<div>The '''free wave equation''' on <math>R^{1+d}</math> is given by<br />
<center><math>\Box f = 0</math></center><br />
where ''f'' is a scalar or vector field on [[Minkowski space]] <math>R^{1+d}</math>.<br />
In coordinates, this becomes<br />
<center><math>- \partial_{tt} f + \Delta f = 0.</math></center><br />
It is the prototype for many [[wave equations|nonlinear wave equations]].<br />
<br />
One can add a mass term to create the [[Klein-Gordon equation]].<br />
<br />
<br />
{{stub}}<br />
[[Category:Wave]]<br />
[[Category:Equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Wave_equations&diff=5216Wave equations2007-09-03T22:14:00Z<p>Oleg Alexandrov: style</p>
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<div>==Non-linear wave equations==<br />
<br />
Nonlinear wave equations arise in physics from two major sources: relativity and [[elasticity]].<br />
<br />
All relativistic field equations in (classical) physics are variants of the [[free wave equation]] or [[Klein-Gordon equation]] on [[Minkowski space]].<br />
<br />
There are several ways to perturb this equation. There are ''linear perturbations'', which include the addition of potential terms, connection terms, and drag terms, as well as the replacement of the flat Minkowski metric with a more general curved metric, or by placing obstacles or otherwise changing the topology of the domain manifold <math>R^{1+d}</math>. <br />
<br />
Here we shall focus more on purely ''non-linear'' perturbations, which collapse to a constant-coefficient wave equation in the [[small amplitude limit]].<br />
In the fullest generality, this would mean studying equations of the form<br />
<br />
<center><math>F(f, Df, D^2_{}f) = 0</math></center><br />
<br />
where <math>D</math> denotes differentiation in space or time and the Taylor expansion of <math>F</math> to first order is the free wave or Klein-Gordon equation. Such fully non-linear equations, though, are very difficult to study, and have only really been analyzed in the one-dimensional case (in which case it can be subsumed into the general theory of 1+1-dimensional hyperbolic systems). In higher dimensions the only known tool to analyze this case is to differentiate the equation, turning it into a quasi-linear system. As such we do not discuss fully non-linear wave equations here. Instead, we consider three less general types of equations, which in increasing order of complexity are the [[NLW|semi-linear]], [[DNLW|semi-linear with derivatives]], and [[QNLW|quasi-linear]] equations.<br />
<br />
Non-linear wave equations are often the [[Euler-Lagrange equation]] for some [[variational problem]]. This usually generates the conserved [[stress-energy tensor]], which is of fundamental importance in the analysis of such equations, especially for the global-in-time theory.<br />
<br />
The principle of relativity asserts that the equations of physics are covariant with respect to the underlying geometry of spacetime. This can be exploited in a number of ways. One is via [[stress-energy tensor]] mentioned previously. Another is via [[conformal transformation]] of spacetime. A third is via [[finite speed of propagation]]. The covariance also generates some important [[null structure]]s in the nonlinear components of the equation.<br />
<br />
The perturbative theory for nonlinear wave equations rests on various linear, bilinear, and nonlinear [[wave estimates|estimates for the linear wave equation]].<br />
<br />
===Dependence on dimension===<br />
<br />
The one-dimensional case <math>d=1</math> is special for several reasons. Firstly, there is the very convenient null co-ordinate system <math>u = t+x, v = t-x</math> which can be used to factorize <math>\Box</math>. Also, the stress-energy tensor often becomes trace-free, which leads to better conformal invariance properties. There are a vastly larger number of conformal transformations, indeed anything of the form <math>(u,v) \rightarrow ( F (u), Y (v))</math> is conformal. Also, the one-dimensional wave equation has no decay, local smoothing, or dispersion properties, and its solutions are essentially travelling waves. Finally, there are a much larger range of spaces beyond Sobolev spaces which are available for well-posedness theory, because the free wave evolution operator preserves all translation-invariant spaces. (In two and higher dimensions only <math>L^2</math>-based spaces such as Sobolev spaces H^s are preserved, because waves can focus at a point (or defocus from a point)).<br />
<br />
The higher-dimensional case <math>d>1</math> is usually quite different from the one-dimensional case, although in spherically symmetric situations one can obtain similar behaviour, especially when viewed in the null co-ordinates <math>u = t+r, v = t-r</math>. Indeed one can think of spherically symmetric wave equations as one-dimensional wave equations with a singular drag term <math>(n-1) f _r / r</math>.<br />
<br />
===Specific wave equations===<br />
<br />
* [[semilinear NLW|Semilinear wave equations]] ([[sine-Gordon]], etc.)<br />
* [[DNLW|NLW with derivatives]]<br />
** [[linear-derivative nonlinear wave equations]] ([[YM|Yang-Mills]], [[YMH|Yang-Mills-Higgs]], [[MKG|Maxwell-Klein-Gordon]])<br />
** [[DDNLW|quadratic-derivative nonlinear wave equations]] ([[wave maps]])<br />
* [[Dirac equations|Dirac-type equations]]<br />
* [[QNLW|Quasilinear wave equations]] ([[Einstein]], [[minimal surface equation|minimal surface]], etc.)<br />
<br />
[[Category:Wave]]<br />
[[Category:Equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Nonlinear_Dirac_equation&diff=5215Nonlinear Dirac equation2007-09-03T22:12:42Z<p>Oleg Alexandrov: bypass double redir</p>
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<div>#REDIRECT [[Dirac equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Maxwell-Dirac_equation&diff=5214Maxwell-Dirac equation2007-09-03T22:12:36Z<p>Oleg Alexandrov: bypass double redir</p>
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<div>#redirect [[Dirac equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Dirac-Klein-Gordon_equation&diff=5213Dirac-Klein-Gordon equation2007-09-03T22:12:26Z<p>Oleg Alexandrov: bypass double redir</p>
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<div>#redirect [[Dirac equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Dirac_equations&diff=5212Dirac equations2007-09-03T22:11:41Z<p>Oleg Alexandrov: style</p>
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<div>This article describes several equations named after [http://en.wikipedia.org/wiki/Paul_Dirac Paul Dirac]. <br />
