Problem For Quasilinear Hyperbolic Systems Research Articles

Global existence of weakly discontinuous solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems

In this paper, the mixed initial-boundary value problem for general first order quasilinear hyperbolic systems with nonlinear boundary conditions in the domain D = {(t, x) | t ≥ 0, x ≥ 0} is considered. A sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution is given.

Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields

In this paper, we study the global existence and the asymptotic behavior of classical solution of the Cauchy problem for quasilinear hyperbolic system with constant multiple and linearly degenerate characteristic fields. We prove that the global C 1 solution exists uniquely if the BV norm of the initial data is sufficiently small. Based on the existence result on the global classical solution, we show that, when the time t tends to the infinity, the solution approaches a combination of C 1 traveling wave solutions. Finally, we give an application to the equation for time-like extremal surfaces in the Minkowski space–time R 1 + n .

Blow-up of solutions to the initial–boundary value problem for quasilinear hyperbolic systems of conservation laws

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. 66 (1) (2007) 93–124]. In the present paper, we study the global structure instability of the self-similar solution u = U ( x t ) containing shocks, contact discontinuities, and at least one rarefaction wave of the initial–boundary Riemann problem for general n × n quasilinear hyperbolic systems of conservation laws on the quarter plane { ( t , x ) ∣ t ≥ 0 , x ≥ 0 } . We prove the nonexistence of global piecewise C 1 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 { ( t , x ) ∣ t ≥ 0 , x ≥ 0 } . Our result indicates that the kind of Riemann solution u = U ( x t ) mentioned above is globally structurally unstable. As an application of our result, we consider the model proposed by Aw and Rascle [A. Aw, M. Rascle, Resurrection of “second order” models of traffic flow, SIAM J. Appl. Math. 60 (2000) 916–938] describing traffic flow on a road network. Following the work of Garavello and Piccoli [M. Garavello, B. Piccoli, Traffic flow on a road network using the Aw–Rascle model, Comm. Partial Differential Equations 31 (2) (2006) 243–275], we prove the nonexistence of global piecewise C 1 solution containing one rarefaction wave of the initial–boundary value problem for this model.

Blow-up mechanism of classical solutions to quasilinear hyperbolic systems

This paper deals with the blow-up phenomenon of classical solutions to the Cauchy problem for quasilinear hyperbolic systems with general small initial data which possess certain decaying properties. We prove that it is the envelope of the same family of characteristics which yields the blowup of classical solutions to the Cauchy problem. Moreover, the blow-up rate is also discussed.

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 C 1 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 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. We also give an example to show that the uniform damping mechanism is not strong enough to prevent the formation of shock waves.

Blow-up Mechanism of Classical Solutions to Quasilinear Hyperbolic Systems in the Critical Case*

This paper deals with the blow-up phenomenon, particularly, the geometric blow-up mechanism, of classical solutions to the Cauchy problem for quasilinear hyperbolic systems in the critical case. We prove that it is still the envelope of the same family of characteristics which yields the blowup of classical solutions to the Cauchy problem in the critical case.

Global exact shock reconstruction for quasilinear hyperbolic systems of conservation laws

In this paper, we consider the inverse generalized Riemann problem for quasilinear hyperbolic systems of conservation laws on the whole domain $t\geq 0$ and obtain that, under suitable conditions, the initial data on the positive (resp. negative) $x$-axis can be uniquely determined by the position of $n$ non-degenerate shocks and the initial data on the negative (resp. positive) $x$-axis.

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.

Exact shock reconstruction**Supported by the Special Funds for Major State Basic Research Projects of China.

In this paper, the exact shock reconstruction is established by solving the inverse generalized Riemann problem for quasilinear hyperbolic systems of conservation laws.

