Regularity of solutions to regular shock reflection for potential flow |
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Authors: | Myoungjean Bae Gui-Qiang Chen Mikhail Feldman |
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Institution: | (1) Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, USA;(2) School of Mathematical Sciences, Fudan University, Shanghai, 200433, China;(3) Department of Mathematics, Northwestern University, Evanston, IL 60208-2730, USA |
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Abstract: | The shock reflection problem is one of the most important problems in mathematical fluid dynamics, since this problem not
only arises in many important physical situations but also is fundamental for the mathematical theory of multidimensional
conservation laws that is still largely incomplete. However, most of the fundamental issues for shock reflection have not
been understood, including the regularity and transition of different patterns of shock reflection configurations. Therefore,
it is important to establish the regularity of solutions to shock reflection in order to understand fully the phenomena of
shock reflection. On the other hand, for a regular reflection configuration, the potential flow governs the exact behavior
of the solution in C
1,1 across the pseudo-sonic circle even starting from the full Euler flow, that is, both of the nonlinear systems are actually
the same in a physically significant region near the pseudo-sonic circle; thus, it becomes essential to understand the optimal
regularity of solutions for the potential flow across the pseudo-sonic circle (the transonic boundary from the elliptic to
hyperbolic region) and at the point where the pseudo-sonic circle (the degenerate elliptic curve) meets the reflected shock
(a free boundary connecting the elliptic to hyperbolic region). In this paper, we study the regularity of solutions to regular
shock reflection for potential flow. In particular, we prove that the C
1,1-regularity is optimal for the solution across the pseudo-sonic circle and at the point where the pseudo-sonic circle meets the reflected shock.
We also obtain the C
2,α regularity of the solution up to the pseudo-sonic circle in the pseudo-subsonic region. The problem involves two types of
transonic flow: one is a continuous transition through the pseudo-sonic circle from the pseudo-supersonic region to the pseudo-subsonic
region; the other a jump transition through the transonic shock as a free boundary from another pseudo-supersonic region to
the pseudo-subsonic region. The techniques and ideas developed in this paper will be useful to other regularity problems for
nonlinear degenerate equations involving similar difficulties. |
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Keywords: | |
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