__TOC__<br />
==The Maxwell-Dirac equation==<br />
<br />
[More info on this equation would be greatly appreciated. - Ed.]<br />
<br />
This equation essentially reads<br />
<br />
<center><math>D_A y = - y </math></center><br />
<center><math>\Box A + \nabla (\nabla_{x,t} A)= \underline{y} y </math></center><br />
<br />
where <math>y</math> is a spinor field (solving a coupled massive Dirac equation), and <math>D</math> is the Dirac operator with connection A. We put <math>y</math> in <math>H^{s_1}</math> and <math>A</math> in <math>H^{s_2} \times H^{s_2 - 1}</math>.<br />
<br />
* Scaling is <math>(s_1, s_2) = (n/2-3/2, n/2-1)</math>.<br />
* When <math>n=1</math>, there is GWP for small smooth data [[Chd1973]]<br />
* When <math>n=3</math> there is LWP for <math>(s_1, s_2) = (1, 1)</math> in the Coulomb gauge [[Bou1999]], and for <math>(s_1, s_2) = (1/2+, 1+)</math> in the Lorentz gauge [[Bou1996]]<br />
** For <math>(s_1, s_2) = (1,2)</math> in the Coulomb gauge this is in [[Bou1996]]<br />
** This has recently been improved by Selberg to <math>(1/4+, 1)</math>. Note that for technical reasons, lower-regularity results do not automatically imply higher ones when the regularity of one field (e.g. <math>A</math>) is kept fixed.<br />
** LWP for smooth data was obtained in [[Grs1966]]<br />
** GWP for small smooth data was obtained in [[Ge1991]]<br />
* When <math>n=4</math>, GWP for small smooth data is known (Psarelli?)<br />
<br />
In the nonrelativistic limit this equation converges to a Maxwell-Poisson system for data in the energy space [[BecMauSb-p2]]; furthermore one has a local well-posedness result which grows logarithmically in the asymptotic parameter. Earlier work appears in [[MasNa2003]].<br />
<br />
==Dirac-Klein-Gordon equation==<br />
<br />
[More info on this equation would be greatly appreciated. - Ed.]<br />
<br />
This equation essentially reads<br />
<br />
<center><math>D \psi = \phi \psi - \psi </math></center><br />
<center><math>\Box \phi = \overline{\psi} \psi </math></center><br />
<br />
where <math>\psi</math> is a spinor field (solving a coupled massive Dirac equation), <math>D</math> is the Dirac operator and <math>\phi</math> is a scalar (real) field. We put <math>\psi</math> in <math>H^{s_1}</math> and <math>( \phi, \phi_t)</math> in <math>H^{s_2} \times H^{s_2 - 1}</math>.<br />
<br />
The energy class is essentially <math>(s_1,s_2) = (1/2,1)</math>, but the energy density is not positive. However, the <math>L^2</math> norm of <math>y</math> is also positive and conserved..<br />
<br />
* Scaling is <math>(s_1, s_2) = (d/2-3/2, d/2-1)</math>.<br />
* When <math>n=1</math> there is GWP for <math>(s_1,s_2) = (1,1)</math> [[Chd1973]], [[Bou2000]] and LWP for <math>(s_1, s_2) = (0, 1/2)</math> [[Bou2000]].<br />
* When <math>n=2</math> there are some LWP results in [[Bou2001]]<br />
<br />
==Nonlinear Dirac equation==<br />
<br />
This equation essentially reads<br />
<br />
<center><math>D \psi - m \psi = \lambda (\gamma \psi, \psi) \psi</math></center><br />
<br />
where <math>\psi</math> is a spinor field, <math>m > 0</math> is the mass, <math>\lambda</math> is a complex parameter, <math>\gamma</math> is the zeroth Pauli matrix, and <math>(,)</math> is the spinor inner product.<br />
<br />
* Scaling is <math>s_c =1</math> (at least in the massless case <math>m=0</math>).<br />
* In <math>R^3</math>, LWP is known for <math>H^s</math> when <math>s > 1</math> [[EscVe1997]]<br />
** This can be improved to LWP in <math>H^1</math> (and GWP for small <math>H^1</math> data) if an epsilon of additional regularity as assumed in the radial variable [[MacNkrNaOz-p]]; in particular one has GWP for radial <math>H^1</math> data.<br />
* In <math>R^3</math>, GWP is known for small <math>H^s</math> data when <math>s > 1</math> [[MacNaOz-p2]]. Some results on the [[nonrelativistic limit]] of this equation are also obtained in that paper.<br />
<br />
[[Category:Wave]]<br />
[[Category:Equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Dirac_equations&diff=5210Dirac equations2007-09-03T22:08:56Z<p>Oleg Alexandrov: Dirac Equations moved to Dirac equations: Uniformity in naming</p>
<hr />
<div>====The Maxwell-Dirac equation====<br />
<br />
[More info on this equation would be greatly appreciated. - Ed.]<br />
<br />
This equation essentially reads<br />
<br />
<center><math>D_A y = - y </math></center><br />
<center><math>\Box A + \nabla (\nabla_{x,t} A)= \underline{y} y </math></center><br />
<br />
where <math>y</math> is a spinor field (solving a coupled massive Dirac equation), and <math>D</math> is the Dirac operator with connection A. We put <math>y</math> in <math>H^{s_1}</math> and <math>A</math> in <math>H^{s_2} \times H^{s_2 - 1}</math>.<br />
<br />
* Scaling is <math>(s_1, s_2) = (n/2-3/2, n/2-1)</math>.<br />
* When <math>n=1</math>, there is GWP for small smooth data [[Chd1973]]<br />
* When <math>n=3</math> there is LWP for <math>(s_1, s_2) = (1, 1)</math> in the Coulomb gauge [[Bou1999]], and for <math>(s_1, s_2) = (1/2+, 1+)</math> in the Lorentz gauge [[Bou1996]]<br />
** For <math>(s_1, s_2) = (1,2)</math> in the Coulomb gauge this is in [[Bou1996]]<br />
** This has recently been improved by Selberg to <math>(1/4+, 1)</math>. Note that for technical reasons, lower-regularity results do not automatically imply higher ones when the regularity of one field (e.g. <math>A</math>) is kept fixed.<br />
** LWP for smooth data was obtained in [[Grs1966]]<br />
** GWP for small smooth data was obtained in [[Ge1991]]<br />
* When <math>n=4</math>, GWP for small smooth data is known (Psarelli?)<br />
<br />
In the nonrelativistic limit this equation converges to a Maxwell-Poisson system for data in the energy space [[BecMauSb-p2]]; furthermore one has a local well-posedness result which grows logarithmically in the asymptotic parameter. Earlier work appears in [[MasNa2003]].<br />
<br />
<br />
====Dirac-Klein-Gordon equation====<br />
<br />
[More info on this equation would be greatly appreciated. - Ed.]<br />
<br />
This equation essentially reads<br />
<br />
<center><math>D \psi = \phi \psi - \psi </math></center><br />
<center><math>\Box \phi = \overline{\psi} \psi </math></center><br />
<br />
where <math>\psi</math> is a spinor field (solving a coupled massive Dirac equation), <math>D</math> is the Dirac operator and <math>\phi</math> is a scalar (real) field. We put <math>\psi</math> in <math>H^{s_1}</math> and <math>( \phi, \phi_t)</math> in <math>H^{s_2} \times H^{s_2 - 1}</math>.<br />
