Well-posedness and scattering for the Zakharov system in four dimensions

The Cauchy problem for the Zakharov system in four dimensions is considered. Some new well-posedness results are obtained. For small initial data, global well-posedness and scattering results are proved, including the case of initial data in the energy space. None of these results is restricted to radially symmetric data.

[1]  F. Merle.  Blow-up results of viriel type for Zakharov equations , 1996 .

[2]  Nobu Kishimoto.  Local well-posedness for the Zakharov system on multidimensional torus , 2011, 1109.3527.

[3]  F. Merle,et al.  Concentration properties of blow-up solutions and instability results for Zakharov equation in dimension two. Part II , 1994 .

[4]  Hideo Takaoka,et al.  Well-posedness for the Zakharov system with the periodic boundary condition , 1999, Differential and Integral Equations.

[5]  F. Merle,et al.  Existence of self-similar blow-up solutions for Zakhrov equation in dimension two. Part I , 1994 .

[6]  T. Ozawa,et al.  Global existence and asymptotic behavior of solutions for the Zakharov equations in three space dimensions , 1993 .

[7]  K. Nakanishi,et al.  Energy convergence for singular limits of Zakharov type systems , 2008 .

[8]  Catherine Sulem,et al.  The nonlinear Schrödinger equation , 2012 .

[9]  K. Tsugawa,et al.  Scattering and well-posedness for the Zakharov system at a critical space in four and more spatial dimensions , 2015, Differential and Integral Equations.

[10]  Small energy scattering for the Klein-Gordon-Zakharov system with radial symmetry , 2012, 1208.1657.

[11]  V. Zakharov.  Collapse of Langmuir Waves , 1972 .

[12]  Zihua Guo.  Sharp spherically averaged Strichartz estimates for the Schrödinger equation , 2014, 1406.2525.

[13]  Zihua Guo,et al.  Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations , 2010, 1007.4299.

[14]  B. Dodson.  Global well - posedness and scattering for the focusing, energy - critical nonlinear Schr\"odinger problem in dimension $d = 4$ for initial data below a ground state threshold , 2014, 1409.1950.

[15]  A. Shimomura.  SCATTERING THEORY FOR ZAKHAROV EQUATIONS IN THREE-DIMENSIONAL SPACE WITH LARGE DATA , 2004 .

[16]  T. Tao,et al.  Endpoint Strichartz estimates , 1998 .

[17]  K. Nakanishi,et al.  Generalized Strichartz Estimates and Scattering for 3D Zakharov System , 2013, 1305.2990.

[18]  K. Nakanishi,et al.  Small Energy Scattering for the Zakharov System with Radial Symmetry , 2012, 1203.3959.

[19]  J. Ginibre,et al.  On the Cauchy Problem for the Zakharov System , 1997 .

[20]  Michael I. Weinstein,et al.  The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence , 1986 .

[21]  J. Bourgain,et al.  On wellposedness of the Zakharov system , 1996 .

[22]  J. Shatah,et al.  Scattering for the Zakharov System in 3 Dimensions , 2012, 1206.3473.

[23]  C. Sulem,et al.  The nonlinear Schrödinger equation : self-focusing and wave collapse , 2004 .

[24]  K. Nakanishi,et al.  Global dynamics below the ground state energy for the Zakharov system in the 3D radial case , 2012, 1206.2457.

[25]  Carlos E. Kenig,et al.  Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case , 2006 .

[26]  T. Ozawa,et al.  The nonlinear Schrödinger limit and the initial layer of the Zakharov equations , 1992, Differential and Integral Equations.

[27]  J. Holmer,et al.  On the 2D Zakharov system with L2 Schrödinger data , 2008, 0811.3047.

[28]  S. Herr,et al.  Convolutions of singular measures and applications to the Zakharov system , 2010, 1009.3250.

[29]  K. Nakanishi,et al.  Global Dynamics below the Ground State Energy for the Klein-Gordon-Zakharov System in the 3D Radial Case , 2013, 1304.3570.

[30]  R. Killip,et al.  The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher , 2008, 0804.1018.

[31]  J. Holmer,et al.  On the 2 d Zakharov system with L 2 Schrödinger data , 2009 .