An Ill Posed Cauchy Problem for a Hyperbolic System in Two Space Dimensions

TL;DR

This paper constructs a counterexample demonstrating that the Cauchy problem for certain hyperbolic systems in two space dimensions can be ill posed, highlighting unresolved issues in multi-dimensional hyperbolic PDE theory.

Contribution

It provides the first known example showing ill-posedness of the Cauchy problem for a simple class of hyperbolic systems in two dimensions.

Findings

Counterexample of ill-posedness in 2D hyperbolic systems

Challenges in extending well-posedness results to multi-dimensional systems

Highlights open problems in the theory of hyperbolic PDEs

Abstract

The theory of weak solutions for nonlinear conservation laws is now well developed in the case of scalar equations [3] and for one-dimensional hyperbolic systems [1, 2]. For systems in several space dimensions, however, even the global existence of solutions to the Cauchy problem remains a challenging open question. In this note we construct a conterexample showing that, even for a simple class of hyperbolic systems, in two space dimensions the Cauchy problem can be ill posed.