Global solutions with shock waves to the generalized Riemann problem for a system of hyperbolic conservation laws with linear damping

It is proven that a class of the generalized Riemann problem for quasilinear hyperbolic systems of conservation laws with the uniform damping term admits a unique global piecewise C¹ solution u=u(t,x) containing only n shock waves with small amplitude on t?0 and this solution possesses a global structure similar to that of the similarity solution of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data. We also give an example to show that the uniform damping mechanism is not strong enough to prevent the formation of shock waves.     

Global structure stability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: shocks and contact discontinuities

In this paper, the author proves the global structure stability of the Lax's Riemann solution , containing only shocks and contact discontinuities, of general n×n quasilinear hyperbolic system of conservation laws. More precisely, the author proves the global existence and uniqueness of the piecewise C¹ solution u=u(t,x) of a class of generalized Riemann problem, which can be regarded as a perturbation of the corresponding Riemann problem, for the quasilinear hyperbolic system of conservation laws; moreover, this solution has a global structure similar to that of the solution . Combining the results in Kong (Global structure instability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: rarefaction waves, to appear), the author proves that the Lax's Riemann solution of general n×n quasilinear hyperbolic system of conservation laws is globally structurally stable if and only if it contains only non-degenerate shocks and contact discontinuities, but no rarefaction waves and other weak discontinuities.     

The Generalized Riemann Problem for First Order Quasilinear Hyperbolic Systems of Conservation Laws II

In this paper, we are concerned with the existence and uniqueness of the local solution to the generalized Riemann problem for first order quasi-linear hyperbolic systems of conservation laws in the presence of the shock wave with large amplitude and the centered wave. Apart from some exceptions, we prove the problem admits a unique piecewise smooth solution u=u(t,x), and this solution has a structure similar to the similarity solution u=u(x/t) of the corresponding Riemann problem in the neighborhood of the origin, provided that the coefficients of the system and the initial conditions are sufficiently smooth. The application of our results in rich system is also given.     

Shock reflection for a system of hyperbolic balance laws

This paper concerns shock reflection for a system of hyperbolic balance laws in one space dimension. It is shown that the generalized nonlinear initial-boundary Riemann problem for a system of hyperbolic balance laws with nonlinear boundary conditions in the half space admits a unique global piecewise C¹ solution u=u(t,x) containing only shocks with small amplitude and this solution possesses a global structure similar to that of self-similar solution of the corresponding homogeneous Riemann problem, if each characteristic field with positive velocity is genuinely nonlinear and the corresponding homogeneous Riemann problem has only shocks but no centered rarefaction waves and contact discontinuities. This result is also applied to shock reflection for the flow equations of a model class of fluids with viscosity induced by fading memory.     

Global solutions with shock waves to the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws

It is proven that the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws admits a unique global piecewise C¹$C^1$ solution u=u(t,x)$u=u(t,x)$ containing only n$n$ shock waves with small amplitude on t?0$t?0$ and this solution possesses a global structure similar to that of the similarity solution u=U(x/t)$u=U(x/t)$ of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data.     

Global weakly discontinuous solutions to quasilinear hyperbolic systems of conservation laws with damping with a kind of non-smooth initial data

For the Cauchy problem with a kind of non-smooth initial data for weakly linearly degenerate hyperbolic systems of conservation laws with the linear damping term, we prove the existence and uniqueness of global weakly discontinuous solution u = u(t, x) containing only n weak discontinuities with small amplitude on t ≥ 0, and this solution possesses a global structure similar to that of the similarity solution of the corresponding homogeneous Riemann problem. As an application of our result, we obtain the existence and uniqueness of global weakly discontinuous solution, continuous and piecewise C ¹ solution with discontinuous first order derivatives, of the flow equations of a model class of fluids with viscosity induced by fading memory. De-Xing Kong: Supported by the National Science Foundation of China(Grant 10371073), the Special Funds for Major State Basic Research Projects of China (Grant 2000077306), the Qi Ming Xing programme of Shanghai Government, and the Project sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, the Ministry of Education of China.     