<br />
The energy class is essentially <math>(s_1,s_2) = (1/2,1)</math>, but the energy density is not positive. However, the <math>L^2</math> norm of <math>y</math> is also positive and conserved..<br />
<br />
* Scaling is <math>(s_1, s_2) = (d/2-3/2, d/2-1)</math>.<br />
* When <math>n=1</math> there is GWP for <math>(s_1,s_2) = (1,1)</math> [[Chd1973]], [[Bou2000]] and LWP for <math>(s_1, s_2) = (0, 1/2)</math> [[Bou2000]].<br />
* When <math>n=2</math> there are some LWP results in [[Bou2001]]<br />
<br />
<br />
====Nonlinear Dirac equation====<br />
<br />
This equation essentially reads<br />
<br />
<center><math>D \psi - m \psi = \lambda (\gamma \psi, \psi) \psi</math></center><br />
<br />
where <math>\psi</math> is a spinor field, <math>m > 0</math> is the mass, <math>\lambda</math> is a complex parameter, <math>\gamma</math> is the zeroth Pauli matrix, and <math>(,)</math> is the spinor inner product.<br />
<br />
* Scaling is <math>s_c =1</math> (at least in the massless case <math>m=0</math>).<br />
* In <math>R^3</math>, LWP is known for <math>H^s</math> when <math>s > 1</math> [[EscVe1997]]<br />
** This can be improved to LWP in <math>H^1</math> (and GWP for small <math>H^1</math> data) if an epsilon of additional regularity as assumed in the radial variable [[MacNkrNaOz-p]]; in particular one has GWP for radial <math>H^1</math> data.<br />
* In <math>R^3</math>, GWP is known for small <math>H^s</math> data when <math>s > 1</math> [[MacNaOz-p2]]. Some results on the [[nonrelativistic limit]] of this equation are also obtained in that paper.<br />
<br />
<br />
[[Category:wave]]<br />
[[Category:Equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Dirac_Equations&diff=5211Dirac Equations2007-09-03T22:08:56Z<p>Oleg Alexandrov: Dirac Equations moved to Dirac equations: Uniformity in naming</p>
<hr />
<div>#REDIRECT [[Dirac equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Main_Page&diff=5209Main Page2007-09-03T22:07:06Z<p>Oleg Alexandrov: Cleanup for style</p>
<hr />
<div>Welcome to the '''Dispersive PDE Wiki'''! These web pages are intended to present the latest results, [[:Category:Open problems|conjectures]], [[:Category:Bibliography|bibliography]], [[:Category:Concept|concepts]] and other material on the local and global [[well-posedness]] problems (and related questions) for non-linear [[dispersion relation|dispersive]] and [[wave equations]]. (We also have [[DispersiveWiki:About|a more detailed description of this wiki and its purpose]].)<br />
<br />
The information here has only recently been converted into wiki format, and there are still several rough edges to it. We welcome updates, corrections, cleanup, and new contributions &mdash; just [[Special:Userlogin&type=signup|create an account]] and you're ready to edit! <br />
If you are new to wikis, we have some [[Help:Contents|help files]] available. If you are interested in contributing, you might want to visit our [[DispersiveWiki:Community Portal|community portal]] for current discussions and projects.<br />
<br />
== Classes of equations ==<br />
<br />
There are a [[:Category:Equations|large number]] of [[nonlinear]] [[dispersive]] and [[wave]] equations of interest, many of which arise from mathematical physics. We can divide them into four classes:<br />
<br />
# [[Wave equations|Nonlinear wave equations]] ([[semilinear NLW|NLW]], [[semilinear NLW|NLKG]], [[Wave maps]], [[Maxwell-Klein-Gordon equations|Maxwell-Klein-Gordon]], [[Yang-Mills equations|Yang-Mills]], [[Einstein equations]], etc.)<br />
# [[Schrodinger equations|Nonlinear Schrodinger equations]] ([[NLS equation|NLS]], [[Schrodinger maps|Schrodinger maps]], [[Cubic DNLS on R|DNLS]], [[Hartree equation|Hartree]], etc.)<br />
# [[KdV equations|Equations of Korteweg-de Vries type]] ([[Korteweg-de Vries equation|KdV]], [[Modified Korteweg-de Vries equation|mKdV]], [[Generalized Korteweg-de Vries equation|gKdV]], etc.)<br />
# [[Other equations|Other equations and systems]] ([[Benjamin-Ono equation|Benjamin-Ono]], [[Kadomtsev-Petviashvili equation|Kadomtsev-Petviashvili]], [[Zakharov equation|Zakharov]], [[Davey-Stewartson system|Davey-Stewartson]], etc.)<br />
<br />
A complete list of equations discussed on this wiki [[:Category:Equations|can be found here]].<br />
<br />
== Other Web resources ==<br />
<br />
*Wikipedia has a [http://en.wikipedia.org/wiki/Partial_differential_equations partial differential equations page].<br />
*There is a [http://en.wikipedia.org/wiki/Category:Mathematics-related_lists list of math related pages on wikipedia ].<br />
*Check out the [http://www.wikiwaves.org/index.php/Main_Page water waves wiki].</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Maxwell-Klein-Gordon_equations&diff=5135Maxwell-Klein-Gordon equations2007-07-04T16:27:25Z<p>Oleg Alexandrov: up two levels of headings, sectioning</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 <math>V(\phi)</math>. Thus A is now purely imaginary, and <math>\phi</math> 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 <math>\phi</math> 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 />
<center><math>\Delta A_0 = O( \phi \phi_t ) + O( \Phi^3 )</math></center><br />
<center><math> \Box A = \nabla^{-1} Q( \phi , \phi )</math></center><br />
<center><math> \Box \phi = Q( \nabla^{-1} A, \phi ) + O( (A_0)_t \phi ) + O( A_0 \phi_t ) + O( \Phi^3 )</math></center><br />
<br />
where <math>O(\Phi^3)</math> denotes terms that are cubic in <math>(A_0, A, \phi)</math>. 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 />
<center><math>\Box A = \nabla^{-1} Q( \phi , \phi ) </math></center><br />
<center><math>\Box \phi = Q( \nabla^{-1} A, \phi )</math></center><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>1/2 by energy estimates. For s<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<sup>2</sup>==<br />
<br />
* Scaling is s_c = 0.<br />
* Heuristically, one expects X^{s,\delta} methods to give LWP for s > 1/4, but we do not know if this has been done rigorously.<br />
** Strichartz estimates give s > 1/2 ([[PoSi1993]]), while energy methods give s>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<sup>3</sup>==<br />
<br />
* Scaling is s_c = 1/2.<br />
* LWP for s>1/2 in the Coulomb Gauge [[MaStz-p]]<br />
