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On wellposedness of the Zakharov system. (English) Zbl 0909.35125
Consider the Zakharov system \[ iu_t+\Delta u= nu,\quad\nabla n= \partial_{tt}u- \Delta n=\Delta(| u|^2)\tag{\(*\)} \] with space variable \(x\in \mathbb{R}^n\). First for \(n= 2\), the authors construct local solutions applying the contraction principle in spaces introduced by the first author [Geom. Funct. Anal. 3, 107-156 (1993; Zbl 0787.35097)] for nonlinear Schrödinger equations. Estimates on the nonlinear term are obtained by direct analysis of the Fourier transform, closely related to the first author [Duke Math. J. 76, 175-202 (1994; Zbl 0821.35120)], and R. S. Strichartz’s inequality for the linear Schrödinger equations [Duke Math. J. 44, 705-714 (1977; Zbl 0372.35001)].
For \(n=2\) global existence is established in [C. Sulem and P. L. Sulem, C. R. Acad. Sci., Paris, Sér. A 289, 173-176 (1979; Zbl 0431.35077)]. However the method used here permit to deal with a larger class of equations (for instance, considering Zakharov systems with more general nonlinearity, for both \(n= 1\) and \(n=2\)) and to prove global existence of classical solutions provided there is no blowup in the energy norm (in particular, for small data).
Second for \(n=3\), the authors show that the following is equally valid for the system \((*)\): Smooth initial data yield smooth solutions until there is a blowup in the energy norm. In particular, for \(\| u(0)\|_{H^1}\) small, they exist globally in time. This result solves the Cauchy problem for classical solutions for \(n= 3\), left open so far.
Reviewer: A.Tsutsumi (Suita)

35Q55 NLS equations (nonlinear Schrödinger equations)
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