Persistence of the steady planar normal shock structure in 3-D unsteady potential flows

TL;DR

This paper proves the dynamic stability of steady planar normal shocks in 3-D unsteady potential flows, overcoming mathematical challenges posed by edge singularities and extending analysis techniques from 2-D to 3-D.

Contribution

It introduces a modified partial hodograph transformation and improved extension techniques to establish well-posedness and stability results for 3-D shock structures.

Findings

Established the well-posedness of the linearized stability problem.

Proved the nonlinear stability of steady planar normal shocks in 3-D.

Avoided Nash-Moser iteration by controlling regularity loss.

Abstract

This paper concerns the dynamic stability of the steady 3-D wave structure of a planar normal shock front intersecting perpendicularly to a planar solid wall for unsteady potential flows. The stability problem can be formulated as a free boundary problem of a quasi-linear hyperbolic equation of second order in a dihedral-space domain between the shock front and the solid wall. The key difficulty is brought by the edge singularity of the space domain, the intersection curve between the shock front and the solid wall. Different from the 2-D case, for which the singular part of the boundary is only a point, it is a curve for the 3-D case in this paper. This difference brings new difficulties to the mathematical analysis of the stability problem. A modified partial hodograph transformation is introduced such that the extension technique developed for the 2-D case can be employed to establish the well-posed theory for the initial-boundary value problem of the linearized hyperbolic equation of second order in a dihedral-space domain. Moreover, the extension technique is improved in this paper such that loss of regularity in the a priori estimates on the shock front does not occur. Thus the classical nonlinear iteration scheme can be constructed to prove the existence of the solution to the stability problem, which shows the dynamic stability of the steady planar normal shock without applying the Nash-Moser iteration method.