Local Well-posedness of Unsteady Potential Flows Near a Space Corner of Right Angle

TL;DR

This paper investigates the local well-posedness of unsteady potential flows near a right-angle space corner, addressing challenges posed by corner singularities and boundary conditions that lack linear stability.

Contribution

It introduces updated extension methods and new techniques to handle corner singularities and boundary operator issues in hyperbolic PDEs.

Findings

Successfully established local well-posedness near the corner

Developed methods to control boundary terms with co-normal operators

Addressed difficulties due to corner singularity and boundary conditions

Abstract

In this paper we are concerned with the local well-posedness of the unsteady potential flows near a space corner of right angle, which could be formulated as an initial-boundary value problem of a hyperbolic equation of second order in a cornered-space domain. The corner singularity is the key difficulty in establishing the local well-posedness of the problem. Moreover, the boundary conditions on both edges of the corner angle are of Neumann-type and fail to satisfy the linear stability condition, which makes it more difficult to establish a priori estimates on the boundary terms in the analysis. In this paper, extension methods will be updated to deal with the corner singularity, and, based on a key observation that the boundary operators are co-normal, new techniques will be developed to control the boundary terms.