Optimal Decay Rates to Conservation Laws with Diffusion-Type Terms of Regularity-gain and Regularity-loss

TL;DR

This paper studies the decay rates of solutions to nonlinear scalar conservation laws with diffusion-type source terms exhibiting regularity-gain or regularity-loss, providing decay estimates, global existence, and asymptotic behavior analysis.

Contribution

It introduces a unified approach to obtain optimal decay rates for equations with both regularity-gain and regularity-loss features, extending previous results to a broader class of models.

Findings

Established $L^p$-$L^q$ decay estimates for linear solutions.

Proved global existence of small-amplitude solutions.

Analyzed asymptotic convergence to heat diffusion waves.

Abstract

We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term related to an index $s\in \R$ over the whole space $\R^n$ for any spatial dimension $n\geq 1$. Here, the diffusion-type source term behaves as the usual diffusion term over the low frequency domain while it admits on the high frequency part a feature of regularity-gain and regularity-loss for $s< 1$ and $s>1$, respectively. For all $s\in \R$, we not only obtain the $L^p$-$L^q$ time-decay estimates on the linear solution semigroup but also establish the global existence and optimal time-decay rates of small-amplitude classical solutions to the nonlinear Cauchy problem. In the case of regularity-loss, the time-weighted energy method is introduced to overcome the weakly dissipative property of the equation. Moreover, the large-time behavior of solutions asymptotically tending to the heat diffusion waves is also studied. The current results have general applications to several concrete models arising from physics.