Well-posedness For Several Classes Of Generalized Zakharov Equation

Abstract:This thesis deals with the well-posedness for several classes of generalized Zakharov equations by means of the Gelerkin method.In Chapter1, we introduce the background of our research, the context of this thesis and some preliminaries.Chapter2concerns with the well-posedness for modified Zakharov equations. First, in view of a priori estimates and Galerkin method, the existence and uniqueness of golbal smooth solutions for the ini-tial boundary value problem of the modified Zakharov equations have been obtained. Next, the existence and uniqueness of golbal smooth solutions for the Cauchy problem of the modified Zakharov equations in two-dimensions are proved. Then we study the asymptotic behavior of the equations of Langmuir turbulence when H goes to zero. One expects that the sequence of the equations of Langmuir turbulence "converges" to the Zakharov equations. A weak convergence result and a strong convergence result have been obtained.Chapter3studies the well-posedness for two classes of generalized modified Zakharov equations. firstly, we have proved the existence and uniqueness of golbal smooth solutions for the Cauchy problem of the generalized modified Zakharov equations in two-dimensions by means of the Gelerkin method. Secondly, the existence and uniqueness of golbal smooth solutions for the initial boundary value problem of the generalized modified Zakharov equations in two-dimensions are built up.Chapter4is devoted to the well-posedness for three classes of gen- eralized Zakharov equations. First of all, under some conditions the existence and uniqueness of golbal smooth solutions for the Cauchy problem of the generalized Zakharov equations in two-dimensions are established via Galerkin method and a priori estimates. Next, it was shown that for small enough initial conditions, the existence and u-niqueness of golbal smooth solutions for the initial boundary value problem of the generalized Zakharov equations in two-dimensions are established. Finally, the existence of golbal generalized solutions for the initial boundary value problem of the generalized Zakharov equa-tions are given under some conditions.