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.     

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.     

Exact boundary controllability for a kind of second‐order quasilinear hyperbolic systems and its applications

In this paper, by means of a constructive method based on the existence and uniqueness of the semi‐global C² solution, we establish the local exact boundary controllability for a kind of second‐order quasilinear hyperbolic systems. As an application, we obtain the one‐sided local exact boundary controllability for the first‐order quasilinear hyperbolic systems of diagonal form with boundary conditions in which the diagonal variables corresponding to the positive eigenvalues and those corresponding to the negative eigenvalues are decoupled. Copyright © 2009 John Wiley & Sons, Ltd.     

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 exact boundary observability for first‐order quasilinear hyperbolic systems of diagonal form

In this paper, by means of a constructive method based on the theory of the existence and the uniqueness of the C¹ solution to the Cauchy problem and the Goursat problem, the global exact boundary observability for the first‐order quasilinear hyperbolic systems of diagonal form with linearly degenerate characteristics is obtained. In the case that the system has no zero characteristics, we realize the two‐sided and one‐sided global exact boundary observability by the boundary observed values and obtain the observability inequality. Copyright © 2012 John Wiley & Sons, Ltd.     

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

This work is a continuation of our previous work. In the present paper, we study the existence and uniqueness of global piecewise C¹ solutions with shock waves to the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping in the presence of a boundary. It is shown that the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping with nonlinear boundary conditions in the half space {(t, x) | t ≥ 0, x ≥ 0} admits a unique global piecewise C¹ solution u = u (t, x) containing only shock waves with small amplitude and this solution possesses a global structure similar to that of a self‐similar solution u = U (x /t) of the corresponding homogeneous Riemann problem, if each characteristic field with positive velocity is genuinely nonlinear and the corresponding homogeneous Riemann problem has only shock waves but no 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. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations

We establish the existence and stability of multidimensional transonic shocks (hyperbolic‐elliptic shocks) for the Euler equations for steady compressible potential fluids in infinite cylinders. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for velocity, can be written as a second order nonlinear equation of mixed elliptic‐hyperbolic type for the velocity potential. The transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem: The free boundary is the location of the multidimensional transonic shock which divides two regions of C^1,α flow in the infinite cylinder, and the equation is hyperbolic in the upstream region where the C^1,α perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains. Our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region, the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase, and the free boundary is C^1,α, provided that the hyperbolic phase is close in C^1,α to a uniform flow. We further prove that, if the steady perturbation of the hyperbolic phase is C^2,α, the free boundary is C^2,α and stable under the steady perturbation. © 2003 Wiley Periodicals Inc.     

Exact Boundary Controllability for a Kind of Second-Order Quasilinear Hyperbolic Systems

Based on the theory of semi-global C ¹ solution and the local exact boundary controllability for first-order quasilinear hyperbolic systems, the local exact boundary controllability for a kind of second-order quasilinear hyperbolic systems is obtained by a constructive method.     

Global Existence of Smooth Solutions and Convergence to Barenblatt Solutions for the Physical Vacuum Free Boundary Problem of Compressible Euler Equations with Damping

For the physical vacuum free boundary problem with the sound speed being C^1/2‐Hölder continuous near vacuum boundaries of the one‐dimensional compressible Euler equations with damping, the global existence of the smooth solution is proved, which is shown to converge to the Barenblatt self‐similar solution for the porous media equation with the same total mass when the initial datum is a small perturbation of the Barenblatt solution. The pointwise convergence with a rate of density, the convergence rate of velocity in the supremum norm, and the precise expanding rate of the physical vacuum boundaries are also given. The proof is based on a construction of higher‐order weighted functionals with both space and time weights capturing the behavior of solutions both near vacuum states and in large time, an introduction of a new ansatz, higher‐order nonlinear energy estimates, and elliptic estimates.© 2016 Wiley Periodicals, Inc.     

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).     

Exact boundary observability for nonautonomous first-order quasilinear hyperbolic systems

By means of the theory on the semiglobal C¹ solution to the mixed initial-boundary value problem for first-order quasilinear hyperbolic systems, we establish the local exact boundary observability for general nonautonomous first-order quasilinear hyperbolic systems without zero eigenvalues and reveal the essential difference between nonautonomous hyperbolic systems and autonomous hyperbolic systems. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Existence of global smooth solutions for euler equations with symmetry

Eventhough existence of global smooth solutions for one dimensional quasilinear hyperbolic systems has been well established, much less is known about the corresponding results for higher dimensional cases. In this paper, we study the existence of global smoothe solutions for the initial-boundary value problem ofo Euler equtions satisfying γ law with damping and exisymmetry, or spherical symmetry. When the damping is strong enough, we give some sufficient conditions for existence of global smooth solutions as 1<γ< 5 3 and 5 3 <γ<3 . The proof is based on technical estimation of the C ¹ norm of the solutions.     

