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Decay of weak solutions to the Davey-Stewartson systems. (English) Zbl 0808.35142

One considers the initial value problem for the Davey-Stewartson system of elliptic-hyperbolic type which appears quite naturally as the envelope equations in the study of water waves. The aim of the paper is to establish the global existence of weak solutions which satisfy a structural inequality referred to as pseudo-conformal inequality, and to obtain decay estimates of these weak solutions.

The first introductory section states the main result, section 2 proposes a suitable approximate equation which regularizes the DS system and constructs strong solutions to the regularized problem. Section 3 provides a priori estimates, and the last section displays the proof of the main theorem.

MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
35D05Existence of generalized solutions of PDE (MSC2000)
35B40Asymptotic behavior of solutions of PDE