** For the model equation LWP fails for s < 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>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>1 this follows (in any of the three gauges) from Strichartz estimates [[PoSi1993]]<br />
** For s>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 <math>\nabla^{-1} Q( \phi , \phi )</math>, as this is slightly smoother than <math> Q( \nabla^{-1}A, \phi )</math>.<br />
* GWP for s>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 <math>\phi</math> 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>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Benjamin-Ono_equation&diff=4794Benjamin-Ono equation2007-05-05T16:21:42Z<p>Oleg Alexandrov: TeX</p>
<hr />
<div><center>'''Benjamin-Ono equation'''</center><br />
<br />
The Benjamin-Ono equation (BO) [[Bj1967]], [[On1975]], which models one-dimensional internal waves in deep water, is given by<br />
<center><math>u_t + H u_{xx} + u u_x = 0</math></center><br />
where <math>H</math> is the [[Hilbert transform]]. This equation is [[completely integrable]] (see e.g., [[AbFs1983]], [[CoiWic1990]]).<br />
<br />
Scaling is <math>s = -1/2,</math> and the following results are known:<br />
<br />
* LWP in <math>H^s</math> for <math>s \ge 1</math> [[Ta2004]]<br />
** For <math>s > 9/8</math> this is in [[KnKoe2003]]<br />
** For <math>s > 5/4</math> this is in [[KocTz2003]]<br />
** For <math>s \ge 3/2</math> this is in [[Po1991]]<br />
** For <math>s > 3/2</math> this is in [[Io1986]]<br />
** For <math>s > 3</math> this is in [[Sau1979]]<br />
** For no value of s is the solution map uniformly continuous [[KocTz2005]]<br />
*** For <math>s < -1/2</math> this is in [[BiLi2001]]<br />
* Global weak solutions exist for <math>L^2</math> data [[Sau1979]], [[GiVl1989b]], [[GiVl1991]], [[Tom1990]]<br />
* Global well-<span class="SpellE">posedness</span> in <span class="SpellE"><math>H^s</math></span> for <math>s \ge 1</math> [[Ta2004]]<br />
** For <math>s \ge 3/2</math> this is in [[Po1991]]<br />
** For smooth solutions this is in [[Sau1979]]<br />
<br />
== Generalized Benjamin-Ono equation ==<br />
<br />
The ''generalized Benjamin-Ono equation'' is the scalar equation<br />
<center><math>\partial_t u + D_x^{1+a} \partial_x u + u\partial_x u = 0.</math></center><br />
<br />
where <math>D_x = \sqrt{-\Delta}</math> is the positive differentiation operator. When <math>a=1</math> this is [[KdV]]; when <math>a=0</math> this is Benjamin-Ono. Both of these two extreme cases are [[completely integrable]], though the intermediate cases <math>0 < a < 1</math> are not.<br />
<br />
When <math>0 < a < 1,</math> scaling is <math>s = -1/2 - a,</math> and the following results are known:<br />
<br />
* LWP in <math>H^s</math> is known for <math>s > 9/8 - 3a/8</math> [[KnKoe2003]]<br />
** For <math>s \ge 3/4 (2-a)</math> this is in [[KnPoVe1994b]]<br />
* GWP is known when <math>s \ge (a+1)/2</math> when <math>a > 4/5,</math> from the conservation of the Hamiltonian [[KnPoVe1994b]]<br />
* The LWP results are obtained by energy methods; it is known that pure iteration methods cannot work [[MlSauTz2001]]<br />
** However, this can be salvaged by combining the <math>H^s</math> norm <math>|| f ||_{H^s}</math> with a weighted Sobolev space, namely <math>|| xf ||_{H^{s - 2s_*}},</math> where <math>s_* = (a+1)/2</math> is the energy regularity. [[CoKnSt2003]]<br />
<br />
== Benjamin-Ono with power nonlinearity ==<br />
<br />
This is the equation<br />
<center><math> u_t + H u_{xx} + (u^k)_x = 0.</math></center><br />
Thus the original Benjamin-Ono equation corresponds to the case <math>k=2.</math><br />
The scaling exponent is <math>1/2 - 1/(k-1).</math><br />
<br />
* For <math>k=3,</math> one has GWP for large data in <math>H^1</math> [[KnKoe2003]] and LWP for small data in <math>H^s,</math> <math>s > 1/2</math> [[MlRi2004]]<br />
** For small data in <math>H^s,</math> <math>s>1,</math> LWP was obtained in [[KnPoVe1994b]]<br />
** With the addition of a small viscosity term, GWP can also be obtained in <math>H^1</math> by complete integrability methods in [[FsLu2000]], with asymptotics under the additional assumption that the initial data is in <math>L^1.</math><br />
** For <math>s < 1/2,</math> the solution map is not <math>C^3</math> [[MlRi2004]]<br />
* For <math>k=4,</math> LWP for small data in <math>H^s,</math> <math>s > 5/6</math> was obtained in [[KnPoVe1994b]].<br />
* For <math>k>4,</math> LWP for small data in <math>H^s,</math> <math>s \ge 3/4</math> was obtained in [[KnPoVe1994b]].<br />
* For any <math>k \ge 3</math> and <math>s < 1/2 - 1/k</math> the solution map is not uniformly continuous [[BiLi2001]]<br />
<br />
== Other generalizations ==<br />
<br />
The KdV-Benjamin Ono equation is formed by combining the linear parts of the KdV and Benjamin-Ono equations together. It is globally well-posed in <math>L^2</math> [[Li1999]], and locally well-posed in <math>H^{-3/4+}</math> [[KozOgTns2001]] (see also [[HuoGuo2005]] where <math>H^{-1/8+}</math> is obtained). <br />
<br />
Similarly one can generalize the non-linearity to be k-linear, generating for instance the modified KdV-BO equation, which is locally well-posed in <math>H^{1/4+}</math> [[HuoGuo2005]]. For general gKdV-gBO equations one has local well-posedness in <math>H^3</math> and above [[GuoTan1992]]. One can also add damping terms <math>Hu_x</math> to the equation; this arises as a model for ion-acoustic waves of finite amplitude with linear Landau damping [[OttSud1970]].<br />
<br />
[[Category:Integrability]]<br />
[[Category:Equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Main_Page&diff=4793Main Page2007-05-05T15:57:30Z<p>Oleg Alexandrov: Links. Bypass a few redirects. A few small fixes.</p>
<hr />
<div>== Dispersive PDE Wiki ==<br />
<br />
Welcome to the Dispersive PDE Wiki! These web pages are intended to present the latest results, [[:Category:Open problems|conjectures]], [[:Category:Bibliography|bibliography]], [[:Category:Concept|concepts]] and other material on the local and global [[well-posedness]] problems (and related questions) for non-linear [[dispersion relation|dispersive]] and [[wave equations]]. (We also have [[DispersiveWiki:About|a more detailed description of this wiki and its purpose]].)<br />
<br />
The information here has only recently been converted into wiki format, and there are still several rough edges to it. We welcome updates, corrections, cleanup, and new contributions &mdash; just [[Special:Userlogin&type=signup|create an account]] and you're ready to edit! <br />
If you are new to wikis, we have some [[Help:Contents|help files]] available. If you are interested in contributing, you might want to visit our [[DispersiveWiki:Community Portal|community portal]] for current discussions and projects.<br />
