Asymptotic Smoothing Effect of Solutions to Davey–Stewartson Systems on the Whole Plane

Abstract

This paper discusses the Cauchy problem of elliptic-elliptic-type Davey–Stewartson systems with zero-order dissipation on ℝ2. Making use of the Fourier spectral projector, together with a long time comparison between the solutions to the Davey–Stewartson systems and to an auxiliary problem, we prove that the global attractor in H 1(ℝ2) for the addressed Davey–Stewartson systems is a compact subset of H 2(ℝ2), and thus reveal the asymptotic smoothing effect of the solutions for the systems.