Low regularity bounds for mKdV

TL;DR

This paper establishes global in time H^s bounds for the mKdV equation on the real line for -1/8 < s < 1/4, despite the lack of uniform continuity of the data-to-solution map below s=1/4.

Contribution

It extends the understanding of low regularity bounds for mKdV solutions, providing bounds in a regime where well-posedness is not known.

Findings

Global H^s bounds for -1/8 < s < 1/4 depend only on initial data and time.

The results are weaker than global well-posedness due to lack of control on solution differences.

Proof employs Bourgain's Fourier restriction spaces adapted to frequency-dependent time intervals.

Abstract

We study the local well-posedness in the Sobolev space H^s for the modified Korteweg-de Vries (mKdV) equation on the real line. Kenig-Ponce-Vega \cite{KPV2} and Christ-Colliander-Tao established that the data-to-solution map fails to be uniformly continuous on a fixed ball in H^s when s<1/4. In spite of this, we establish that for -1/8 < s < 1/4, the solution satisfies global in time H^s(R) bounds which depend only on the time and on the H^s(R) norm of the initial data. This result is weaker than global well-posedness, as we have no control on differences of solutions. Our proof is modeled on recent work by Christ-Colliander-Tao and Koch-Tataru employing a version of Bourgain's Fourier restriction spaces adapted to time intervals whose length depends on the spatial frequency.