On the local well--posedness of the two component $b$-family of equations

TL;DR

This paper investigates the two component b-family equations on the real line, demonstrating that their solutions depend on initial data in a highly non-regular way, with no local uniform continuity, Lipschitz, or Hölder continuity.

Contribution

The paper reformulates the equations on a Sobolev diffeomorphism group and proves the non-regular dependence on initial data, highlighting their ill-posedness in certain senses.

Findings

Dependence on initial data is nowhere locally uniformly continuous.

Dependence on initial data is nowhere locally Lipschitz.

Dependence on initial data is nowhere locally Hölder continuous.

Abstract

In this paper we consider the two component $b$-family of equations on $\mathbb R$. We write the equations on a Sobolev type diffeomorphism group. As an application of this formulation we show that the dependence on the initial data is nowhere locally uniformly continuous. In particular it is nowhere locally Lipschitz and nowhere locally H\"older continuous.