Global existence of weak discontinuous solutions to the Cauchy problem with small BV initial data for quasilinear hyperbolic systems

2014 ◽  
Vol 38 (5) ◽  
pp. 966-979 ◽  
Author(s):  
Libin Wang
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Global Existence ◽  
Hyperbolic Systems ◽  
Discontinuous Solutions ◽  
Quasilinear Hyperbolic Systems ◽  
The Cauchy Problem

Singularities of solutions for first order quasilinear hyperbolic systems

1983 ◽  
Vol 94 (1-2) ◽  
pp. 137-147 ◽  
Author(s):  
Lee Da-tsin(Li Ta-tsien) ◽  
Shi Jia-hong
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Hyperbolic Systems ◽  
Smooth Solutions ◽  
Small Initial Data ◽  
Quasilinear Hyperbolic Systems ◽  
First Order ◽  
Global Smooth Solutions ◽  
The Cauchy Problem

SynopsisIn this paper, the existence of global smooth solutions and the formation of singularities of solutions for strictly hyperbolic systems with general eigenvalues are discussed for the Cauchy problem with essentially periodic small initial data or nonperiodic initial data. A result of Klainerman and Majda is thus extended to the general case.

Global smooth solutions for a class of quasilinear hyperbolic systems with dissipative terms

1997 ◽  
Vol 127 (6) ◽  
pp. 1311-1324 ◽  
Author(s):  
Tong Yang ◽  
Changjiang Zhu ◽  
Huijiang Zhao
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Existence Theorem ◽  
Hyperbolic Systems ◽  
A Priori ◽  
Smooth Solutions ◽  
Quasilinear Hyperbolic Systems ◽  
Global Smooth Solutions ◽  
The Cauchy Problem ◽  
Dissipative Terms

In this paper we prove an existence theorem of global smooth solutions for the Cauchy problem of a class of quasilinear hyperbolic systems with nonlinear dissipative terms under the assumption that only the C0-norm of the initial data is sufficiently small, while the C1-norm of the initial data can be large. The analysis is based on a priori estimates, which are obtained by a generalised Lax transformation.

scholarly journals A remark on the Glimm scheme for inhomogeneous hyperbolic systems of balance laws

2015 ◽  
Vol 12 (04) ◽  
pp. 787-797 ◽  
Author(s):  
Cleopatra Christoforou
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Weak Solutions ◽  
State Vector ◽  
Equilibrium Solution ◽  
Hyperbolic Systems ◽  
Balance Laws ◽  
Glimm Scheme ◽  
Systems Of Balance Laws ◽  
The Cauchy Problem

General hyperbolic systems of balance laws with inhomogeneous flux and source are studied. Global existence of entropy weak solutions to the Cauchy problem is established for small BV data under appropriate assumptions on the decay of the flux and the source with respect to space and time. There is neither a hypothesis about equilibrium solution nor about the dependence of the source on the state vector as previous results have assumed.

scholarly journals Global existence and asymptotic behavior for a generalized Boussinesq type equation without dissipation

2021 ◽  
Author(s):  
Guowei Liu ◽  
Wei Wang ◽  
Qiuju Xu
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Initial Data ◽  
Global Existence ◽  
Linear Group ◽  
Type Equation ◽  
Small Initial Data ◽  
Dispersive Estimate ◽  
The Cauchy Problem

In this paper, we study the Cauchy problem for a generalized Boussinesq type equation in $\mathbb{R}^n$. We establish a dispersive estimate for the linear group associated with the generalized Boussinesq type equation. As applications, the global existence, decay and scattering of solutions are established for small initial data.

Existence and uniqueness of discontinuous solutions for a hyperbolic system

1997 ◽  
Vol 127 (6) ◽  
pp. 1193-1205 ◽  
Author(s):  
Feimin Huang
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Hyperbolic System ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solutions ◽  
Discontinuous Solutions ◽  
The Cauchy Problem ◽  
Global Existence And Uniqueness

In this paper, we prove the global existence and uniqueness of solutions to the Cauchy problem of a hyperbolic system, which probably contains so-called δ-waves.

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