<br />
== Classes of equations ==<br />
<br />
There are a [[:Category:Equations|large number]] of [[nonlinear]] [[dispersive]] and [[wave]] equations of interest, many of which arise from mathematical physics. We can divide them into four classes:<br />
<br />
# [[Wave equations|Nonlinear wave equations]] ([[semilinear NLW|NLW]], [[semilinear NLW|NLKG]], [[Wave maps]], [[Maxwell-Klein-Gordon equations|Maxwell-Klein-Gordon]], [[Yang-Mills equations|Yang-Mills]], [[Einstein equations]], etc.)<br />
# [[Schrodinger equations|Nonlinear Schrodinger equations]] ([[NLS equation|NLS]], [[Schrodinger maps|Schrodinger maps]], [[Cubic DNLS on R|DNLS]], [[Hartree equation|Hartree]], etc.)<br />
# [[KdV equations|Equations of Korteweg-de Vries type]] ([[Korteweg-de Vries equation|KdV]], [[Modified Korteweg-de Vries equation|mKdV]], [[Generalized Korteweg-de Vries equation|gKdV]], etc.)<br />
# [[Other equations|Other equations and systems]] ([[Benjamin-Ono equation|Benjamin-Ono]], [[Kadomtsev-Petviashvili equation|Kadomtsev-Petviashvili]], [[Zakharov equation|Zakharov]], [[Davey-Stewartson system|Davey-Stewartson]], etc.)<br />
<br />
A complete list of equations discussed on this wiki [[:Category:Equations|can be found here]].<br />
<br />
== Other Web resources ==<br />
<br />
*Wikipedia has a [http://en.wikipedia.org/wiki/Partial_differential_equations partial differential equations page].<br />
*There is a [http://en.wikipedia.org/wiki/Category:Mathematics-related_lists list of math related pages on wikipedia ].<br />
*Check out the [http://www.wikiwaves.org/index.php/Main_Page water waves wiki].</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Sine-Gordon_equation&diff=4792Sine-Gordon equation2007-05-05T15:51:14Z<p>Oleg Alexandrov: TeX</p>
<hr />
<div>[Contributions to this section are sorely needed!]<br />
<br />
The '''sine-Gordon equation'''<br />
<br />
<center><math>\Box u = \sin(u)</math></center><br />
<br />
in <math>R^{1+1}</math> arises in the study of optical pulses, or from the Scott model of a continuum of pendula hanging from a wire. It is a [[completely integrable]] equation, and has many interesting solutions, including "breather" solutions.<br />
<br />
Because the non-linearity is bounded, GWP is easily obtained for <math>L^2</math> or even <math>L^1</math> data.<br />
<br />
[[Category:Integrability]]<br />
[[Category:wave]]<br />
[[Category:Equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Sine-Gordon_equation&diff=4790Sine-Gordon equation2007-05-05T15:50:44Z<p>Oleg Alexandrov: Sine-Gordon moved to Sine-Gordon equation: proper name</p>
<hr />
<div>[Contributions to this section are sorely needed!]<br />
<br />
The '''sine-Gordon equation'''<br />
<br />
<center><math>\Box u = sin(u)</math></center><br />
<br />
in <math>R^{1+1}</math> arises in the study of optical pulses, or from the Scott model of a continuum of pendula hanging from a wire. It is a [[completely integrable]] equation, and has many interesting solutions, including "breather" solutions.<br />
<br />
Because the non-linearity is bounded, GWP is easily obtained for <math>L^2</math> or even <math>L^1</math> data.<br />
<br />
[[Category:Integrability]]<br />
[[Category:wave]]<br />
[[Category:Equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Sine-Gordon&diff=4791Sine-Gordon2007-05-05T15:50:44Z<p>Oleg Alexandrov: Sine-Gordon moved to Sine-Gordon equation: proper name</p>
<hr />
<div>#REDIRECT [[Sine-Gordon equation]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Einstein&diff=4789Einstein2007-05-05T15:49:55Z<p>Oleg Alexandrov: #REDIRECT Einstein equations, fix double redir</p>
<hr />
<div>#REDIRECT [[Einstein equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Einstein_equations&diff=4787Einstein equations2007-05-05T15:49:20Z<p>Oleg Alexandrov: Einstein Equations moved to Einstein equations: consistency with other equation pages. No need for "equation" in upper case. Also per Wikipedia naming conventions.</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 />
The (vacuum) Einstein equations take the form<br />
<br />
<center><math>R_{ \alpha \beta } = C R g_{ \alpha \beta }</math></center><br />
<br />
where <math>g</math> is the metric for a 3+1-dimensional manifold, <math>R</math> is the Ricci curvature tensor, and <math>C</math> is an absolute 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-p]]; 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 />
** This result has now been improved further to <math>s=2</math> ([[KlRo-p6]], [[KlRo-p7]], [[KlRo-p8]]).<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]]).<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 />
== Open problems ==<br />
<br />
* [[Cosmic Censorship Hypothesis]]<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 />
[[Category:Geometry]]<br />
[[Category:wave]]<br />
[[Category:Equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Einstein_Equations&diff=4788Einstein Equations2007-05-05T15:49:20Z<p>Oleg Alexandrov: Einstein Equations moved to Einstein equations: consistency with other equation pages. No need for "equation" in upper case. Also per Wikipedia naming conventions.</p>
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<div>#REDIRECT [[Einstein equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Maxwell-Klein-Gordon&diff=4786Maxwell-Klein-Gordon2007-05-05T15:47:53Z<p>Oleg Alexandrov: #REDIRECT Maxwell-Klein-Gordon equations</p>
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<div>#REDIRECT [[Maxwell-Klein-Gordon equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=MKG&diff=4785MKG2007-05-05T15:47:47Z<p>Oleg Alexandrov: #REDIRECT Maxwell-Klein-Gordon equations, fix double redir</p>
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<div>#REDIRECT [[Maxwell-Klein-Gordon equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Maxwell-Klein-Gordon_system&diff=4784Maxwell-Klein-Gordon system2007-05-05T15:47:38Z<p>Oleg Alexandrov: #REDIRECT Maxwell-Klein-Gordon equations</p>
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<div>#REDIRECT [[Maxwell-Klein-Gordon equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Maxwell-Klein-Gordon_equation&diff=4783Maxwell-Klein-Gordon equation2007-05-05T15:47:32Z<p>Oleg Alexandrov: #REDIRECT Maxwell-Klein-Gordon equations, fix double redir</p>
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<div>#REDIRECT [[Maxwell-Klein-Gordon equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Maxwell-Klein-Gordon_equations&diff=4779Maxwell-Klein-Gordon equations2007-05-05T15:46:43Z<p>Oleg Alexandrov: Maxwell-Klein-Gordon Equations moved to Maxwell-Klein-Gordon equations: consistency with other equation pages</p>
