Nontrivial weak solutions of the stationary KdV equation in sharp $L^p$ spaces

TL;DR

This paper constructs non-trivial solutions to the stationary KdV equation in certain $L^p$ spaces using convex integration, and shows that solutions in $L^2$ are necessarily smooth, establishing sharpness of the result.

Contribution

It introduces a convex integration method to find weak solutions in $L^p$ spaces for $p<2$ and proves solutions in $L^2$ are smooth, highlighting the sharpness of the space boundary.

Findings

Existence of non-trivial solutions in $L^p$ for $p<2$

Solutions in $L^2$ are necessarily smooth

Sharpness of the $L^p$ space boundary for solutions

Abstract

In this paper we utilize a convex integration scheme to construct non-trivial solutions to the stationary KdV equation which lie in $L^p(\mathbb{T})$, $p < 2$. In addition, we demonstrate this result is sharp in the sense that if $u \in L^2(\mathbb{T})$ is a weak solution then $u \in C^\infty(\mathbb{T})$.