On the Global Regularity for a Wave-Klein–Gordon Coupled System
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摘要
In this paper we consider a coupled Wave-Klein–Gordon system in 3 D, and prove global regularity and modified scattering for small and smooth initial data with suitable decay at infinity. This system was derived by Wang and LeFloch–Ma as a simplified model for the global nonlinear stability of the Minkowski space-time for self-gravitating massive fields.
In this paper we consider a coupled Wave-Klein–Gordon system in 3 D, and prove global regularity and modified scattering for small and smooth initial data with suitable decay at infinity. This system was derived by Wang and LeFloch–Ma as a simplified model for the global nonlinear stability of the Minkowski space-time for self-gravitating massive fields.
In this paper we consider a coupled Wave-Klein–Gordon system in 3 D, and prove global regularity and modified scattering for small and smooth initial data with suitable decay at infinity. This system was derived by Wang and LeFloch–Ma as a simplified model for the global nonlinear stability of the Minkowski space-time for self-gravitating massive fields.
引文
[1]Alinhac,S.:The null condition for quasilinear wave equations in two space dimensions I.Invent.Math.,145,597-618(2001)
[2]Alinhac,S.:The null condition for quasilinear wave equations in two space dimensions.II.Amer.J.Math.,123,1071-1101(2001)
[3]Bieri,L.,Zipser,N.:Extensions of the stability theorem of the Minkowski space in general relativity.AMS/IP Studies in Advanced Mathematics,45.American Mathematical Society,Providence,RI;International Press,Cambridge,MA,2009
[4]Christodoulou,D.:Global solutions of nonlinear hyperbolic equations for small initial data.Comm.Pure Appl.Math.,39,267-282(1986)
[5]Christodoulou,D.,Klainerman,S.:The global nonlinear stability of the Minkowski space.Princeton Mathematical Series,41.Princeton University Press,Princeton,NJ,1993
[6]Delort,J.M.:Existence globale et comportement asymptotique pour l’′equation de Klein-Gordon quasilin′eairèa donn′ees petites en dimension 1.Ann.Sci.′Ecole Norm.Sup.,34,1-61(2001)
[7]Delort,J.M.,Fang,D.:Almost global existence for solutions of semilinear Klein-Gordon equations with small weakly decaying Cauchy data.Comm.Partial Differential Equations,25,2119-2169(2000)
[8]Delort,J.M.,Fang,D.,Xue,R.:Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions.J.Funct.Anal.,211 288-323(2004)
[9]Deng,Y.:Multispeed Klein-Gordon systems in dimension three,Int.Math.Res.Not.IMRN,rnx038
[10]Deng,Y.,Ionescu,A.D.,Pausader,B.:The Euler-Maxwell system for electrons:global solutions in 2D.Arch.Ration.Mech.Anal.,225(2),771-871(2017)
[11]Deng,Y.,Ionescu,A.D.,Pausader,B.,et al.:Global solutions of the gravity-capillary water wave system in 3 dimensions.Acta Math.,219(2),213-402(2017)
[12]Friedrich,H.:On the existence of n-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure.Commun.Math.Phys.,107,587-609(1986)
[13]Georgiev,V.:Global solution of the system of wave and Klein-Gordon equations.Math.Z.,203,683-698(1990)
[14]Germain,P.,Masmoudi,N.:Global existence for the Euler-Maxwell system.Ann.Sci.′Ec.Norm.Sup′er.,47,469-503(2014)
[15]Germain,P.,Masmoudi,N.,Shatah,J.:Global solutions for 3D quadratic Schrodinger equations.Int.Math.Res.Not.,414-432(2009)
[16]Germain,P.,Masmoudi,N.,Shatah,J.:Global solutions for the gravity water waves equation in dimension3.Ann.of Math.(2),175,691-754(2012)
[17]Guo,Y.,Ionescu,A.D.,Pausader,B.:Global solutions of the Euler-Maxwell two-fluid system in 3D.Ann.of Math.(2),183,377-498(2016)
[18]Guo,Y.,Pausader,B.:Global smooth ion dynamics in the Euler-Poisson system.Comm.Math.Phys.,303,89-125(2011)
[19]Gustafson,S.,Nakanishi,K.,Tsai,T.P.:Scattering theory for the Gross-Pitaevskii equation in three dimensions.Commun.Contemp.Math.,11,657-707(2009)
[20]Ionescu,A.D.,Pausader,B.:The Euler-Poisson system in 2D:global stability of the constant equilibrium solution.Int.Math.Res.Not.,2013,761-826(2013)