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<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 <math>V(\phi)</math>. Thus A is now purely imaginary, and <math>\phi</math> is complex.<br />
<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 <math>\phi</math> 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 />
<center><math>\Delta A_0 = O( \phi \phi_t ) + O( \Phi^3 )</math></center><br />
<center><math> \Box A = \nabla^{-1} Q( \phi , \phi )</math></center><br />
<center><math> \Box \phi = Q( \nabla^{-1} A, \phi ) + O( (A_0)_t \phi ) + O( A_0 \phi_t ) + O( \Phi^3 )</math></center><br />
<br />
where <math>O(\Phi^3)</math> denotes terms that are cubic in <math>(A_0, A, \phi)</math>. 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 />
<center><math>\Box A = \nabla^{-1} Q( \phi , \phi ) </math></center><br />
<center><math>\Box \phi = Q( \nabla^{-1} A, \phi )</math></center><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 />
<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>1/2 by energy estimates. For s<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 />
<br />
====Maxwell-Klein-Gordon on R<sup>2</sup>====<br />
<br />
* Scaling is s_c = 0.<br />
* Heuristically, one expects X^{s,\delta} methods to give LWP for s > 1/4, but we do not know if this has been done rigorously.<br />
** Strichartz estimates give s > 1/2 ([[PoSi1993]]), while energy methods give s>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 />
<br />
<br />
====Maxwell-Klein-Gordon on R<sup>3</sup>====<br />
<br />
<br />
<br />
* Scaling is s_c = 1/2.<br />
* LWP for s>1/2 in the Coulomb Gauge [[MaStz-p]]<br />
** For the model equation LWP fails for s < 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>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>1 this follows (in any of the three gauges) from Strichartz estimates [[PoSi1993]]<br />
** For s>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 <math>\nabla^{-1} Q( \phi , \phi )</math>, as this is slightly smoother than <math> Q( \nabla^{-1}A, \phi )</math>.<br />
* GWP for s>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 <math>\phi</math> 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>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Maxwell-Klein-Gordon_Equations&diff=4780Maxwell-Klein-Gordon Equations2007-05-05T15:46:43Z<p>Oleg Alexandrov: Maxwell-Klein-Gordon Equations moved to Maxwell-Klein-Gordon equations: consistency with other equation pages</p>
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<div>#REDIRECT [[Maxwell-Klein-Gordon equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Talk:Maxwell-Klein-Gordon_equations&diff=4781Talk:Maxwell-Klein-Gordon equations2007-05-05T15:46:43Z<p>Oleg Alexandrov: Talk:Maxwell-Klein-Gordon Equations moved to Talk:Maxwell-Klein-Gordon equations: consistency with other equation pages</p>
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<div>This page was derived from an earlier version which was improved with comments from Jacob Sterbenz.</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Talk:Maxwell-Klein-Gordon_Equations&diff=4782Talk:Maxwell-Klein-Gordon Equations2007-05-05T15:46:43Z<p>Oleg Alexandrov: Talk:Maxwell-Klein-Gordon Equations moved to Talk:Maxwell-Klein-Gordon equations: consistency with other equation pages</p>
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<div>#REDIRECT [[Talk:Maxwell-Klein-Gordon equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=YM&diff=4778YM2007-05-05T15:45:32Z<p>Oleg Alexandrov: #REDIRECT Yang-Mills equations</p>
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<div>#REDIRECT [[Yang-Mills equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Yang-Mills&diff=4777Yang-Mills2007-05-05T15:45:27Z<p>Oleg Alexandrov: #REDIRECT Yang-Mills equations, bypass double redir</p>
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<div>#REDIRECT [[Yang-Mills equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=YMH&diff=4776YMH2007-05-05T15:45:18Z<p>Oleg Alexandrov: #REDIRECT Yang-Mills equations</p>
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<div>#REDIRECT [[Yang-Mills equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Yang-Mills_equation&diff=4775Yang-Mills equation2007-05-05T15:45:09Z<p>Oleg Alexandrov: #REDIRECT Yang-Mills equations, bypass double redir</p>
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<div>#REDIRECT [[Yang-Mills equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Yang-Mills_equations&diff=4771Yang-Mills equations2007-05-05T15:44:32Z<p>Oleg Alexandrov: Yang-Mills Equations moved to Yang-Mills equations: Consistency with other equation pages</p>
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<div>{{equation<br />
| name = Yang-Mills <br />
| equation = <math>D_\alpha F^{\alpha \beta} = 0</math><br />
| fields = <math>A_\alpha: \R^{1+d} \to \mathfrak{g}</math><br />
| data = <math>A_\alpha[0] \in H^s(\R^d) \times H^{s-1}(\R^d)</math><br />
| hamiltonian = [[Hamiltonian]]<br />
| linear = [[free wave equation|wave]]<br />
| nonlinear = [[semilinear|semilinear with derivatives]]<br />
| critical = <math>\dot H^{d/2 - 1}(\R^d)</math><br />
| criticality = energy critical for d=4<br />
| covariance = [[Lorentzian]], [[gauge]]<br />
| lwp = varies | gwp = varies<br />
| parent = [[DNLW]]<br />
| special = Yang-Mills on R^2, R^3, R^4<br />
| related = [[MKG]], [[Cubic NLW/NLKG|Cubic NLW]], [[YMH|Yang-Mills-Higgs]]<br />
}}<br />
<br />
====The Yang-Mills equation====<br />
<br />
Let <math>A</math> be a connection on <math>R^{d+1}</math> which takes values in the Lie algebra g of a compact Lie group G. Formally, the connection A is said to obey the ''Yang-Mills equation'' if it is a critical point for the Lagrangian functional<br />
<br />
<center><math>\int F^{\alpha \beta} F_{\alpha \beta}</math></center><br />
<br />
where <math>F:=dA + [A,A]</math> is the curvature of the connection <math>A</math>. The Euler-Lagrange equations for this functional have the schematic form<br />
<br />
<center><math>\Box A + \nabla (\nabla_{x,t} A) = [A, \nabla A] + [A, [A,A]]</math></center><br />
<br />
where <math>\nabla_{x,t} A = \partial_ a A^ a</math> is the spacetime divergence of <math>A</math>. A more succinct (but less tractable) formulation of this equation is<br />
<br />
<center><math>D_\alpha F^{\alpha \beta} = 0</math>.</center><br />
<br />
