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 C¹ 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 C¹ traveling wave solutions. Finally, we give an application to the equation for time-like extremal surfaces in the Minkowski space-time R1+n.     

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 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.     

Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems

This paper is concerned with the asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems with linearly degenerate characteristic fields. Based on the existence results of global classical solutions, we prove that when t tends to infinity, the solution approaches a combination of C¹ travelling wave solutions, provided that L¹ ∩ L^∞ norm of the initial data as well as its derivative are bounded. Application is given for the time‐like extremal surface in Minkowski space. Copyright © 2006 John Wiley & Sons, Ltd.     

Asymptotic behavior of global classical solutions to Goursat problem for quasilinear hyperbolic systems with small BV data

This paper is concerned with the asymptotic behavior of global C ¹ solutions of the Goursat problem for quasilinear hyperbolic systems. Based on the existence result on the global classical solution, we prove that when t tends to the infinity, the solution approaches a combination of Lipschitz continuous and piecewise C ¹ traveling wave solutions, provided that the C ¹ norm of the boundary data is bounded but possibly large, and the BV norm of the boundary data is sufficiently small. Applications include the 1D compressible Euler equations for Chaplygin gases.     

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.     

Mechanism of singularity formation for quasilinear hyperbolic systems with linearly degenerate characteristic fields

For the Cauchy problem of 1-D first order quasilinear hyperbolic linearly degenerate systems, a new mechanism of singularity formation is given to show that all the W1,p(1<p?+∞) norms of the C¹ solution should blow up simultaneously. It gives a way to verify the property of ODE singularity by directly using the energy method in the framework of C¹ solution.     

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.     

Asymptotic Behavior of Global Classical Solutions of Quasilinear Non-strictly Hyperbolic Systems with Weakly Linear Degeneracy

In this paper, we study the asymptotic behavior of global classical solutions of the Cauchy problem for general quasilinear hyperbolic systems with constant multiple and weakly linearly degenerate characteristic fields. Based on the existence of global classical solution proved by Zhou Yi et al., we show that, when t tends to infinity, the solution approaches a combination of C1 travelling wave solutions, provided that the total variation and the L1 norm of initial data are sufficiently small.     

Global existence of classical solutions of Goursat problem for quasilinear hyperbolic systems with BV data

We investigate the existence of a global classical solution to the Goursat problem for linearly degenerate quasilinear hyperbolic systems. As the result in [A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J. 37 (1988) 409–421] suggests that one may achieve global smoothness even if the C¹ norm of the initial data is large, we prove that, if the C¹ norm of the boundary data is bounded but possibly large, and the BV norm of the boundary data is sufficiently small, then the solution remains C¹ globally in time. Applications include the equation of time‐like extremal surfaces in Minkowski space R^{1 + (1 + n)} and the one‐dimensional Chaplygin gas equations. Copyright © 2013 John Wiley & Sons, Ltd.     

Generalized solutions to partial differential equations of evolution type

This paper deals with a new solution concept for partial differential equations in algebras of generalized functions. Introducing regularized derivatives for generalized functions, we show that the Cauchy problem is wellposed backward and forward in time for every system of linear partial differential equations of evolution type in this sense. We obtain existence and uniqueness of generalized solutions in situations where there is no distributional solution or where even smooth solutions are nonunique. In the case of symmetric hyperbolic systems, the generalized solution has the classical weak solution as macroscopic aspect. Two extensions to nonlinear systems are given: global solutions to quasilinear evolution equations with bounded nonlinearities and local solutions to quasilinear symmetric hyperbolic systems.     

Asymptotic behavior of classical solutions of reducible quasilinear hyperbolic systems with characteristic boundaries

In this paper, we investigate the asymptotic behavior of classical solutions of reducible quasilinear hyperbolic systems with characteristic boundaries. Under some suitable assumptions, we prove that the solution approaches a combination of Lipschitz continuous and piecewise C¹ traveling wave solution. As an application, we apply the result to the equation for time-like extremal surfaces in the Minkowski space-time R1+(1+n).     

Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems

By introducing the concept of the weak linear degeneracy the authors give a complete result for the global existence and for the life span of C¹ solutions to the Cauchy problem for general first order quasilinear hyperbolic systems with initial data small in C¹ norm and with compact support.     

Breakdown of classical solutions to Cauchy problem for inhomogeneous quasilinear hyperbolic systems

This paper deals with the asymptotic behavior of the life-span of classical solutions to Cauchy problem for inhomogeneous quasilinear strictly hyperbolic systems with weaker decaying initial data. Under the assumption that the source term satisfies the corresponding matching condition, we obtains a blow-up result for C ¹ solution to the Cauchy problem.     

Long time behavior of classical solutions to the generalized Goursat problem of quasilinear hyperbolic systems

We investigate the existence of a global classical solution to the generalized Goursat problem. Under some degenerate assumptions of boundary conditions, we prove that the solution approaches a combination of Lipschitz continuous and a piecewise C¹ traveling wave solution.     

拟线性双曲型方程组Cauchy问题行波解的稳定性

当拟线性双曲系统线性退化时,其Cauchy问题最左族和最右族行波解是稳定的.而其中间族行波解未必稳定.我们在弱线性退化条件下,证明了拟线性双曲系统Cauchy问题适当小的W~(1,1)∩L~∞范数适当小的行波解是稳定的,并将此稳定性应用于可对角化的拟线性双曲系统和Chaplygin气体动力学方程组.     

Global exact controllability for quasilinear hyperbolic systems of diagonal form with linearly degenerate characteristics

This paper deals with the global exact controllability for first-order quasilinear hyperbolic systems of diagonal form with linearly degenerate characteristics. When the system has no zero characteristics, we establish the global exact boundary controllability from one arbitrarily preassigned C¹$C^1$ data to another by means of a constructive method, in which the desired boundary controls can be acted either on both sides or only on one side. Sharp estimates on the exact controllable time are given in both cases. When the system has some zero characteristics, the global exact controllability is also established.     

On the breakdown of classical solutions of the planar motion of an elastic string

We prove a result of formation of singularities for the classical solutions of the planar motion of a nonlinear elastic string. In a particular, but physically relevant, case we give a characterization of the global C¹ solutions with positive tension.     

Positive solutions of quasilinear parabolic systems with Dirichlet boundary condition

Coupled systems for a class of quasilinear parabolic equations and the corresponding elliptic systems, including systems of parabolic and ordinary differential equations are investigated. The aim of this paper is to show the existence, uniqueness, and asymptotic behavior of time-dependent solutions. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients Di(ui) may have the property Di(0)=0 for some or all i=1,…,N, and the boundary condition is ui=0. Using the method of upper and lower solutions, we show that a unique global classical time-dependent solution exists and converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a scalar polynomial growth problem, a coupled system of polynomial growth problem, and a two component competition model in ecology.     

Global existence of classical solutions to the Cauchy problem for a kind of quasilinear hyperbolic systems in divergence form

This paper deals with the global existence of classical solutions to a kind of second order quasilinear hyperbolic systems subject to a null condition, with the linear elastodynamic system as its principal part and the nonlinear terms depending on the product of u² and the derivatives of u.