Growth of Fourier--Lebesgue norms for mKdV

TL;DR

This paper shows that solutions to the focusing mKdV equation can exhibit unbounded growth in Fourier--Lebesgue norms over time, even when starting from initial data that converges to zero in these spaces.

Contribution

It constructs explicit solutions demonstrating norm inflation in Fourier--Lebesgue spaces for the focusing mKdV, highlighting new phenomena in the equation's dynamics.

Findings

Solutions with initial data tending to zero can diverge in norm over time.

Norm inflation occurs for all real s and p ≠ 2 in Fourier--Lebesgue spaces.

Explicit sequences of solutions illustrating this inflation are constructed.

Abstract

We demonstrate inflation of Fourier--Lebesgue norms for solutions to the focusing modified Korteweg--de Vries equation posed on the real line. For $p\neq 2$ and all $s\in \mathbb{R}$, we construct a sequence of solutions $u_n$ whose initial data $u_n(0)$ converges to zero in the Fourier--Lebesgue spaces $\mathcal F L^p_s(\mathbb{R})$, but whose evolutions at later times $t_n$ diverge to infinity.