Decay properties of solutions toward a multiwave pattern to the Cauchy problem for the scalar conservation law with degenerate flux and viscosity

TL;DR

This paper analyzes the decay rates over time of solutions to a scalar conservation law with degenerate viscosity, focusing on multiwave patterns involving rarefaction waves and contact discontinuities, and extends understanding of their asymptotic behavior.

Contribution

It provides the first detailed decay rate analysis for solutions approaching multiwave patterns with degenerate flux and viscosity, including higher order derivatives.

Findings

Decay rates for solutions approaching multiwave patterns are established.

Higher order derivatives of solutions also exhibit specific decay behaviors.

The methods used include L1, L2-energy, and time-weighted Lq-energy techniques.

Abstract

In this paper, we study the precise decay rate in time to solutions of the Cauchy problem for the one-dimensional conservation law with a nonlinearly degenerate viscosity where the far field states are prescribed. Especially, we deal with the case when the flux function is convex or concave but linearly degenerate on some interval. As the corresponding Riemann problem admits a Riemann solution as a multiwave pattern which consists of the rarefaction waves and the contact discontinuity, it has already been proved by Yoshida that the solution to the Cauchy problem tends toward the linear combination of the rarefaction waves and contact wave for p-Laplacian type viscosity as the time goes to infinity. We investigate that the decay rate in time of the corresponding solutions toward the multiwave pattern. Furthermore, we also investigate that the decay rate in time of the solution for the higher order derivative. The proof is given by L1, L2-energy and time-weighted Lq-energy methods under the use of the precise asymptotic properties of the interactions between the nonlinear waves.