It is often convenient to split <math>A</math> into temporal and spatial components as <math>A = (A_0, A_i)</math>.<br />
<br />
As written, the Yang-Mills equation is under-determined because of the gauge invariance<br />
<br />
<center><math>A -> U^{-1} dU + U^{-1} A U</math></center><br />
<center><math>F -> U^{-1} F U</math></center><br />
<br />
in the equation, where U is an arbitrary function taking values in <math>G</math>. In order to correctly formulate a Cauchy problem, one must impose a further constraint on the gauge. There are three standard ones:<br />
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<center>[[Temporal gauge]]: <math>A^0 = 0</math></center><br />
<center>[[Coulomb gauge]]: <math>\partial_i A_i = 0</math></center><br />
<center>[[Lorenz gauge]]: <math>\nabla_{x,t} A = 0</math></center><br />
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There are also several other useful gauges, such as the Cronstrom gauge [[Cs1980]] centered around a point in spacetime.<br />
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The Lorentz gauge has the advantage of being invariant under conformal transformations, but it appears that the Yang-Mills equation is not well-behaved in this gauge for rough data. (For smooth data one can obtain local well-posedness in this gauge by energy estimates). The Coulomb gauge is the simplest to work with technically, and in this gauge the bilinear expression <math>[A, \nabla A]</math> acquires a null structure [[KlMa1995]] which allows for a satisfactory analysis of the equation. Unfortunately there are often difficulties in creating a global Coulomb gauge, and one often has to rely instead on local Coulomb gauges pieced together using finite speed of propagation; see [[KlMa1995]]. The Temporal gauge is fairly close to the Coulomb gauge, and one can develop a parallel theory for this gauge. The temporal gauge has the advantage of being easy to establish globally, but the null form structure is less obvious (one needs to partition the connection into divergence-free and curl-free components). See e.g. [[Ta2003]].<br />
<br />
In the Coulomb or Temporal gauges, one can create a model equation for the Yang-Mills system by ignoring cubic terms and any contribution from the "elliptic" portion of the gauge (<math>A_0</math> in the Coulomb gauge, or the curl-free portion of <math>A_i</math> in the Temporal gauge). The resulting model equation is<br />
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<center><math>\Box A = \nabla ^{-1} Q(A,A) + Q( \nabla ^{-1}A, A)</math></center><br />
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where <math>Q(A,A')</math> is some null form such as<br />
<br />
<center><math>Q(A,A') := \partial_i A \partial_j A' - \partial_j A \partial_i A'</math>.</center><br />
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The results known for the model equation are slightly better than those known for the actual Yang-Mills or Maxwell-Klein-Gordon equations.<br />
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The Yang-Mills equations come with a positive definite conserved Hamiltonian<br />
<br />
<center><math>\int |F_{0,i}|^2 + |F_{i,j}|^2 dx</math></center><br />
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which mostly controls the <math>H^1</math> norm of <math>A</math> and the <math>L^2</math> norm of <math>A_t</math>. However, there are some portions of the <math>H^1 \times L^2</math> norm which are not controlled by the Hamiltonian (in the Coulomb gauge, it is <math>\partial_t A_0</math>; in the Temporal gauge, it is the <math>H^1</math> norm of the curl-free part of <math>A_i</math>). This causes some technical difficulties in the global well-posedness theory.<br />
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The Yang-Mills equations can also be coupled with a g-valued scalar field <math>f</math>, with the Lagrangian functional of the form<br />
<br />
<center><math>\int F^{\alpha \beta} F_{\alpha \beta} + D_\alpha f \cdot D^\alpha f + V( f )</math></center><br />
<br />
where <math>D_\alpha := \partial_\alpha + [A_\alpha, .]</math> are covariant derivatives and <math>V</math> is some potential function (e.g. <math>V( f ) = | f |^{k+1})</math>. The corresponding Euler-Lagrange equations have the schematic form<br />
<br />
<center><math>\Box A + \nabla (\nabla_{x,t} A) = [A, \nabla A] + [A, [A,A]] + [ f , D f ], D_\alpha D^\alpha f = V'( f )</math></center><br />
<br />
and are generally known as the ''[[Yang-Mills-Higgs]] system of equations''. This system may be thought of as a Yang-Mills equation coupled with a semi-linear wave equation. [[MKG|The Maxwell-Klein-Gordon system]] is a special case of Yang-Mills-Higgs.<br />
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The theory of Yang-Mills connections is considerably more advanced in the elliptic case (when the Minkowski metric is replaced by a Riemannian one), especially in the critical case of four dimensions, but a discussion of this topic is beyond our expertise.<br />
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Attention has mostly focussed on the three and four dimensional cases; the one-dimensional case is trivial (e.g. in the temporal gauge it collapses to <math>A_{tt} = 0</math>). In higher dimensions n=5,7,9 singularities can develop from large smooth radial data [[CaSaTv1998]] (see also [[Biz-p]]). Numerics suggest this phenomenon is generic, and also one appears to have blowup also at the critical dimension [[BizTb2001]], [[Biz-p]].<br />
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The Yang-Mills equations can also be coupled with a spinor field. In the <math>U(1)</math> case this becomes the Maxwell-Dirac equation.<br />
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The Yang-Mills equations in dimension n have many formal similarities with the wave maps equation at dimension d-2 (see e.g. [[CaSaTv1998]] for a discussion).<br />
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====Yang-Mills on <math>R^2</math>====<br />
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* Scaling is <math>s_c = 0</math>.<br />
* One can use the method of descent and finite speed of propagation to infer R<sup>2</sup> results from the R^3 results. Thus, for instance, one has LWP for s > 3/4 in the temporal gauge and GWP in the temporal gauge for <math>s\geq 1</math>. These results are almost certainly non-optimal, however, and probably have much simpler proofs (for instance, one can obtain the LWP result from the general theory of DNLW without using any null form structure).<br />
