On the growth of high Sobolev norms for certain one-dimensional Hamiltonian PDEs

2015/01/01 by Joseph Thirouin, Thirouin, Joseph
Mathematics · #Advanced Mathematical Physics Problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1506.04181

Abstract

This paper is devoted to the study of large time bounds for the Sobolev norms of the solutions of the following fractional cubic Schrödinger equation on the torus :i ∂_t u = |D|^αu+|u|² u, u(0, ⋅)=u_0,where α is a real parameter. We show that, apart from the case α= 1, which corresponds to a half-wave equation with no dispersive property at all, solutions of this equation grow at a polynomial rate at most. We also address the case of the cubic and quadratic half-wave equations.

Citations

• Bilinear eigenfunction estimates and the nonlinear Schr�dinger equation on surfaces

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