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.

Asymptotic behavior of BV solutions to hyperbolic systems of balance laws with relaxation

2015 ◽  
Vol 12 (02) ◽  
pp. 277-292 ◽  
Author(s):  
Constantine M. