Loss of regularity for supercritical nonlinear Schrödinger equations

Abstract

Abstract. We consider the nonlinear Schrödinger equation with defocusing, smooth, nonlinearity. Below the critical Sobolev regularity, it is known that the Cauchy problem is ill-posed. We show that this is even worse, namely that there is a loss of regularity, in the spirit of the result due to G. Lebeau in the case of the wave equation. As a consequence, the Cauchy problem for energy-supercritical equations is not well-posed in the sense of Hadamard. We reduce the problem to a supercritical WKB analysis. For super-cubic, smooth nonlinearity, this analysis is new, and relies on the introduction of a modulated energy functionalà la Brenier.

Geometric Optics and Boundary Layers for Nonlinear-Schrödinger Equations

“…In this section, we consider a smooth WKB approximate solution Ψ a = a ε exp i ϕ ε ε of (1) such that…”

“…We are interested in the semiclassical limit ε → 0. The nonlinear Schrödinger equation (1) appears, for instance, in optics, and also as a model for Bose-Einstein condensates, with f (ρ) = ρ − 1, and the equation is termed Gross-Pitaevskii equation, or also with f (ρ) = ρ 2 (see [13]). Some more complicated nonlinearities are also used especially in low dimensions, see [12].…”

“…The additional difficulty is that for such nonlinearities, the system (7) is only weakly hyperbolic at a = 0 and in particular the symmetrizer S is not anymore positive definite at a = 0. In more recent works, see [19], [14], [1] it was proven that for every weak solution of (1) with f (ρ) = ρ − 1 or f (ρ) = ρ, the limits as ε → 0…”

“…Consequently, our result can also be applied to nonlinearities like f (ρ) = ρ σ 1 − ρ σ 2 for every σ 2 > σ 1 provided |a 0 | 2 ≤ β < 1. Note that when σ 2 is too large, the classical global existence result of weak solutions (see [8]) for (1) is not valid and hence it does not seem possible to use the modulated energy method of [1], [14] to investigate the semi-classical limit.…”

“…The limit (8) was proven with (ρ, u) the solution of the compressible Euler equation with boundary condition u · n /∂Ω = 0, n being the normal to the boundary. Note that the result of [14] is restricted to the two-dimensional case only in order to have a global solution in the energy space of (1). By using more recent results on the Cauchy problem, [3], one can also get the result in the three-dimensional case at least when u ∞ = 0.…”