Local Well-Posedness of the Nonlinear Wave System Near a Space Corner of Right Angle

Abstract

We are concerned with the well-posedness of the nonlinear wave system, which is a first-order hyperbolic system, in the vicinity of a right-angled spatial corner. The problem can be expressed as an initial boundary value problem (IBVP) involving a second-order hyperbolic equation in a spatial domain with a corner. The main difficulty in establishing the local well-posedness of the problem arises from the lack of smoothness in the spatial domain due to the presence of the corner point. Additionally, the Neumann-type boundary conditions on both edges of the corner angle do not satisfy the linear stability condition, posing challenges in obtaining higher-order a priori estimates for the boundary terms in the analysis. To address the corner singularity, modified extension methods will be employed in this paper. Furthermore, new techniques will be developed to control the boundary terms, leveraging the observation that the boundary operators are co-normal.