[21]Ionescu,A.D.,Pausader,B.:Global solutions of quasilinear systems of Klein-Gordon equations in 3D.J.Eur.Math.Soc.(JEMS),16,2355-2431(2014)
[22]Ionescu,A.D.,Pusateri,F.:Nonlinear fractional Schrodinger equations in one dimension.J.Funct.Anal.,266,139-176(2014)
[23]Ionescu,A.D.,Pusateri,F.:Global solutions for the gravity water waves system in 2d.Invent.Math.,199,653-804(2015)
[24]John,F.:Blow-up of solutions of nonlinear wave equations in three space dimensions.Manuscripta Math.,28,235-268(1979)
[25]John,F.,Klainerman,S.:Almost global existence to nonlinear wave equations in three space dimensions.Comm.Pure Appl.Math.,37,443-455(1984)
[26]Katayama,S.:Global existence for coupled systems of nonlinear wave and Klein-Gordon equations in three space dimensions.Math.Z.,270,487-513(2012)
[27]Kato,T.:The Cauchy problem for quasi-linear symmetric hyperbolic systems.Arch.Rational Mech.Anal.,58,181-205(1975)
[28]Kato,J.,Pusateri,F.:A new proof of long range scattering for critical nonlinear Schrodinger equations.Differential and Integral Equations,24,923-940(2011)
[29]Klainerman,S.:Long time behaviour of solutions to nonlinear wave equations.Proceedings of the International Congress of Mathematicians,Vol.1,2(Warsaw,1983),1209-1215,PWN,Warsaw,(1984)
[30]Klainerman,S.:Uniform decay estimates and the Lorentz invariance of the classical wave equation.Comm.Pure Appl.Math.,38,321-332(1985)
[31]Klainerman,S.:Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions.Comm.Pure Appl.Math.,38,631-641(1985)
[32]Klainerman,S.:The null condition and global existence to nonlinear wave equations,Nonlinear systems of partial differential equations in applied mathematics,Part 1(Santa Fe,N.M.,1984),293-326,Lectures in Appl.Math.,23,Amer.Math.Soc.,Providence,RI,1986
[33]Klainerman,S.,Nicolo,F.:The evolution problem in general relativity.Progress in Mathematical Physics,25.Birkhauser Boston,Inc.,Boston,MA,2003
[34]LeFloch,P.G.,Ma,Y.:The hyperboloidal foliation method,Series in Applied and Computational Mathematics,2.World Scientific Publishing Co.Pte.Ltd.,Hackensack,NJ,2014
[35]LeFloch,P.G.,Ma,Y.:The global nonlinear stability of Minkowski space for self-gravitating massive fields.The Wave-Klein-Gordon model.Comm.Math.Phys.,346,603-665(2016)
[36]LeFloch,P.G.,Ma,Y.:The global nonlinear stability of Minkowski space for self-gravitating massive fields,Series in Applied and Computational Mathematics,3.World Scientific Publishing Co.Pte.Ltd.,Hackensack,NJ,2017
[37]Lindblad,H.,Rodnianski,I.:Global existence for the Einstein vacuum equations in wave coordinates.Commun.Math.Phys.,256,43-110(2005)
[38]Lindblad,H.,Rodnianski,I.:The global stability of Minkowski space-time in harmonic gauge.Ann.of Math.(2),171,1401-1477(2010)
[39]Shatah,J.:Normal forms and quadratic nonlinear Klein-Gordon equations.Comm.Pure Appl.Math.,38,685-696(1985)
[40]Simon,J.:A wave operator for a nonlinear Klein-Gordon equation.Lett.Math.Phys.,7,387-398(1983)
[41]Speck,J.:The global stability of the Minkowski spacetime solution to the Einstein-nonlinear system in wave coordinates.Anal.PDE,7,771-901(2014)
[42]Wang,Q.:Global Existence for the Einstein equations with massive scalar fields,Lecture at the workshop Mathematical Problems in General Relativity,January 19-23,2015,http://scgp.stonybrook.edu/videoportal/video.php?id=1420seminar
[43]Wang,Q.:An intrinsic hyperboloid approach for Einstein Klein-Gordon equations,preprint(2016)arXiv:1607.01466
1)The linear scattering here is likely due to the very simple semilinear equation for u.In the case of the full Einstein-Klein-Gordon system,one expects modified scattering for the metric components as well.
2)In general,one should think of the Z norm as being connected to the location and the shape of the set of space-time resonances of the system,as in[10,11,17,21].In our case here there are no nontrivial space-time resonances and the construction is simpler.
