Persistence of the Steady Normal Shock Structure for the Unsteady Potential Flow

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.

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

“…Remark It is worth pointing out that, since the solid boundary is perturbed and no longer flat, the symmetry assumptions proposed in [39, 63] fail to be valid in this problem. Therefore, new ideas and methods must be developed to deal with the dihedral singularity, which is also completely different from the one caused by the corner singularity in [37]. These are the main new ingredients of this paper.…”

“…Remark 2.1. In [37], 𝑝(𝐱) does not appear in the partial hodograph transformation. While in this paper, 𝑝(𝐱) plays an essential role, as it is used to match the perturbations on the 𝑥 2direction and 𝑥 3 -direction.…”

“…) [39,63] fail to be valid in this problem. Therefore, new ideas and methods must be developed to deal with the dihedral singularity, which is also completely different from the one caused by the corner singularity in [37]. These are the main new ingredients of this paper.…”

“…Motivated by the extension techniques developed in [37] for 2-D case, we shall look for an appropriate transformation, under which it is possible to extend the linearized initial-boundary value problem in the dihedral-space domain into an initial-boundary value problem in the halfspace domain. To make it, a modified partial hodograph transformation (see (2.5) for details), different from the transformation employed for the 2-D case, is introduced.…”

“…Moreover, as the space dimension increases, the analysis needed for the a priori estimates for higher order derivatives is more complicated than the 2‐D case and it should be dealt with more carefully. Finally, it is worth mentioning that a transformation (see Section 4) is introduced to reformulate the nonlinear problem (NLP), which helps to improve the extension argument develop in [37], such that the loss‐of‐regularity for the a priori estimates on the shock‐front will not occur. Hence, instead of the Nash–Moser iteration scheme employed in [37], a classical nonlinear iteration scheme is sufficient to prove the existence of the solutions to the NLP.…”