On the Cauchy problem for scalar conservation laws on the Bohr compactification of $\R^n$

TL;DR

This paper investigates the existence, uniqueness, and long-term behavior of entropy solutions to scalar conservation laws on the Bohr compactification of ^n, extending classical results to a more general setting with continuous flux.

Contribution

It establishes the existence and uniqueness of entropy solutions for multidimensional scalar conservation laws on the Bohr compactification with merely continuous flux.

Findings

Proves existence and uniqueness of entropy solutions.

Provides necessary and sufficient conditions for solution decay as time approaches infinity.

Extends classical conservation law results to the setting of the Bohr compactification.

Abstract

We study the Cauchy problem for a multidimensional scalar conservation law on the Bohr compactification of $\R^n$. The existence and uniqueness of entropy solutions are established in the general case of merely continuous flux vector. We propose also the necessary and sufficient condition for the decay of entropy solutions as time $t\to+\infty$.