In this paper, we investigate the formation and propagation of singularities for the system for one-dimensional Chaplygin gas, which is described by a quasilinear hyperbolic system with linearly degenerate characteristic fields. The phenomena of concentration and the formation of "δ-shock" waves are identified and analyzed systematically for this system under suitably large initial data. In contrast to the Rankine-Hogoniot conditions for classical shock, the generalized Rankine-Hogoniot conditions for "δ-shock" waves are established. Finally, it is shown that the total mass and momentum related to the solution are independent of time.

We investigate the formation and propagation of singularities for the system of one-dimensional Chaplygin gas. Under suitable assumptions we construct a physically meaningful solution containing a new type of singularities called "delta-like" solution for this kind of quasilinear hyperbolic system with linearly degenerate characteristics. By a careful analysis, we study the behavior of the solution in a neighborhood of a blow-up point. The formation of this new kind of singularities is related to the envelop of different characteristic families, instead of characteristics of the same family in the standard situation. This shows that the blow-up phenomenon for systems with linearly degenerate characteristics is quite different from the problem of shock formation for the system with genuinely nonlinear characteristic fields. Different initial data can lead to different delta-like singularities: the delta-like singularity with point-shape and the delta-like singularity with line-shape.

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In this paper, we investigate the smooth spherically symmetric solutions to 3D relativistic Chaplygin gases with variable entropy, whose initial data is assumed to be a small smooth perturbation to a constant state. Due to the structure of the equation, we can still take advantage of the "null condition" which is satisfied by the potential equation for isentropic and spacetime irrotational relativistic Chaplygin gases and obtain the global existence of smooth solution by continuity method. This generalizes the result for classical Chaplygin gases of Godin [J Math Pures Appl 87 (9): 2007] to the relativistic case.

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We establish a global existence theory for the equation governing the evolution of a relativistic membrane in a (possibly curved) Lorentzian manifold, when the spacetime metric is a perturbation of the Minkowski metric. Relying on the Hyperboloidal Foliation Method introduced by LeFloch and Ma in 2014, we revisit a theorem established earlier by Lindblad (who treated membranes in the flat Minkowski spacetime) and we provide a simpler proof of existence, which is also valid in a curved spacetime and, most importantly, leads to the important property that the total energy of the membrane is globally bounded in time. 1Our proof uses LeFloch and Ma's Hyperboloidal Foliation Method [15]- [18], which allows one to treat coupled systems of wave and Klein-Gordon equations. This method was built upon earlier work by Klainerman [12] and Hormander [9] on the quasilinear Klein-Gordon equation. The use of the hyperboloidal foliation of Minkowski spacetime to study coupled systems of wave and Klein-Gordon equation was investigated in [15] and several classes of such systems were then treated by this method, including the Einstein equations of general relativity [15,17,18]. Estimates in the hyperboloidal foliation are often more precise and it was observed in [15] that, for certain classes of equations, the energy of the solution is bounded globally in time. See LeFloch-Ma's theorem in [15, Chap. 6]. Our aim in the present paper is to extend this observation to the membrane equation (1.1). Our proof will, in addition, rely on a "double null" property of the membrane equation which was first observed by Lindblad [20]. Statement of the theoremBy the property of finite speed of propagation of the relativistic membranes, we at first specify the domain of the nonvanishing functions, which is the interior of the future light cone from the point (1, 0, 0) K := {(t, x) | r < t − 1}, and the following domain limited by two hyperboloids (with s 0 < s 1 )

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