Existence and smoothing effect of solutions for the Zakharov equations. (English) Zbl 0842.35116

Ozawa, Tohru; Tsutsumi, Yoshio

Publ. Res. Inst. Math. Sci. 28, No. 3, 329-361 (1992).

The paper under review concerns the Zakharov equations \[ (1) \quad i {\partial E \over \partial t} + \Delta E = nE, \qquad (2) \quad {\partial^2n \over \partial t^2} - \Delta n = \Delta |E |^2,\;t > 0,\;x \in \mathbb{R}^N \] with initial data \[ E(0,x) = E_0 (x), \quad n(0,x) = n_0 (x), \quad {\partial \over \partial t} n(0,x) = n_1 (0,x). \tag{3} \] Here, \(E\) is a function from \(\mathbb{R}^+_t \times \mathbb{R}^N_x\) to \(\mathbb{C}^N\), \(n\) is a function from \(\mathbb{R}^+_t \times \mathbb{R}^N_x \) to \(\mathbb{R}\) and \(1 \leq N \leq 3\). The system of equations (1)–(3) describes the long wave Langmuir turbulence in a plasma. A certain limiting case of this system is related to the nonlinear Schrödinger equation \[ i {\partial E \over \partial t} + \Delta E = - |E |^2 E. \tag{4} \] It is well known that equation (4) possesses nice smoothing properties, and it is natural to conjecture whether some of these properties are also shared by the system (1)–(3). The authors study the local solvability and the smoothing effect of the (1)–(3). They establish local existence and uniqueness of strong solutions in certain Sobolev and weighted Sobolev spaces. Their results include the case of initial data \((E_0, n_0, n_1) \in H^2 \oplus H^1 \oplus L^2\). Concerning the study of the smoothing properties, the authors prove a nice result that clarifies the similarities and differences between the system (1)–(3) and the nonlinear Schrödinger equation (4).

Reviewer: J.P.Zubelli (Santa Cruz)

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
76X05	Ionized gas flow in electromagnetic fields; plasmic flow

Keywords:

Zakharov equations; nonlinear Schrödinger type equations; local existence; uniqueness