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.     

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.     

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 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 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.     

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.     

Asymptotic stability of Riemann solutions in BGK approximations to certain multidimensional systems of conservation laws

We prove the asymptotic stability of nonplanar two-states Riemann solutions in BGK approximations of a class of multidimensional systems of conservation laws. The latter consists of systems whose flux-functions in different directions share a common complete system of Riemann invariants, the level surfaces of which are hyperplanes. The asymptotic stability to which the main result refers is in the sense of the convergence as t→∞ in of the space of directions ζ=x/t. That is, the solution z(t,x,ξ) of the perturbed Cauchy problem for the corresponding BGK system satisfies as t→∞, in , where R(ζ) is the self-similar entropy solution of the two-states nonplanar Riemann problem for the system of conservation laws.     

Global classical solutions of mixed initial-boundary value problem for quasilinear hyperbolic systems

In this paper, we consider the mixed initial-boundary value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant . Under the assumptions that the system is strictly hyperbolic and linearly degenerate or weakly linearly degenerate, the global existence and uniqueness of C¹ solutions are obtained for small initial and boundary data. We also present two applications for physical models.     

Wave patterns, stability, and slow motions in inviscid and viscous hyperbolic equations with stiff reaction terms

We study the behavior of solutions to the inviscid (A=0) and the viscous (A>0) hyperbolic conservation laws with stiff source terms

     

Nonlinear stability of strong rarefaction waves for the generalized KdV-Burgers-Kuramoto equation with large initial perturbation

In was shown in Ruan et al. (2008) [3] that rarefaction waves for the generalized KdV-Burgers-Kuramoto equation are nonlinearly stable provided that both the strength of the rarefaction waves and the initial perturbation are sufficiently small. The main purpose of this work is concerned with nonlinear stability of strong rarefaction waves for the generalized KdV-Burgers-Kuramoto equation with large initial perturbation. In our results, we do not require the strength of the rarefaction waves to be small and when the smooth nonlinear flux function satisfies certain growth condition at infinity, the initial perturbation can be chosen arbitrarily in , while for a general smooth nonlinear flux function, we need to ask for the L²-norm of the initial perturbation to be small but the L²-norm of the first derivative of the initial perturbation can be large and, consequently, the -norm of the initial perturbation can also be large.     

The mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic systems in the first quadrant

In this paper, we investigate the mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic systems with nonlinear boundary conditions on a half-unbounded domain . Under the assumptions that system is strictly hyperbolic and linearly degenerate, we obtain the global existence and uniqueness of C¹ solutions with the bounded L¹∩L^∞ norm of the initial data as well as their derivatives and appropriate boundary condition. Based on the existence results of global classical solutions, we also prove that when t tends to infinity, the solutions approach a combination of C¹ travelling wave solutions. Under the appropriate assumptions of initial and boundary data, the results can be applied to the equation of time-like extremal surface in Minkowski space R^1+(1+n).     

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.     

A higher-dimensional partial Legendre transform, and regularity of degenerate Monge-Ampère equations

In dimension n?3, we define a generalization of the classical two-dimensional partial Legendre transform, that reduces interior regularity of the generalized Monge-Ampère equation to regularity of a divergence form quasilinear system of special form. This is then used to obtain smoothness of C^2,1 solutions, having n-1 nonvanishing principal curvatures, to certain subelliptic Monge-Ampère equations in dimension n?3. A corollary is that if k?0 vanishes only at nondegenerate critical points, then a C^2,1 convex solution u is smooth if and only if the symmetric function of degree n-1 of the principal curvatures of u is positive, and moreover, u fails to be when not smooth.     

Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum

The asymptotic behavior of solutions of the damped compressible Euler equations is conjectured to obey to the famous porous media equations (PMES). The previous works on this topic concern the case away from vacuum where the system is strictly hyperbolic. In present paper, we prove that the L^∞ entropy weak solution with vacuum, obtained by the compensated compactness theory, converges strongly in space to the unique similarity solution of the related PME, as time goes to infinity.     

Finite time blow up for critical wave equations in high dimensions

We prove that solutions to the critical wave equation (1.1) with dimension n?4 can not be global if the initial values are positive somewhere and nonnegative. This completes the solution to the famous Strauss conjecture about semilinear wave equations of the form . The rest of the cases, the lower-dimensional case n?3, and the sub or super critical cases were settled many years earlier by the work of several authors.     

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.