Corner Singularity Dynamics and Regularity of Compressible Viscous Navier--Stokes Flows

Abstract

We show unique existence (global in time) and regularity of solutions to the Navier--Stokes equations for compressible viscous flows with nonzero initial and Dirichlet boundary data on polygonal domains. The solution is constructed by the inverse of the sum of two operators taken from Da Prato and Grisvard [J. Math. Pures Appl. (9), 54 (1975), pp. 305--387] and, applying to the inverse the corner singularity result of the Lamé system and the Laplace with parameters, is split into singular and regular parts. The orders of the leading corner singularities for the velocity are the same as those of the Lamé system, and the one for the temperature is the same as that for the Laplace operator. The coefficients of the singularities, called the stress intensity functions, are constructed, and the remainders are shown to have increased regularities. Consequently the derivatives of the flow variables may blow up along the trajectory curves starting near the nonconvex vertices, so the dynamical behaviors will be steeply changed there. The corner singularities are also propagated into the region by the transport character of the continuity equation. The explicit expression of singularities near the corners may be employed in a direct estimation of compressible cavity flow oscillations, for instance, the resonant amplitude tones near corners.