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E-H Type Mach Configuration and its Stability

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Abstract

This paper is devoted to the study of the classification and stability of Mach configurations. In the Mach reflection of a shock wave by a rigid wall the flow behind the reflected shock front can be supersonic or subsonic. Correspondingly, the Mach configuration can be classified as E-E type and E-H type. In this paper we proved the stability of E-H type Mach configurations provided the reflected shock is weak. The result is the complement of the stability of E-E type Mach configurations in our earlier work.

The stability of E-H Mach configurations is reduced to a generalized Tricomi problem of a nonlinear mixed type equation. The linearization of this problem is a generalized Tricomi problem of a linear Lavrentiev-Bitsadze’s mixed type equation. In order to prove the existence of such a problem we first use the equation in the hyperbolic region and a condition on the reflected shock to establish a relation of the unknown function and its normal derivative on the sonic line. Then the generalized Tricomi problem is reduced to a nonlocal boundary value problem in the elliptic region. Careful analysis gives necessary estimates and the solvability of the linearized problem. It leads to the existence of the solution to the corresponding nonlinear problem, which implies the required stability of E-H Mach configurations.

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  1. Institute of Mathematics, Fudan University, Shanghai, 200433, P.R. China

    Shuxing Chen

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  1. Shuxing Chen
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Correspondence to Shuxing Chen.

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Communicated by P. Constantin

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Chen, S. E-H Type Mach Configuration and its Stability. Commun. Math. Phys. 315, 563–602 (2012). https://doi.org/10.1007/s00220-012-1570-4

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  • DOI: https://doi.org/10.1007/s00220-012-1570-4

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