2014/11/24 by Natsumi Yoshida, Yoshida, Natsumi
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.1502.04602
openalex publication_date 2014/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study the decay rate in time to solutions of the Cauchy problem for the one-dimensional viscous conservation law where the far field states are prescribed. Especially, we deal with the case that the flux function which is convex and also the viscosity is a nonlinearly degenerate one (p-Laplacian type viscosity). As the corresponding Riemann problem admits a Riemann solution as the constant state or the single rarefaction wave, it has already been proved by Matsumura-Nishihara that the solution to the Cauchy problem tends toward the constant state or the single rarefaction wave as the time goes to infinity. We investigate that the decay rate in time of the corresponding solutions. Furthermore, we also investigate that the decay rate in time of the solution for the higher order derivative. These are the first result concerning the asymptotic decay of the solutions to the Cauchy problem of the scalar conservation law with nonlinear viscosity. The proof is given by L1, L2-energy and time-weighted Lq-energy methods.