Existence and Stability of Supersonic Euler Flows Past Lipschitz Wedges
 GuiQiang Chen,
 Yongqian Zhang,
 Dianwen Zhu
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It is well known that, when the vertex angle of a straight wedge is less than the critical angle, there exists a shockfront emanating from the wedge vertex so that the constant states on both sides of the shockfront are supersonic. Since the shockfront at the vertex is usually strong, especially when the vertex angle of the wedge is large, then a global flow is physically required to be governed by the isentropic or adiabatic Euler equations. In this paper, we systematically study twodimensional steady supersonic Euler (i.e. nonpotential) flows past Lipschitz wedges and establish the existence and stability of supersonic Euler flows when the total variation of the tangent angle functions along the wedge boundaries is suitably small. We develop a modified Glimm difference scheme and identify a Glimmtype functional, by naturally incorporating the Lipschitz wedge boundary and the strong shockfront and by tracing the interaction not only between the boundary and weak waves, but also between the strong shockfront and weak waves, to obtain the required BV estimates. These estimates are then employed to establish the convergence of both approximate solutions to a global entropy solution and corresponding approximate strong shockfronts emanating from the vertex to the strong shockfront of the entropy solution. The regularity of strong shockfronts emanating from the wedge vertex and the asymptotic stability of entropy solutions in the flow direction are also established.
 Title
 Existence and Stability of Supersonic Euler Flows Past Lipschitz Wedges
 Journal

Archive for Rational Mechanics and Analysis
Volume 181, Issue 2 , pp 261310
 Cover Date
 200607
 DOI
 10.1007/s0020500504123
 Print ISSN
 00039527
 Online ISSN
 14320673
 Publisher
 SpringerVerlag
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 Authors

 GuiQiang Chen ^{(1)} ^{(2)}
 Yongqian Zhang ^{(1)}
 Dianwen Zhu ^{(2)}
 Author Affiliations

 1. Institute of Mathematics, Fudan University, Shanghai, 200433, PRC
 2. Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL, 60208, USA