On the Riemann problem for 2D compressible Euler equations in three pieces

The Riemann problem for two-dimensional isentropic Euler equations is considered. The initial data are three constants in three fan domains forming different angles. Under the assumption that only a rarefaction wave, shock wave or contact discontinuity connects two neighboring constant initial states, it is proved that the cases involving three shock or rarefaction waves are impossible. For the cases involving one rarefaction (shock) wave and two shock (rarefaction) waves, only the combinations when the three elementary waves have the same sign are possible (impossible).     

Stability of the Riemann solutions for a nonstrictly hyperbolic system of conservation laws

We prove that the Riemann solutions are stable for a nonstrictly hyperbolic system of conservation laws under local small perturbations of the Riemann initial data. The proof is based on the detailed analysis of the interactions of delta shock waves with shock waves and rarefaction waves. During the interaction process of the delta shock wave with the rarefaction wave, a new kind of nonclassical wave, namely a delta contact discontinuity, is discovered here, which is a Dirac delta function supported on a contact discontinuity and has already appeared in the interaction process for the magnetohydrodynamics equations [M. Nedeljkov and M. Oberguggenberger, Interactions of delta shock waves in a strictly hyperbolic system of conservation laws, J. Math. Anal. Appl. 344 (2008) 1143-1157]. Moreover, the global structures and large time asymptotic behaviors of the solutions are constructed and analyzed case by case.     

Global structure of Riemann solutions to a system of two-dimensional hyperbolic conservation laws

The Riemann problem for a two-dimensional nonstrictly hyperbolic system of conservation laws is considered. Without the restriction that each jump of the initial data projects one planar elementary wave, ten topologically distinct solutions are obtained by applying the method of generalized characteristic analysis. Some of these solutions involve the nonclassical waves, i.e., the delta shock wave and the delta contact discontinuity, for which we explicitly give the expressions of their strengths, locations and propagation speeds. Moreover, we demonstrate that the nature of our solutions is identical with that of solutions to the corresponding one-dimensional Cauchy problem, which provides a verification that our construction produces the correct unique global solutions.     

Global transonic conic shock wave for the symmetrically perturbed supersonic flow past a cone

In this paper, we are concerned with the global existence and stability of a steady transonic conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Theoretically, as indicated in [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, 1948], it follows from the Rankine-Hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the vertex of the sharp cone in terms of the different pressure states at infinity behind the shock surface, which correspond to the supersonic shock and the transonic shock respectively. In the references [Shuxing Chen, Zhouping Xin, Huicheng Yin, Global shock wave for the supersonic flow past a perturbed cone, Comm. Math. Phys. 228 (2002) 47-84; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Polytropic case, preprint, 2006; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Isothermal case, Pacific J. Math. 233 (2) (2007) 257-289] and [Zhouping Xin, Huicheng Yin, Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone, Anal. Appl. 4 (2) (2006) 101-132], the authors have established the global existence and stability of a supersonic shock for the perturbed hypersonic incoming flow past a sharp cone when the pressure at infinity is appropriately smaller than that of the incoming flow. At present, for the supersonic symmetric incoming flow, we will study the global transonic shock problem when the pressure at infinity is appropriately large.     

Cauchy problem with general discontinuous initial data along a smooth curve for 2-d Euler system

We study the Cauchy problems for the isentropic 2-d Euler system with discontinuous initial data along a smooth curve. All three singularities are present in the solution: shock wave, rarefaction wave and contact discontinuity. We show that the usual restrictive high order compatibility conditions for the initial data are automatically satisfied. The local existence of piecewise smooth solution containing all three waves is established.     

Global supersonic conic shock wave for the steady supersonic flow past a cone: Polytropic gas

In this paper, we establish the global existence and stability of a steady conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body as long as the vertex angle is less than a critical value. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Based on the delicate asymptotic expansion of the background solution, one can verify that the boundary conditions on the shock and the conic surface satisfy the “dissipative” property. From this property, by use of the reflected characteristics method and the special form of the shock equation, we show that the conic shock attached at the vertex of the cone exists globally in the whole space when the speed of the supersonic coming flow is appropriately large. On the other hand, we remove the smallness restriction on the sharp vertex angle in order to establish the global existence of a shock or a global weak solution, moreover, our proof approach is different from that in [Shuxing Chen, Zhouping Xin, Huicheng Yin, Global shock wave for the supersonic flow past a perturbed cone, Comm. Math. Phys. 228 (2002) 47-84] and [Zhouping Xin, Huicheng Yin, Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone, Anal. Appl. 4 (2) (2006) 101-132].     