Differential difference inequalities related to hyperbolic functional differential systems and applications

Initial boundary value problems for quasilinear hyperbolic systems are transformed by discretization in space variables into systems of ordinary functional differential equations. Sufficient conditions for the convergence of the method of lines are given. An implicit difference method is proposed for the numerical solving of systems thus obtained. This leads to an implicit difference method for the original problem. A comparison technique is used. We give a complete convergence analysis for the methods and we show by an example that the new methods are considerable better than the classical schemes. Mathematics subject classification (2000): 35R10, 65M20.

Global classical solutions to a kind 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 a half-unbounded domain {$(t,x)|\ t\geq 0,x\geq 0$}. Under the assumption that the positive eigenvalues are weakly linearly degenerate, we obtain the global existence and uniqueness of $C^1$ solution with small and decaying initial data. Some applications are given for the system of the planar motion of an elastic string.

Blowup of solutions to generalized Riemann problem for quasilinear hyperbolic systems of conservation laws

In this paper we study the global structure instability of Riemann solution u = U(x/t), containing at least one centred rarefaction wave, for the general n×n quasilinear hyperbolic system of conservation laws. We prove the non‐existence of the global piecewise C1 solution to a class of generalized Riemann problems, which can be regarded as a perturbation of the corresponding Riemann problem. This result shows that this kind of Riemann solution mentioned above is globally structurally unstable. Applications to quasilinear hyperbolic systems arising from physics and mechanics, particularly to surface waves in hyperelastic materials, are also given.

Pointwise Estimates of Solutions to Cauchy Problem for Quasilinear Hyperbolic Systems

This paper considers the pointwise estimate of the solutions to Cauchy problem for quasilinear hyperbolic systems, which bases on the existence of the solutions by using the fundamental solutions. It gives a sharp pointwise estimates of the solutions on domain under consideration. Specially, the estimate is precise near each characteristic direction.

Local Exact Boundary Controllability for Nonlinear Vibrating String Equations

Based on the theory of semiglobal C1 solutions to a class of nonlocal mixed initial-boundary value problems for quasilinear hyperbolic systems, we establish the local exact boundary controllability for a class of nonlinear vibrating string problems with boundary condition of the third type on one end.

Cauchy Problem for Quasilinear Hyperbolic Systems with Higher Order Dissipative Terms

In this paper, the author studies the global existence, singularities and life span of smooth solutions of the Cauchy problem for a class of quasilinear hyperbolic systems with higher order dissipative terms and gives their applications to nonlinear wave equations with higher order dissipative terms.

Solution [formula omitted] semi-globale et contrôlabilité exacte frontière de systèmes hyperboliques quasi linéaires

In this Note, we first get the existence and uniqueness of semi-global C 1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems with general nonlinear boundary conditions, and then we establish the corresponding exact boundary controllability, provided that the C 1 norm of the initial and final states is small enough.

SEMI-GLOBAL C1SOLUTION TO THE MIXED INITIAL-BOUNDARY VALUE PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS

By means of an equivalent invariant form of boundary conditions, the author get the existence and uniqueness of semi-global C1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems with general nonlinear boundary conditions.

A NECESSARY AND SUFFICIENT CONDITION FOR GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS TO CAUCHY PROBLEM OF QUASILINEAR HYPERBOLIC SYSTEMS IN DIAGONAL FORM

This article considers Cauchy problem for quasilinear hyperbolic systems in diagonal form. A necessary and sufficient condition in guaranteeing that Cauchy problem admits a unique global classical solution on t ≥ 0 is obtained, and a sharp estimate of the life span for the classical solution is given.

Asymptotic blow-up time of classical solutions for general quasilinear hyperbolic systems and its applications

By means of determining the asymptotic blow-up time, which follows Hörmander (Report, Institute Mittag-Leffler, 1985), for classical solutions to Cauchy problem for quasilinear hyperbolic systems with small decay initial data, we give a limit formula on the life span of classical solutions. Applications to quasilinear hyperbolic systems arising from physics and mechanics are given.

Mixed problems for quasilinear hyperbolic systems

Mixed problems for quasilinear hyperbolic systems