Convergence rate to the nonlinear waves for viscous conservation laws on the half line
The authors in this study investigate the convergence rate of solutions to the initial-boundary value problem for scalar viscous conservation laws on the half line. Especially, they deal with the case where the Riemann problem for the corresponding hyperbolic equation admits transonic rarefaction waves. In this case, it is known that the solution tends toward a linear superposition of the stationary solution and the rarefaction wave. We show that the convergence rate is \((1+t)^{-\frac 1 2 (1-\frac 1 p)}\log ^2(2+t)\) in \(L^p\) norm \((1 \leq p < \infty)\) and \((1 + t)^{-\frac 1 2 +\varepsilon} \) in \(L^\infty\) norm if the initial perturbation from the corresponding superposition is located in \(H ^1 \cap L^1\). In this article, the proof is given by a combination of the weighted \(L^p\) energy method and the \(L^1\) estimate.
scientific article; zbMATH DE number 5761643