A Crank‐Nicolson Petrov‐Galerkin method with quadrature for semi‐linear parabolic problems

We propose and analyze an application of a fully discrete C² spline quadrature Petrov‐Galerkin method for spatial discretization of semi‐linear parabolic initial‐boundary value problems on rectangular domains. We prove second order in time and optimal order H¹ norm convergence in space for the extrapolated Crank‐Nicolson quadrature Petrov‐Galerkin scheme. We demonstrate numerically both L² and H¹ norm optimal order convergence of the scheme even if the nonlinear source term is not smooth. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy

We study the asymptotic time behavior of global smooth solutions to general entropy, dissipative, hyperbolic systems of balance laws in m space dimensions, under the Shizuta‐Kawashima condition. We show that these solutions approach a constant equilibrium state in the L^p‐norm at a rate O(t^{? (m/2)(1 ? 1/p)}) as t → ∞ for p ∈ [min{m, 2}, ∞]. Moreover, we can show that we can approximate, with a faster order of convergence, the conservative part of the solution in terms of the linearized hyperbolic operator for m ≥ 2, and by a parabolic equation, in the spirit of Chapman‐Enskog expansion in every space dimension. The main tool is given by a detailed analysis of the Green function for the linearized problem. © 2007 Wiley Periodicals, Inc.     

A Priori Error Estimates of Crank–Nicolson Finite Volume Element Method for a Hyperbolic Optimal Control Problem

In this article, a Crank–Nicolson linear finite volume element scheme is developed to solve a hyperbolic optimal control problem. We use the variational discretization technique for the approximation of the control variable. The optimal convergent order O(h² + k²) is proved for the numerical solution of the control, state and adjoint‐state in a discrete L²‐norm. To derive this result, we also get the error estimate (convergent order O(h² + k²)) of Crank–Nicolson finite volume element approximation for the second‐order hyperbolic initial boundary value problem. Numerical experiments are presented to verify the theoretical results.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1331–1356, 2016     

Point‐wise decay estimate for the global classical solutions to quasilinear hyperbolic systems

In this paper, we first consider the Cauchy problem for quasilinear strictly hyperbolic systems with weak linear degeneracy. The existence of global classical solutions for small and decay initial data was established in (Commun. Partial Differential Equations 1994; 19 :1263–1317; Nonlinear Anal. 1997; 28 :1299–1322; Chin. Ann. Math. 2004; 25B :37–56). We give a new, very simple proof of this result and also give a sharp point‐wise decay estimate of the solution. Then, we consider the mixed initial‐boundary‐value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant. Under the assumption that the positive eigenvalues are weakly linearly degenerate, the global existence of classical solution with small and decay initial and boundary data was established in (Discrete Continuous Dynamical Systems 2005; 12 (1):59–78; Zhou and Yang, in press). We also give a simple proof of this result as well as a sharp point‐wise decay estimate of the solution. Copyright © 2008 John Wiley & Sons, Ltd.     

On a wave map equation arising in general relativity

We consider a class of space‐times for which the essential part of Einstein's equations can be written as a wave map equation. The domain is not the standard one, but the target is hyperbolic space. One ends up with a 1 + 1 nonlinear wave equation, where the space variable belongs to the circle and the time variable belongs to the positive real numbers. The main objective of this paper is to analyze the asymptotics of solutions to these equations as t → ∞. For each point in time, the solution defines a loop in hyperbolic space, and the first result is that the length of this loop tends to 0 as t^?1/2 as t → ∞. In other words, the solution in some sense becomes spatially homogeneous. However, the asymptotic behavior need not be similar to that of spatially homogeneous solutions to the equations. The orbits of such solutions are either a point or a geodesic in the hyperbolic plane. In the nonhomogeneous case, one gets the following asymptotic behavior in the upper half‐plane (after applying an isometry of hyperbolic space if necessary):

• 1 The solution converges to a point.
• 2 The solution converges to the origin on the boundary along a straight line (which need not be perpendicular to the boundary).
• 3 The solution goes to infinity along a curve y = const.
• 4 The solution oscillates around a circle inside the upper half‐plane.

Thus, even though the solutions become spatially homogeneous in the sense that the spatial variations die out, the asymptotic behavior may be radically different from anything observed for spatially homogeneous solutions of the equations. This analysis can then be applied to draw conclusions concerning the associated class of space‐times. For instance, one obtains the leading‐order behavior of the functions appearing in the metric, and one can conclude future causal geodesic completeness. © 2004 Wiley Periodicals, Inc.