Lie-group theoretic method for analyzing interaction of discontinuous waves in a relaxing gas

A Lie group of transformations method is used to establish self-similar solutions to the problem of shock wave propagation through a relaxing gas and its interaction with the weak discontinuity wave. The forms of the equilibrium value of the vibrational energy and the relaxation time, varying with the density and pressure are determined for which the system admits self-similar solutions. A particular solution to the problem has been found out and used to study the effects of specific heat ratio and ambient density exponent on the flow parameters. The coefficients of amplitudes of reflected and transmitted waves after the interaction are determined.     

Interactions of delta shock waves for a class of nonstrictly hyperbolic system of conservation laws

In this paper, we study the perturbed Riemann problem for a class of nonstrictly hyperbolic system of conservation laws, and focuse on the interactions of delta shock waves with the shock waves and the rarefaction waves. The global solutions are constructed completely with the method of splitting delta function. In solutions, we find a new kind of nonclassical wave, which is called delta contact discontinuity with Dirac delta function in both components. It is quite different from the previous ones on which only one state variable contains the Dirac delta function. Moreover, by letting perturbed parameter $\varepsilon$ tend to zero, we analyze the stability of Riemann solutions.     

Global solutions for initial-boundary value problem of quasilinear wave equations

This work investigates the existence of globally Lipschitz continuous solutions to a class of initial-boundary value problem of quasilinear wave equations. Applying the Lax's method and generalized Glimm's method, we construct the approximate solutions of initial-boundary Riemann problem near the boundary layer and perturbed Riemann problem away from the boundary layer. By showing the weak convergence of residuals for the approximate solutions, we establish the global existence for the derivatives of solutions and obtain the existence of global Lipschitz continuous solutions of the problem.     

Stability of viscous contact wave for compressible navier-stokes system of general gas with free boundary

In this paper, we study the large time behavior of solutions to the nonisentropic Navier-Stokes equations of general gas, where polytropic gas is included as a special case, with a free boundary. First we construct a viscous contact wave which approximates to the contact discontinuity, which is a basic wave pattern of compressible Euler equation, in finite time as the heat conductivity tends to zero. Then we prove the viscous contact wave is asymptotic stable if the initial perturbations and the strength of the contact wave are small. This generalizes our previous result [6] which is only for polytropic gas.     

On global transonic shocks for the steady supersonic Euler flows past sharp 2-D wedges

In this paper, under certain downstream pressure condition at infinity, we study the globally stable transonic shock problem for the perturbed steady supersonic Euler flow past an infinitely long 2-D wedge with a sharp angle. As described in the book of Courant and Friedrichs [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948] (pages 317-318): when a supersonic flow hits a sharp wedge, it follows from the Rankine-Hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the edge of the sharp wedge in terms of the different pressure states in the downstream region, which correspond to the supersonic shock and the transonic shock respectively. It has frequently been stated that the strong shock is unstable and that, therefore, only the weak shock could occur. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand this open problem. More concretely, we will establish the global existence and stability of a transonic shock solution for 2-D full Euler system when the downstream pressure at infinity is suitably given. Meanwhile, the asymptotic state of the downstream subsonic solution is determined.     

Stability of reflection and refraction of shocks on interface

In this paper we study the stability of the nonlinear wave structure caused by the attack of an incident shock on an interface of two different kinds of media. The attack will produce a reflected wave and a refracted wave, and also let the interface deflected. In this paper we will mainly study the case, when the reflected wave is a shock, and the flow between the reflected wave and the refracted shock is relatively subsonic. Our result indicates that the wave structure and the flow field for the reflection-refraction problem in this case is conditionally stable.To describe the motion of the fluid we use the inviscid Euler system as the mathematical model. The reflection-refraction problem can be reduced to a free boundary value problem, where the unknown reflected shock and refracted shock are free boundaries, and the deflected interface is also to be determined. In the proof of the existence and the stability of the corresponding wave structure we apply the Lagrange transformation to fix the interface and the decoupling technique to decouple the elliptic-hyperbolic composite system in its principal part. Meanwhile, some efficient weighted Sobolev estimates are established to derive the existence for corresponding nonlinear problems.     

Globally Lipschitz continuous solutions to a class of quasilinear wave equations

This work investigates the existence of globally Lipschitz continuous solutions to a class of Cauchy problem of quasilinear wave equations. Applying Lax's method and generalized Glimm's method, we construct the approximate solutions of the corresponding perturbed Riemann problem and establish the global existence for the derivatives of solutions. Then, the existence of global Lipschitz continuous solutions can be carried out by showing the weak convergence of residuals for the source term of equation.     

