Asymptotic Behavior of Solutions Toward the Viscous Shock Waves to the Cauchy Problem for the Scalar Conservation Law with Nonlinear Flux and Viscosity

Abstract

In this paper, we investigate the global structure of solutions to the Cauchy problem for the scalar viscous conservation law where the far field states are prescribed. Especially, we deal with the case when the viscous/diffusive flux σ(v) ∼ | v | p is of non-Newtonian type (i.e., p > 0), including a pseudo-plastic case (i.e., p < 0). When the corresponding Riemann problem for the hyperbolic part admits a Riemann solution which consists of single rarefaction wave, under a condition on nonlinearity of the viscosity, it has been recently proved by that the solution of the Cauchy problem tends toward the rarefaction wave as time goes to infinity for the case p > 3/7 without any smallness conditions. The new ingredients we obtained are the extension to the stability results in [38] to the case p > 1/3 (also without any smallness conditions), and furthermore their precise time-decay estimates.Keywords: viscous conservation law, asymptotic behavior, convex flux, pseudoplastic-type viscosity, rarefaction wave AMS subject classifications: 35K55, 35B40, 35L65The decay results correspond to Theorem 1.1 are the following.Theorem 1.2. Under the same assumptions as in Theorem 1.1, the unique global in time solution u of the Cauchy problem (1.1) has the following timedecay estimatesfor q ∈ [ 2, ∞) and any ǫ > 0.Theorem 1.3. Under the same assumptions as in Theorem 1.1, if the initial data further satisfies u 0 −ũ ∈ L 1 , then the unique global in time solution u of the Cauchy problem (1.1) has the following time-growth and time-decay estimatesfor q ∈ (1, ∞) and any ǫ > 0.Theorem 1.4. Under the same assumptions as in Theorem 1.1, the unique global in time solution u of the Cauchy problem (1.1) has the following timedecay estimates for the derivativesfor q ∈ (1, ∞) and any ǫ > 0.

Asymptotic behavior of solutions toward the rarefaction waves to the Cauchy problem for the scalar diffusive dispersive conservation laws

“…it is investigated by Harabetian [14], Matsumura-Nishihara [27], [28], Matsumura-Yoshida [29], [30], and Yoshida [37], [38], [39], [40], [41], [43], that the asymptotic stability and time-decay estimates of various nonlinear waves, such as rarefaction wave, viscous shock wave, viscous contact wave (see also [22], [44]), and multiwave pattern which consists of the rarefaction waves and the viscous contact wave. For the rarefaction problem of the Korteweg-de Vries equation…”