Norm inflation for the viscous nonlinear wave equation

TL;DR

This paper investigates the ill-posedness of the viscous nonlinear wave equation in negative Sobolev spaces, demonstrating norm inflation and failure of certain continuity properties above critical regularity levels.

Contribution

It establishes norm inflation results and continuity failure for the viscous nonlinear wave equation in negative Sobolev spaces, extending understanding of its ill-posedness.

Findings

Norm inflation occurs above the scaling critical regularity.

Failure of $C^k$-continuity up to a certain regularity threshold.

Ill-posedness results hold for polynomial nonlinearities.

Abstract

In this article, we study the ill-posedness of the viscous nonlinear wave equation for any polynomial nonlinearity in negative Sobolev spaces. In particular, we prove a norm inflation result above the scaling critical regularity in some cases. We also show failure of $C^k$-continuity, for $k$ the power of the nonlinearity, up to some regularity threshold.