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 Br0(0)⊂ Rn for n≥ 3 and some positive constant r0>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 Hashimoto-Matsumura-JDE-2019 that a sufficient condition to guarantee the existence of such a stationary wave is v+<0, v-≤ |v+|+μ(n-1)/r0. Since the stationary wave is no longer monotonic, its nonlinear stability is justified only recently in Hashimoto-Matsumura-JDE-2019 for the case when v<0, v-≤ v++μ(n-1)/r0. 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 satisfying v+<0, v-≤ |v+|+μ(n-1)/r0. 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 Kawashima-Matsumura-CMP-1985 and Yin-Zhao-KRM-2009.