Stability properties and dynamics of solutions to viscous conservation laws with mean curvature operator

Abstract

In this paper we study the long time dynamics of the solutions to an initial-boundary value problem for a scalar conservation law with a saturating nonlinear diffusion. After discussing the existence of a unique stationary solution and its asymptotic stability, we focus our attention on the phenomenon of metastability, whereby the time-dependent solution develops into a layered function in a relatively short time, and subsequently approaches a steady state in a very long time interval. Numerical simulations illustrate the results.

On the speed rate of convergence of solutions to conservation laws with nonlinear diffusions

“…Motivated by the behavior of the solutions to the viscous Burgers equation with linear diffusion (1.7) (see, for instance, [14,20,21,23,27,29]), in Section 4 we investigate the phenomenon of metastability (whereby the time dependent solution reaches its asymptotic configuration in an exponentially, with respect to ε 1, long time interval) in the case of a nonlinear diffusion. In particular, we show that a metastable behavior also appears for the solutions to (1.1) with dissipation fluxes (1.4), (1.5) (the case of the mean curvature operator in the Euclidean space (1.3) has been recently studied in [11]); in these cases condition (1.10) becomes…”

“…Note that in the model example (1.3) previously studied in [11] (the mean curvature operator in Euclidean space), we have T > 0. The IBVP for the equation (1.8) and Q given by (1.4) has been studied in [17,Section 2].…”

“…To start with, we study the existence of smooth stationary solutions to (1.1) with Q satisfying (1.2). We stress that the special case of a mean curvature type diffusion in the Euclidean space (1.3) has been already studied in [11].…”

“…• Q ∈ C 2 (R) monotone increasing and bounded: precisely, we assume The model example (already studied in [11]) we have in mind is…”

“…The goal of this paper is twofold: on the one side, we generalize the results of [11] to the general setting (1.2); moreover, the model we consider here is slightly different with respect to the one proposed in [11], so that we are able to overcome the smallness condition on the boundary data stated in [11], as will we see in details later on. On the other side, we investigate existence, stability and metastability properties of the steady states also in the cases of the dissipation flux functions (1.4) and (1.5).…”