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.     

A note on the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems

This paper is concerned with the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems with linearly degenerate characteristic fields. On the basis of the existence result for the global classical solution, we prove that when t tends to the infinity, the solution approaches a combination of C¹ traveling wave solutions, provided that the C¹ norm and the BV norm of the initial data are bounded but possibly large. In contrast to former results obtained by Liu and Zhou [J. Liu, Y. Zhou, Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems, Math. Methods Appl. Sci. 30 (2007) 479-500], ours do not require their assumption that the system is rich in the sense of Serre. Applications include that to the one-dimensional Born-Infeld system arising in string theory and high energy physics.     

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.     

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.     

Blow-up of periodic solutions to reducible quasilinear hyperbolic systems

Under general assumption, we prove that the smooth solution to reducible quasilinear hyperbolic system with periodic initial data in general develop singularities in finite time, provided that the total variation on one period of the initial data is small.     

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.     

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).     

Characteristic boundary layers for parabolic perturbations of quasilinear hyperbolic problems

In this paper, we study the existence and nonlinear stability of the totally characteristic boundary layer for the quasilinear equations with positive definite viscosity matrix under the assumption that the boundary matrix vanishes identically on the boundary x=0. We carry out a series of weighted estimates to the boundary layer equations—Prandtl type equations to get the regularity and the far field behavior of the solutions. This allows us to perform a weighted energy estimate for the error equation to prove the stability of the boundary layers. The stability result finally implies the asymptotic limit of the viscous solutions.     

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.     

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.     

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.     

Multiplicity of solutions for quasilinear elliptic systems with critical growth

We study the existence of multiple solutions for a quasilinear elliptic system of gradient type with critical growth and the possibility of coupling on the subcritical term. The solutions are obtained from a version of the Symmetric Mountain Pass Theorem. The Concentration-Compactness Principle allows to verify that the Palais-Smale condition is satisfied below a certain level. The authors were partially supported by CNPq/Brazil     

Generalized Glimm scheme to the initial boundary value problem of hyperbolic systems of balance laws

In this paper we provide a generalized version of the Glimm scheme to establish the global existence of weak solutions to the initial-boundary value problem of 2×2 hyperbolic systems of conservation laws with source terms. We extend the methods in [J.B. Goodman, Initial boundary value problem for hyperbolic systems of conservation laws, Ph.D. Dissertation, Stanford University, 1982; J.M. Hong, An extension of Glimm’s method to inhomogeneous strictly hyperbolic systems of conservation laws by “weaker than weak” solutions of the Riemann problem, J. Differential Equations 222 (2006) 515-549] to construct the approximate solutions of Riemann and boundary Riemann problems, which can be adopted as the building block of approximate solutions for our initial-boundary value problem. By extending the results in [J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965) 697-715] and showing the weak convergence of residuals, we obtain stability and consistency of the scheme.     

Long time asymptotics for 2 by 2 hyperbolic systems

The goal of this paper is to study the behavior of the energy for 2 by 2 strictly hyperbolic systems. On the one hand we are interested in generalized energy conservation which excludes blow-up and decay of the energy for t→∞. On the other hand we present scattering results which take account of terms of order zero.     

Long-time behavior of solutions to a nonlinear hyperbolic relaxation system

We study the large time behavior of solutions of a one-dimensional hyperbolic relaxation system that may be written as a nonlinear damped wave equation. First, we prove the global existence of a unique solution and their decay properties for sufficiently small initial data. We also show that for some large initial data, solutions blow-up in finite time. For quadratic nonlinearities, we prove that the large time behavior of solutions is given by the fundamental solution of the viscous Burgers equation. In some other cases, the convection term is too weak and the large time behavior is given by the linear heat kernel.     

BREAKDOWN OF CLASSICAL SOLUTIONS TO QUASILINEAR HYPERBOLIC SYSTEMS

This paper deals with the asymptotic behavior of the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with weaker decaying initial data, and obtains a blow-up result for C1 solution to Cauchy problem.     

Existence of solutions for a class of singular quasilinear elliptic resonance problems

The paper concerns a resonance problem for a class of singular quasilinear elliptic equations in weighted Sobolev spaces. The equation set studied is one of the most useful sets of Navier-Stokes equations; these describe the motion of viscous fluid substances such as liquids, gases and so on. By using Galerkin-type techniques, the Brouwer fixed point theorem, and a new weighted compact Sobolev-type embedding theorem established by Shapiro, we show the existence of a nontrivial solution.     

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.