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

Abstract

We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term related to an index $s\in \mathbb{R}$ over the whole space ${\mathbb{R}}^n$ for any spatial dimension $n\ge 1$. 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 $s\in \mathbb{R}$, 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.

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• Global Smooth Flows for the Compressible Euler-Maxwell System: Relaxation Case