Decay properties of solutions to the Cauchy problem for the scalar conservation law with nonlinearly degenerate viscosity

TL;DR

This paper investigates the decay rates over time of solutions to a one-dimensional scalar conservation law with nonlinear degenerate viscosity, providing the first results on asymptotic decay for such equations.

Contribution

It establishes the decay rates of solutions and their higher derivatives for the first time in the context of nonlinear viscosity in scalar conservation laws.

Findings

Solutions tend toward constant states or rarefaction waves as time increases.

Decay rates are quantified using L1, L2-energy, and time-weighted Lq-energy methods.

First results on asymptotic decay for equations with nonlinear viscosity.

Abstract

In this paper, we study the decay rate in time to solutions of the Cauchy problem for the one-dimensional viscous conservation law where the far field states are prescribed. Especially, we deal with the case that the flux function which is convex and also the viscosity is a nonlinearly degenerate one (p-Laplacian type viscosity). As the corresponding Riemann problem admits a Riemann solution as the constant state or the single rarefaction wave, it has already been proved by Matsumura-Nishihara that the solution to the Cauchy problem tends toward the constant state or the single rarefaction wave as the time goes to infinity. We investigate that the decay rate in time of the corresponding solutions. Furthermore, we also investigate that the decay rate in time of the solution for the higher order derivative. These are the first result concerning the asymptotic decay of the solutions to the Cauchy problem of the scalar conservation law with nonlinear viscosity. The proof is given by L1, L2-energy and time-weighted Lq-energy methods.