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

2011/04/07 by Duan, Renjun, Ruan, Lizhi, Zhu, Changjiang · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1104.1271

Abstract

We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term related to an index s∈ \R over the whole space \Rⁿ for any spatial dimension n≥ 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∈ \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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