Constraints on possible singularities for the unsteady transonic small disturbance (UTSD) equations

Irene M. Gamba, Rodolfo R. Rosales, Esteban G. Tabak

Research output: Contribution to journalArticle

Abstract

We discuss the singular behavior of solutions to two-dimensional, general second-order, uniformly elliptic equations in divergence form, with bounded measurable coefficients and discontinuous Dirichlet data along a portion of a Lipschitz boundary. We show that the conjugate to the solution develops a singularity that is at least logarithmic along the boundary at the points of discontinuity in the boundary data. A problem like this arises in the study of self-similar solutions to the hyperbolic conservation laws in two space dimensions given by the unsteady transonic small disturbance (UTSD) flow equations. These solutions model the reflection of a weak shock wave upon a thin wedge in the regime where the von Neumann paradox applies. The present result provides a step in the direction of understanding the nature of the solutions to the UTSD equations near a triple point. It shows that the flow behind the triple point cannot be strictly subsonic under some mild assumptions on the solutions.

Original languageEnglish (US)
Pages (from-to)763-779
Number of pages17
JournalCommunications on Pure and Applied Mathematics
Volume52
Issue number6
DOIs
StatePublished - Jun 1999

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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