Global classical discontinuous solutions to a class of generalized riemann problem for general quasilinear hyperbolic systems of conservation laws

In this paper, the authors prove the global existence and uniqueness of piecewise C¹ solution u = u(t, x) containing only n contact discontinuities with small amplitude to the generalized Riemann problem for general linearly degenerate quasilinear hyperbolic systems of conservation laws with small decay initial data. This solution has a global structure similar to the similarity solution u=U(x/t) to the corresponding Riemann problem. The result shows that the similarity solution u=U(x/t) possesses a global nonlinear structural stability.     

The generalized nonlinear initial-boundary Riemann problem for linearly degenerate quasilinear hyperbolic systems of conservation laws

This work is a continuation of our previous work, in the present paper we study the generalized nonlinear initial-boundary Riemann problem with small BV data for linearly degenerate quasilinear hyperbolic systems of conservation laws with nonlinear boundary conditions in a half space . We prove the global existence and uniqueness of piecewise C¹ solution containing only contact discontinuities to a class of the generalized nonlinear initial-boundary Riemann problem, which can be regarded as a small BV perturbation of the corresponding nonlinear initial-boundary Riemann problem, for general n×n linearly degenerate quasilinear hyperbolic system of conservation laws; moreover, this solution has a global structure similar to the one of the self-similar solution to the corresponding nonlinear initial-boundary Riemann problem. Some applications to quasilinear hyperbolic systems of conservation laws arising in the string theory and high energy physics are also given.     

Global structure instability of Riemann solutions for general quasilinear hyperbolic systems of conservation laws in the presence of a boundary

This work is a continuation of our previous work [Z.-Q. Shao, D.-X. Kong, Y.-C. Li, Shock reflection for general quasilinear hyperbolic systems of conservation laws, Nonlinear Anal. TMA 66 (1) (2007) 93-124]. In this paper, we study the global structure instability of the Riemann solution containing shocks, at least one rarefaction wave for general n×n quasilinear hyperbolic systems of conservation laws in the presence of a boundary. We prove the nonexistence of global piecewise C¹ solution to a class of the mixed initial-boundary value problem for general n×n quasilinear hyperbolic systems of conservation laws on the quarter plane. Our result indicates that this kind of Riemann solution mentioned above for general n×n quasilinear hyperbolic systems of conservation laws in the presence of a boundary is globally structurally unstable. Some applications to quasilinear hyperbolic systems of conservation laws arising from physics and mechanics are also given.     

Global weakly discontinuous solutions for hyperbolic conservation laws in the presence of a boundary

This work is a continuation of our previous work, in the present paper we study the mixed initial-boundary value problem for general n×n quasilinear hyperbolic systems of conservation laws with non-linear boundary conditions in the half space . Under the assumption that each characteristic with positive velocity is linearly degenerate, we prove the existence and uniqueness of global weakly discontinuous solution u=u(t,x) with small amplitude, and this solution possesses a global structure similar to that of the self-similar solution of the corresponding Riemann problem. Some applications to quasilinear hyperbolic systems of conservation laws arising in physics and other disciplines, particularly to the system describing the motion of the relativistic string in Minkowski space R1+n, are also given.     

Global structure instability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: Rarefaction waves

This work is a continuation of our previous work (Kong, J. Differential Equations 188 (2003) 242-271) “Global structure stability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: shocks and contact discontinuities”. In the present paper we prove the global structure instability of the Lax's Riemann solution , containing rarefaction waves, of general n×n quasilinear hyperbolic system of conservation laws. Combining the results in (Kong, 2003), we prove that the Lax's Riemann solution of general n×n quasilinear hyperbolic system of conservation laws is globally structurally stable if and only if it contains only non-degenerate shocks and contact discontinuities, but no rarefaction waves and other weak discontinuities.     

Stabilité de la structure d'une solution de Riemann à deux grands chocs

Sunto Costruiamo una soluzione debole di un problema di Cauchy iperbolico conservativo globale in tempo, vicino a un problema di Riemann che presenta due urti di grande ampiezza. Sotto delle ipotesi di stabilità e di smorzamento, un adattamento dello schema di Bressan permette di conservare la struttura a due urti, con delle stime fini della soluzione nei settori definiti da questi urti.

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Stability of the structure of a Riemann wave with two large shocks. We construct a global weak solution of an hyperbolic system of conservation laws, close to a Riemann problem which contains two large shock waves. Under some stability and damping conditions, we define a version of the scheme of A. Bressan which maintains the structure with two shocks in the solution and leads to refined estimates.