3)In fact,we will not prove optimal decay for the full function v,but we will decompose v=v∞+v2 where v∞has optimal pointwise decay and v2 is suitably small in L2.
4)One should think of(n1,n2,n)=(0,N1,N1)as the worst case.In this case the only available bounds for the profiles Ukg,ι2L2are the L2bounds in(4.2).
5)Here it is important thatμ=(kg,+), so the phase is nonresonant. The nonlinear correction(7.4)was done precisely to weaken the corresponding resonant contribution of the profile Vkg,+.
[2]Alinhac,S.:The null condition for quasilinear wave equations in two space dimensions.II.Amer.J.Math.,123,1071-1101(2001)
[3]Bieri,L.,Zipser,N.:Extensions of the stability theorem of the Minkowski space in general relativity.AMS/IP Studies in Advanced Mathematics,45.American Mathematical Society,Providence,RI;International Press,Cambridge,MA,2009
[4]Christodoulou,D.:Global solutions of nonlinear hyperbolic equations for small initial data.Comm.Pure Appl.Math.,39,267-282(1986)
[5]Christodoulou,D.,Klainerman,S.:The global nonlinear stability of the Minkowski space.Princeton Mathematical Series,41.Princeton University Press,Princeton,NJ,1993
[6]Delort,J.M.:Existence globale et comportement asymptotique pour l’′equation de Klein-Gordon quasilin′eairèa donn′ees petites en dimension 1.Ann.Sci.′Ecole Norm.Sup.,34,1-61(2001)
[7]Delort,J.M.,Fang,D.:Almost global existence for solutions of semilinear Klein-Gordon equations with small weakly decaying Cauchy data.Comm.Partial Differential Equations,25,2119-2169(2000)
[8]Delort,J.M.,Fang,D.,Xue,R.:Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions.J.Funct.Anal.,211 288-323(2004)
[9]Deng,Y.:Multispeed Klein-Gordon systems in dimension three,Int.Math.Res.Not.IMRN,rnx038
[10]Deng,Y.,Ionescu,A.D.,Pausader,B.:The Euler-Maxwell system for electrons:global solutions in 2D.Arch.Ration.Mech.Anal.,225(2),771-871(2017)
[11]Deng,Y.,Ionescu,A.D.,Pausader,B.,et al.:Global solutions of the gravity-capillary water wave system in 3 dimensions.Acta Math.,219(2),213-402(2017)
[12]Friedrich,H.:On the existence of n-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure.Commun.Math.Phys.,107,587-609(1986)
[13]Georgiev,V.:Global solution of the system of wave and Klein-Gordon equations.Math.Z.,203,683-698(1990)
[14]Germain,P.,Masmoudi,N.:Global existence for the Euler-Maxwell system.Ann.Sci.′Ec.Norm.Sup′er.,47,469-503(2014)
[15]Germain,P.,Masmoudi,N.,Shatah,J.:Global solutions for 3D quadratic Schrodinger equations.Int.Math.Res.Not.,414-432(2009)
[16]Germain,P.,Masmoudi,N.,Shatah,J.:Global solutions for the gravity water waves equation in dimension3.Ann.of Math.(2),175,691-754(2012)
[17]Guo,Y.,Ionescu,A.D.,Pausader,B.:Global solutions of the Euler-Maxwell two-fluid system in 3D.Ann.of Math.(2),183,377-498(2016)
[18]Guo,Y.,Pausader,B.:Global smooth ion dynamics in the Euler-Poisson system.Comm.Math.Phys.,303,89-125(2011)
[19]Gustafson,S.,Nakanishi,K.,Tsai,T.P.:Scattering theory for the Gross-Pitaevskii equation in three dimensions.Commun.Contemp.Math.,11,657-707(2009)
[20]Ionescu,A.D.,Pausader,B.:The Euler-Poisson system in 2D:global stability of the constant equilibrium solution.Int.Math.Res.Not.,2013,761-826(2013)
[21]Ionescu,A.D.,Pausader,B.:Global solutions of quasilinear systems of Klein-Gordon equations in 3D.J.Eur.Math.Soc.(JEMS),16,2355-2431(2014)
[22]Ionescu,A.D.,Pusateri,F.:Nonlinear fractional Schrodinger equations in one dimension.J.Funct.Anal.,266,139-176(2014)
[23]Ionescu,A.D.,Pusateri,F.:Global solutions for the gravity water waves system in 2d.Invent.Math.,199,653-804(2015)
[24]John,F.:Blow-up of solutions of nonlinear wave equations in three space dimensions.Manuscripta Math.,28,235-268(1979)
[25]John,F.,Klainerman,S.:Almost global existence to nonlinear wave equations in three space dimensions.Comm.Pure Appl.Math.,37,443-455(1984)