Global existence of shock front solutions in 1-dimensional piston problem in the relativistic Euler equations

The paper studies the 1-D piston problem of the relativistic Euler equations when the speed of the piston is a perturbation of a constant. A sequence of approximate solutions constructed by a modified Glimm scheme is proved to be convergent to the weak solution (which includes a strong leading shock) to the piston problem. In particular, we give the precise estimates on the reflection of the perturbed waves on the piston and the leading shock. The paper is supported by the National Natural Science Foundation of China (Grant 10626034) and the Special Research Fund for Selecting Excellent Young Teachers of the Universities in Shanghai.     

Transonic shock wave in an infinite nozzle asymptotically converging to a cylinder

We construct a single transonic shock wave pattern in an infinite nozzle asymptotically converging to a cylinder, which is close to a uniform transonic shock wave. In other words, suppose there is a uniform transonic shock wave in an infinite cylinder nozzle which can be constructed easily, if we perturbed the supersonic incoming flow and the infinite nozzle a little bit, we can obtain a transonic wave near the uniform one. As a consequence, we can show that the uniform transonic wave is stable with respect to the perturbation of the incoming flow and nozzle wall. Based on the theory of [G.Q. Chen, M. Feldman, Existence and stability of multi-dimensional transonic flows through an infinite nozzle of arbitrary cross-sections, Arch. Ration. Mech. Anal. 184 (2007) 185-242], the crucial parts of this paper are to derive the uniform Schauder estimates of the linear elliptic equation for the infinite nozzle asymptotically converging to a cylinder.     

Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle

In this paper, we study the stability of supersonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle of varying cross-sections. We formulate the problem as an initial–boundary value problem with the contact discontinuity as a free boundary. To deal with the free boundary value problem, we employ the Lagrangian transformation to straighten the contact discontinuity and then the free boundary value problem becomes a fixed boundary value problem. We develop an iteration scheme and establish some novel estimates of solutions for the first order of hyperbolic equations on a cornered domain. Finally, by using the inverse Lagrangian transformation and under the assumption that the incoming flows and the nozzle walls are smooth perturbations of the background state, we prove that the original free boundary problem admits a unique weak solution which is a small perturbation of the background state and the solution consists of two smooth supersonic flows separated by a smooth contact discontinuity.     

Dynamically adapted grids for interacting discontinuous solutions

Further development of the dynamic adaptation method for gas dynamics problems that describe multiple interactions of shock waves, rarefaction waves, and contact discontinuities is considered. Using the Woodward-Colella problem and a nonuniformly accelerating piston as examples, the efficiency of the proposed method is demonstrated for the gas dynamics problems with shock wave and contact discontinuity tracking. The grid points are distributed under the control of the diffusion approximation. The choice of the diffusion coefficient for obtaining both quasi-uniform and strongly nonuniform grids for each subdomain of the solution is validated. The interaction between discontinuities is resolved using the Riemann problem for an arbitrary discontinuity. Application of the dynamic adaptation method to the Woodward-Colella problem made it possible to obtain a solution on a grid consisting of 420 cells that is almost identical to the solution obtained using the WENO5m method on a grid consisting of 12 800 cells. In the problem for a nonuniformly accelerating piston, a proper choice of the diffusion coefficient in the transformation functions makes it possible to generate strongly nonuniform grids, which are used to simulate the interaction of a series of shock waves using shock wave and contact discontinuity tracking.     

扰动色谱方程黎曼解的极限-delta激波的行成和转换

本文主要讨论扰动色谱方程delta激波解的行成和转换,并讨论上述方程的黎曼问题.当扰动参数趋于零时,通过研究黎曼解的极限,我们可以观察到如下两个重要现象:激波和接触间断重合行成delta激波,一类激波(一个变量含有delta函数).     

Solutions of the Cauchy problem for the 2-D Euler system with fan-shaped structure

We study the Cauchy problem for the isentropic 2-D Euler system with initial data having discontinuity on a smooth curve. A local existence of a solution is established, which consists of shock wave, rarefaction wave and contact discontinuity.     

一维Chaplygin气体欧拉方程组中波的相互作用

研究一维Chaplygin气体欧拉方程组中波的相互作用.方程组的波包含接触间断和在密度变量以及内能变量上同时具有狄拉克函数的狄拉克激波.根据这些波的不同组合,问题被分成了7种情形.通过详细地构造每种情形的整体解,获得了各种波相互作用的完整结果.特别地,对于一类初值,两个接触间断相互作用后,产生了一个狄拉克激波;然而,对于另外一类初值,一个狄拉克激波与一个接触间断相互作用后,狄拉克激波消失.这些都是波相互作用中非常特别的现象.