     

Solutions to quasilinear hyperbolic conservation laws with initial discontinuities

We study the singular structure of a family of two dimensional non-self-similar global solutions and their interactions for quasilinear hyperbolic conservation laws. For the case when the initial discontinuity happens only on two disjoint unit circles and the initial data are two different constant states, global solutions are constructed and some new phenomena are discovered. In the analysis, we first construct the solution for 0 ≤ t T~*.Then, when T~*≤ t T′, we get a new shock wave between two rarefactions, and then, when t T′,another shock wave between two shock waves occurs. Finally, we give the large time behavior of the solution when t →∞. The technique does not involve dimensional reduction or coordinate transformation.     

Global Perturbation of the Riemann Problem for the System of Compressible Flow Through Porous Media

In this paper we consider the unperturbatcd and perturbated Riemann problem for the damped quasiliuear hyperbolic system {v_t - u_x = 0 u_t + p(v)_x = -αu, α > 0, p'(v} < 0 with initial structure of two rarefaction waves or one rarefaction wave plus one shock wave. Under certain restrictions, it admits a unique global discontinuous solution in a class of piecewise continuous and piecewise smooth functions and keeps the initial structure. Moreover, the shock strength is found decaying exponentially due to damping for the later case.     

Global Exisienoe of Classical Solutions to the Typical Free Boundary Problem for General Quasilinear Hyperbolic Systems and Its Applications

In this paper the authors prove the existence and uniqueness of global classical solutions to the typical free boundary problem for general quasilinear hyperbolic systems. As an application, a unique global discontinuous solution only containing n shocks on t \leq 0 is obtained for a class of generalized Riemann problem for the quasilinear hyperbolic system of n conservation laws.     

Existence in the Large for Pressure-Gradient System

In this paper, the authors use Glimm scheme to study the global existence of BV solutions to Cauchy problem of the pressure-gradient system with large initial data. To this end, some important properties of the shock curves of the pressure-gradient system in the Riemann invariant coordinate system and verify that the shock curves satisfy Diperna’s conditions (see [Diperna, R. J., Existence in the large for quasilinear hyperbolic conservation laws, Arch. Ration. Mech. Anal., 52(3), 1973, 244–257]) are studied. Then they construct the approximate solution sequence through Glimm scheme. By establishing accurate local interaction estimates, they prove the boundedness of the approximate solution sequence and its total variation.     

The generalized nonlinear initial–boundary Riemann problem for quasilinear hyperbolic systems of conservation laws

In this paper, we study the nonlinear initial–boundary Riemann problem and the generalized nonlinear initial–boundary Riemann problem for quasilinear hyperbolic systems of conservation laws with nonlinear boundary conditions on the domain $\{(t,x)|t⩾0,x⩾0\}$. Under the assumption that each positive eigenvalue is either linearly degenerate or genuinely nonlinear, we get the existence and uniqueness of the self-similar solution to the nonlinear initial–boundary Riemann problem and of the global piecewise $C^1$ solution containing only shocks and (or) contact discontinuities to the corresponding generalized nonlinear initial–boundary Riemann problem. It shows that the self-similar solution to the nonlinear initial–boundary Riemann problem possesses the global structural stability.     

Solutions globales avec discontinuités de contact pour les systèmes hyperboliques quasi linéaires de lois de conservation

Under certain conditions we get the global structural stability of the similarity solution with ii contact discontinuities to the Riemann problem for quasilinear hyperbolic systems of n conservation laws.     

Asymptotic Behaviors of the Solutions to Scalar Viscous Conservation Laws on Bounded Interval

Abstract This paper concerns the asymptotic behaviors of the solutions to the initial-boundary value prob-lem for scalar viscous conservations laws u_t+f(u)_x=u_(xx) on[0,1],with the boundary condition u(0,t) =u_,u(1,t)=u_+ and the initial data u(x,0)=u_0(x,0)=u_0(x),where u_≠u_+ and f is a given function satisfyingf'(u)>0 for u under consideration.By means of energy estimates method and under some more regular condi-tions on the initial data,both the global existence and the asymptotic behavior are obtained.When u_u_+, which corresponds to shock waves in inviscid conservation laws, it is established for weak shockwaves,which means that │u_-u_+│is small.Moreover,exponential decay rates are both given.