On the existence of periodic solutions to the modified Korteweg-de Vries equation below $H^{1/2}(\mathbb{T})$

Abstract

Existence and a priori estimates for real-valued periodic solutions to the modified Korteweg-de Vries equation with initial data in $H^s$ are established for $s>0$. The short-time Fourier restriction norm method is employed to overcome the derivative loss. Further, non-existence of solutions below $L^2$ is proved conditional upon conjectured linear Strichartz estimates.

45 pages, minor corrections, explanations added

References in corpus (3)

• Global well-posedness of the KP-I initial-value problem in the energy space
• Strichartz estimates for the fractional Schrödinger and wave equations on compact manifolds without boundary
• On the wellposedness of the defocusing mKdV equation below $L^2$