[26]Katayama,S.:Global existence for coupled systems of nonlinear wave and Klein-Gordon equations in three space dimensions.Math.Z.,270,487-513(2012)
[27]Kato,T.:The Cauchy problem for quasi-linear symmetric hyperbolic systems.Arch.Rational Mech.Anal.,58,181-205(1975)
[28]Kato,J.,Pusateri,F.:A new proof of long range scattering for critical nonlinear Schrodinger equations.Differential and Integral Equations,24,923-940(2011)
[29]Klainerman,S.:Long time behaviour of solutions to nonlinear wave equations.Proceedings of the International Congress of Mathematicians,Vol.1,2(Warsaw,1983),1209-1215,PWN,Warsaw,(1984)
[30]Klainerman,S.:Uniform decay estimates and the Lorentz invariance of the classical wave equation.Comm.Pure Appl.Math.,38,321-332(1985)
[31]Klainerman,S.:Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions.Comm.Pure Appl.Math.,38,631-641(1985)
[32]Klainerman,S.:The null condition and global existence to nonlinear wave equations,Nonlinear systems of partial differential equations in applied mathematics,Part 1(Santa Fe,N.M.,1984),293-326,Lectures in Appl.Math.,23,Amer.Math.Soc.,Providence,RI,1986
[33]Klainerman,S.,Nicolo,F.:The evolution problem in general relativity.Progress in Mathematical Physics,25.Birkhauser Boston,Inc.,Boston,MA,2003
[34]LeFloch,P.G.,Ma,Y.:The hyperboloidal foliation method,Series in Applied and Computational Mathematics,2.World Scientific Publishing Co.Pte.Ltd.,Hackensack,NJ,2014
[35]LeFloch,P.G.,Ma,Y.:The global nonlinear stability of Minkowski space for self-gravitating massive fields.The Wave-Klein-Gordon model.Comm.Math.Phys.,346,603-665(2016)
[36]LeFloch,P.G.,Ma,Y.:The global nonlinear stability of Minkowski space for self-gravitating massive fields,Series in Applied and Computational Mathematics,3.World Scientific Publishing Co.Pte.Ltd.,Hackensack,NJ,2017
[37]Lindblad,H.,Rodnianski,I.:Global existence for the Einstein vacuum equations in wave coordinates.Commun.Math.Phys.,256,43-110(2005)
[38]Lindblad,H.,Rodnianski,I.:The global stability of Minkowski space-time in harmonic gauge.Ann.of Math.(2),171,1401-1477(2010)
[39]Shatah,J.:Normal forms and quadratic nonlinear Klein-Gordon equations.Comm.Pure Appl.Math.,38,685-696(1985)
[40]Simon,J.:A wave operator for a nonlinear Klein-Gordon equation.Lett.Math.Phys.,7,387-398(1983)
[41]Speck,J.:The global stability of the Minkowski spacetime solution to the Einstein-nonlinear system in wave coordinates.Anal.PDE,7,771-901(2014)
[42]Wang,Q.:Global Existence for the Einstein equations with massive scalar fields,Lecture at the workshop Mathematical Problems in General Relativity,January 19-23,2015,http://scgp.stonybrook.edu/videoportal/video.php?id=1420seminar
[43]Wang,Q.:An intrinsic hyperboloid approach for Einstein Klein-Gordon equations,preprint(2016)arXiv:1607.01466
1)The linear scattering here is likely due to the very simple semilinear equation for u.In the case of the full Einstein-Klein-Gordon system,one expects modified scattering for the metric components as well.
2)In general,one should think of the Z norm as being connected to the location and the shape of the set of space-time resonances of the system,as in[10,11,17,21].In our case here there are no nontrivial space-time resonances and the construction is simpler.
3)In fact,we will not prove optimal decay for the full function v,but we will decompose v=v∞+v2 where v∞has optimal pointwise decay and v2 is suitably small in L2.
4)One should think of(n1,n2,n)=(0,N1,N1)as the worst case.In this case the only available bounds for the profiles Ukg,ι2L2are the L2bounds in(4.2).
5)Here it is important thatμ=(kg,+), so the phase is nonresonant. The nonlinear correction(7.4)was done precisely to weaken the corresponding resonant contribution of the profile Vkg,+.