Asymptotics of Radially Symmetric Solutions for the Exterior Problem of Multidimensional Burgers Equation

Abstract

We are concerned with the large-time behavior of the radially symmetric solution for multidimensional Burgers equation on the exterior of a ball ${\mathbb{B}}_{r_0}(0)\subset {\mathbb{R}}^n$ for $n\ge 3$ and some positive constant $r_0>0$, where the boundary data $v_{-}$ and the far field state $v_{+}$ of the initial data are prescribed and correspond to a stationary wave. It is shown in \cite{Hashimoto-Matsumura-JDE-2019} that a sufficient condition to guarantee the existence of such a stationary wave is $v_{+}<0,v_{-}\le |v_{+}|+\mu (n-1)/r_0$. Since the stationary wave is no longer monotonic, its nonlinear stability is justified only recently in \cite{Hashimoto-Matsumura-JDE-2019} for the case when $v_{\pm }<0,v_{-}\le v_{+}+\mu (n-1)/r_0$. The main purpose of this paper is to verify the time asymptotically nonlinear stability of such a stationary wave for the whole range of $v_{\pm }$ satisfying $v_{+}<0,v_{-}\le |v_{+}|+\mu (n-1)/r_0$. Furthermore, we also derive the temporal convergence rate, both algebraically and exponentially. Our stability analysis is based on a space weighted energy method with a suitable chosen weight function, while for the temporal decay rates, in addition to such a space weighted energy method, we also use the space-time weighted energy method employed in \cite{Kawashima-Matsumura-CMP-1985} and \cite{Yin-Zhao-KRM-2009}.