Multi-dimensional shock interaction for a Chaplygin gas

A Chaplygin gas is an inviscid, compressible fluid in which the acoustic fields are linearly degenerate. We analyse the multi-dimensional shocks, which turn out to be sonic. Two shocks in general position interact rather simply. We investigate several twodimensional Riemann problems and prove the existence of a unique solution. Among them is the supersonic reflection of a planar shock against a wedge: we remark that the solution cannot be a Mach Reflection, contrary to what happens for other gases, and that there always exists a solution in the form of a Regular Reflection. Plan of the paper ; main results We define a Chaplygin gas in the first section, where we prove that the pressure (also called acoustic) waves are characteristic. In the sequel, such waves are called ‘shocks’, despite the fact that they are contact discontinuities for such a gas. We also observe that the entropy remains constant across shocks, contrary to what happens in other gases. This explains why the solution must be isentropic when the initial data is so. The local analysis of multidimensional shocks is made at Section 2. For steady flows, we show the lack of vorticity generation, even across a curved shock (Theorem 2.1). We give a fivelines proof of the non-existence of a three-shocks pattern (Paragraph 2.2), a result that holds true in a much more general context, see [12]. We then show that there are a lot of four-shocks patterns, and that they provide the interaction of two incoming shocks given in rather general position (Paragraphs 2.3 and 2.4). We also describe the shock reflection against an infinite wall (Paragraph 2.5). When the shock strength is large enough, we observe a concentration phenomenon of the mass along the wall; this is due to the boundedness of the pressure at inifinite density. Section 3 is devoted to self-similar flows, which obey to a system that differs from the steady Euler equations by zero-order terms only. We prove again the lack of vorticity generation. This explains why the solution must be irrotational when the self-similar initial data is so. We show that shock curves bounded by a constant state are a priori known, because they are ∗UMPA, UMR CNRS–ENS Lyon # 5669. École Normale Supérieure de Lyon, 46, allée d’Italie, F–69364 Lyon, cedex 07.