Growth of Sobolev norms for coupled lowest Landau level equations

Abstract

We study coupled systems of nonlinear lowest Landau level equations, for which we prove global existence results with polynomial bounds on the possible growth of Sobolev norms of the solutions. We also exhibit explicit unbounded trajectories which show that these bounds are optimal.

On multi-solitons for coupled Lowest Landau Level equations

“…Introduction and main results. In this paper, we continue the study of a system of coupled Lowest Landau Level (LLL) equations which was initiated in [26].…”

“…In the case σ = −1, we have constructed in [26] traveling-waves (solitons) solutions to (1) and the aim of the present work is to show the existence of multi-solitons and study some of their properties. When σ = 1, such solutions are excluded, because their existence would contradict the conservation laws of the system (see [26,Proposition 1.4]). Therefore, from now on, we assume that σ = −1 and we consider the system…”

“…Global existence results for the system (2). We first recall the global well-posedness result for (2), which is contained in [26,Theorem 1.1].…”

“…Theorem 1.1 (Theorem 1.1, [26]). For every (u 0 , v 0 ) ∈ E × E, there exists a unique solution (u, v) ∈ C ∞ (R, E × E) to the system (2), and this solution depends smoothly on (u 0 , v 0 ).…”

“…We can also prove polynomial bounds on the possible growth of Sobolev norms for (2), we refer to [26,Theorem 1.5] for details.…”