Well-posedness for a class of generalized Zakharov system

Abstract

In this paper, we study the existence and uniqueness of the global smooth solution for the initial value problem of generalized Zakharov equations in dimension two. By means of a priori integral estimates and Galerkin method, we first construct the existence of global solution with some conditions. Furthermore, we prove that the global solution is unique.

Approximation to the global solution of generalized Zakharov equations in R 2 $\mathbf{R}^{2}$

“…We know that if , , , , and with small, there exists a unique global solution for system ( 3 )–( 5 ) satisfying [ 17 ] Moreover, if , , , , and small, system ( 1 ), 2 , ( 5 ) has a unique global solution satisfying [ 20 ] …”

“…In [ 17 ], You studied the following generalized Zakharov system in space dimension two, and established the global existence for Cauchy problem. …”

Global well-posedness of strong solution for the three dimensional dynamic Cahn-Hilliard-Stokes model

“…The following estimates are valid. 25) in which B 1 and B 4 are time independent and B 2 , B 3 , and B 5 are time dependent constants.…”

Schrödinger-improved Boussinesq system in two space dimensions

“…In spite of a large literature on (S-iB) and (Z) in the whole space-time R×R n (see [3,[10][11][12]34,[43][44][45][46] for (S-iB) and [1,2,4,5,13,[16][17][18][19][20][24][25][26]28,29,35,36,[39][40][41][42] for (Z)), there are few papers treating those systems in R×Ω (see [1,33,[48][49][50] for (Z)). A major reason consists in the lack of the Fourier transform, by which Strichartz estimates, Bourgain's method, and I-method are available.…”