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====Yang-Mills on R<sup>3</sup>====<br />
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* Scaling is s_c = 1/2.<br />
* LWP for s > 3/4 in the Temporal gauge if the norm is sufficiently small [[Ta2003]]. The main tools are bilinear estimates involving both <math>X^{s,\theta}</math> spaces and product Sobolev spaces.<br />
** Presumably the small data assumption can be removed, but the usual methods to do this fail because there are too many time derivatives in the non-linearity in the temporal gauge.<br />
** For <math>s \geq 1 </math>in the Temporal or Coulomb gauges LWP for large data was shown in [[KlMa1995]].<br />
** For s > 1 LWP for the Temporal, Coulomb, or Lorentz gauges follows from Strichartz estimates [[PoSi1993]].<br />
** For s > 3/2 LWP for the Temporal, Coulomb, or Lorentz gauges follows from energy estimates [[EaMc1982]].<br />
** There is a tentative conjecture that one in fact has ill-posedness in the energy class for the Lorentz gauge.<br />
** For the model equation LWP fails for s < 3/4 [[MaStz-p]]<br />
** The endpoint s = 1/2 looks extremely difficult, even for Besov space variants.<br />
* GWP is known for data with finite Hamiltonian (morally, this is for <math> s \geq 1 </math>) in the Coloumb or Temporal gauges [[KlMa1995]].<br />
** For smooth data this was proven in [[EaMc1982]].<br />
*** This result was extended to curved space in [[CcSa1997]]<br />
** It seems likely that one can improve this to something like s>7/8, in analogy with the [[MKG|theory for the Maxwell-Klein-Gordon equation]].<br />
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====MKG and Yang-Mills in R^4====<br />
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* Scaling is s_c = 1.<br />
* For the MKG equations in the [[Coulomb gauge]], LWP is known for s > 1 [[Sb-p5]]. This is still not known for Yang-Mills.<br />
** For the model equations this is in [[KlTt1999]]<br />
*** For general quadratic DNLW this is only known for s > 5/4 (e.g. by the estimates in [[FcKl2000]]). Strichartz estimates need s > 3/2 [[PoSi1993]], while energy estimates need s > 2.<br />
** The latter two results (Strichartz and energy) easily extend to the actual MKG and YM equations in all three standard gauges.<br />
* It is conjectured that one has global well-posedness results for small energy, but this is open.<br />
** For small smooth compactly supported data, one can obtain global existence from the [[QNLW|general theory of quasi-linear equations]].<br />
** For large data Yang-Mills, numerics suggest that blowup does occur, with the solution resembling a rescaled instanton at each time [[BizTb2001]], [[Biz-p]].<br />
*** Further numerics suggests that the radius of the instanton in fact decays like <math>C t / \sqrt(\log t)</math> [[BizOvSi-p]].<br />
** GWP for small <math>B^{1,1}</math> data (with an additional angular derivative of regularity) in the [[Lorentz gauge]] is in [[Stz-p2]].<br />
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====MKG and Yang-Mills in R^d, d>4====<br />
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* Scaling is s_c = d/2 - 1.<br />
* LWP is almost certainly true for MKG-CG for s > s_c by adapting the results in [[Sb-p5]]. The corresponding question for Yang-Mills is still open.<br />
** For the model equations one can probably achieve this by adapting the results in [[Tt1999]]<br />
* For dimensions <math>d\geq 6</math>, GWP for small H^{d/2} data in MKG-CG is in [[RoTa-p]]. The corresponding question for Yang-Mills is still open, but a Besov result follows (in the Lorentz gauge) from [[Stz-p3]].<br />
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====Yang-Mills-Higgs on R<sup>3</sup>====<br />
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* Suppose the potential energy V( f ) behaves like <math>| f |^{p+1}</math> (i.e. defocussing p^th power non-linearity). When <math>p\leq 3</math>, the Higgs term is negligible, and the theory mimics that of the ordinary Yang-Mills equation. The most interesting case is p=5, since the Higgs component is then H^1-critical.<br />
* There is no perfect scale-invariance to this equation (unless p=3); the critical regularity is <math>s_c = max(1/2, 3/2 - 2/(p-1))</math>.<br />
* In the sub-critical case p<5 one has GWP for smooth data [[EaMc1982]], [[GiVl1982b]]. This can be pushed to H^1 by the results in [[Ke1997]]. The local theory might be pushed even further.<br />
* In the critical case p=5 one has GWP for <math>s \geq 1</math> [[Ke1997]].<br />
* In the supercritical case p>5 one probably has LWP for <math>s \geq s_c </math>(because this is true for the Yang-Mills and NLW equations separately), but this has not been rigorously shown. No large data global results are known, but this is also true for the supposedly simpler supercritical NLW. It seems possible however that one could obtain small-data GWP results.<br />
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[[Category:Equations]] [[Category:Geometry]] [[Category:Wave]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Yang-Mills_Equations&diff=4772Yang-Mills Equations2007-05-05T15:44:32Z<p>Oleg Alexandrov: Yang-Mills Equations moved to Yang-Mills equations: Consistency with other equation pages</p>
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<div>#REDIRECT [[Yang-Mills equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Talk:Yang-Mills_equations&diff=4773Talk:Yang-Mills equations2007-05-05T15:44:32Z<p>Oleg Alexandrov: Talk:Yang-Mills Equations moved to Talk:Yang-Mills equations: Consistency with other equation pages</p>
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<div>This page was derived from an earlier version which had corrections and suggestions from Jacob Sterbenz.</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=Talk:Yang-Mills_Equations&diff=4774Talk:Yang-Mills Equations2007-05-05T15:44:32Z<p>Oleg Alexandrov: Talk:Yang-Mills Equations moved to Talk:Yang-Mills equations: Consistency with other equation pages</p>
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<div>#REDIRECT [[Talk:Yang-Mills equations]]</div>Oleg Alexandrovhttps://dispersivewiki.org/DispersiveWiki/index.php?title=User:Oleg_Alexandrov&diff=4770User:Oleg Alexandrov2007-05-05T15:43:49Z<p>Oleg Alexandrov: kill redlink</p>
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<div>I contribute primarily to Wikipedia, see [http://en.wikipedia.org/wiki/User:Oleg_Alexandrov my page there].</div>Oleg Alexandrov