On the initial value problem for the Zakharov equations. (English) Zbl 0875.35111

Ozawa, Tohru; Tsutsumi, Yoshio

Mat. Contemp. 3, 149-164 (1992).

Summary: The following three problems for the Zakharov equations \[ i{\partial E\over \partial t}+\Delta E=nE,\quad {\partial^2n\over \partial t^2}-\Delta n=\Delta|E|^2,\quad t>0,\quad x\in\mathbb{R}^N, \]
\[ E(0,x)=E_0(x),\quad n(0,x)=n_0(x),\quad {\partial\over \partial t} n(0,x)=n_1(0,x), \] where \(E(t,x)\) is a function from \(\mathbb{R}^+_t\times\mathbb{R}^N_x\) to \(\mathbb{C}^N\), \(n(t,x)\) is a function from \(\mathbb{R}^+_t\times\mathbb{R}^N_x\) to \(\mathbb{R}\) and \(1\leq N\leq 3\) are considered: (i) local solvability, (ii) smoothing effect of solutions, (iii) the nonlinear Schrödinger limit.
The authors present the results concerning the above three problems, which have been obtained in the papers [Publ. Res. Inst. Math. Sci. 28, No. 3, 329-361 (1992; Zbl 0842.35116) and in Differ. Integral Equa. 5, No. 4, 721-745 (1992; Zbl 0754.35163)], and the proofs of those results are illustrated.

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35B65	Smoothness and regularity of solutions to PDEs
35A05	General existence and uniqueness theorems (PDE) (MSC2000)

Keywords:

Zakharov equations; solvability; smoothing effect of solutions; nonlinear Schrödinger limit

Citations:

Zbl 0842.35116; Zbl 0754.35163