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

2010 ◽  
Vol 72 (2) ◽  
pp. 635-650 ◽  
Author(s):  
John M. Hong ◽  
Ying-Chin Su
Keyword(s):  
Boundary Value Problem ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Balance Laws ◽  
Glimm Scheme ◽  
Systems Of Balance Laws ◽  
Initial Boundary Value

Definitions of solutions to the IBVP for multi-dimensional scalar balance laws

2018 ◽  
Vol 15 (02) ◽  
pp. 349-374 ◽  
Author(s):  
Elena Rossi
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
General Framework ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Balance Laws ◽  
Initial Boundary Value

We consider four definitions of solution to the initial-boundary value problem (IBVP) for a scalar balance laws in several space dimensions. These definitions are extended to the same most general framework and then compared. The first aim of this paper is to detail differences and analogies among them. We focus then on the ways the boundary conditions are fulfilled according to each definition, providing also connections among these various modes. The main result is the proof of the equivalence among the presented definitions of solution.

GLOBAL WEAKLY DISCONTINUOUS SOLUTIONS TO THE MIXED INITIAL–BOUNDARY VALUE PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS

2009 ◽  
Vol 19 (07) ◽  
pp. 1099-1138 ◽  
Author(s):  
ZHI-QIANG SHAO
Keyword(s):  
Boundary Value Problem ◽  
A Priori Estimates ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Local Existence ◽  
Discontinuous Solution ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Quasilinear Hyperbolic Systems ◽  
Initial Boundary Value

In this paper, we consider the mixed initial–boundary value problem for first-order quasilinear hyperbolic systems with general nonlinear boundary conditions in the half space {(t, x) | t ≥ 0, x ≥ 0}. Based on the fundamental local existence results and global-in-time a priori estimates, we prove the global existence of a unique weakly discontinuous solution u = u(t, x) with small and decaying initial data, provided that each characteristic with positive velocity is weakly linearly degenerate. Some applications to quasilinear hyperbolic systems arising in physics and other disciplines, particularly to the system describing the motion of the relativistic closed string in the Minkowski space R1+n, are also given.

THE TRACE THEOREMS IN ANISOTROPIC SOBOLEV SPACES AND THEIR APPLICATIONS TO THE CHARACTERISTIC INITIAL BOUNDARY VALUE PROBLEM FOR SYMMETRIC HYPERBOLIC SYSTEMS

1995 ◽  
Vol 05 (08) ◽  
pp. 1079-1092 ◽  
Author(s):  
YASUSHI SHIZUTA ◽  
KÔZÔ YABUTA
Keyword(s):  
Boundary Value Problem ◽  
Sobolev Spaces ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Trace Theorems ◽  
Characteristic Boundary ◽  
Anisotropic Sobolev Spaces ◽  
Initial Boundary Value

Anisotropic Sobolev spaces are introduced in order to study the initial boundary value problem for first-order symmetric hyperbolic systems with characteristic boundary of constant multiplicity. A trace theorem is given and used for showing the necessity of the compatibility condition for the existence of solution that lies in the anisotroropic Sobolev space.

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