### 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 language | English (US) |
---|---|

Pages (from-to) | 763-779 |

Number of pages | 17 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 52 |

Issue number | 6 |

State | Published - Jun 1999 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*52*(6), 763-779.

**Constraints on possible singularities for the unsteady transonic small disturbance (UTSD) equations.** / Gamba, Irene M.; Rosales, Rodolfo R.; Tabak, Esteban.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 52, no. 6, pp. 763-779.

}

TY - JOUR

T1 - Constraints on possible singularities for the unsteady transonic small disturbance (UTSD) equations

AU - Gamba, Irene M.

AU - Rosales, Rodolfo R.

AU - Tabak, Esteban

PY - 1999/6

Y1 - 1999/6

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0033455213&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033455213&partnerID=8YFLogxK

M3 - Article

VL - 52

SP - 763

EP - 779

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 6

ER -