Global weak solutions for a variation of the Whitham equation

TL;DR

This work analyzes a variation of the Whitham equation $${\partial_t u = -(-\Delta)^{1/2}{\mathcal M}(u) + {\partial_x}\left(\frac{u^2}{2}\right)},$$ where ${\mathcal M}$ is a Fourier multiplier with symbol ${\mathfrak m}(\xi)=\sqrt{(1+\xi^2)\frac{\tanh(\xi)}{\xi}}$. The authors first prove global existence of weak solutions by introducing a hyperviscosity perturbation, deriving an energy inequality in the energy space ${\mathcal N}$, and passing to the limit through compactness aided by a nonlinear interpolation lemma. They then establish a regularity criterion: if the initial data lies in $L^2\cap\dot H^{\rho}$ with $\rho>0$ and the solution remains bounded in $L^{\infty}_tL^{\infty}_x$, then it gains Sobolev regularity up to $\dot H^{\sigma}$ with $0<\sigma<1/2$ and further up to $\dot H^{\alpha}$ with $\alpha<7/4$, via semigroup smoothing and fractional product estimates. Under these regularity assumptions, they prove uniqueness in the class of weak solutions, and they also develop local-in-time results that relax the global boundedness condition, including a Serrin-type criterion implying space-time boundedness and enabling local regularity conclusions.

Abstract

We study in this article a variation of the Whitham equation which was introduced as an alternative to the KdV equation. We first prove the global existence of weak solutions, then we establish a regularity criterion from which we deduce the uniqueness of weak solutions. Local in time criterions for regularity and uniqueness are also given.

Theorems & Definitions (13)

• Theorem 1: Global Solutions
• Theorem 2: Regularity Criterion
• Theorem 3: Uniqueness
• Proposition 1
• Lemma 1
• Proposition 2
• Proposition 3: Energy inequality
• Lemma 2
• Remark 1
• Remark 2