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Quasilinear Hyperbolic Systems (quasilinear + hyperbolic_system)

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Mathematics and Statistics100%

Selected Abstracts

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

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2010
Lixin Yu
Abstract 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. [source]

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

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2009
Yi Zhou
Abstract 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. [source]

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

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 2 2008
Ta-Tsien Li
Abstract One often believes that there is no shock formation for the Cauchy problem of quasilinear hyperbolic systems (of conservation laws) with linearly degenerate characteristic fields. It has been a conjecture for a long time (see Arch. Rational Mech. Anal. 2004; 172:65,91; Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer: New York, 1984) and it is still an open problem in the general situation up to now. In this paper, a framework to justify this conjecture is proposed, and, by means of the concept such as the strict block hyperbolicity, the part richness and the successively block-closed system, some general kinds of quasilinear hyperbolic systems, which verify the conjecture, are given. Copyright © 2007 John Wiley & Sons, Ltd. [source]

The Cauchy problem for quasilinear SG-hyperbolic systems

MATHEMATISCHE NACHRICHTEN, Issue 7 2007
Marco Cappiello
Abstract We study the Cauchy problem for a class of quasilinear hyperbolic systems with coefficients depending on (t, x) , [0, T ] × ,n and presenting a linear growth for |x | , ,. We prove well-posedness in the Schwartz space ,, (,n). The result is obtained by deriving an energy estimate for the solution of the linearized problem in some weighted Sobolev spaces and applying a fixed point argument. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Exact boundary controllability of unsteady flows in a network of open canals

MATHEMATISCHE NACHRICHTEN, Issue 3 2005
Tatsien Li
Abstract By means of the general results on the exact boundary controllability for quasilinear hyperbolic systems, the author establishes the exact boundary controllability of unsteady flows in both a single open canal and a network of open canals with star configuration respectively. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]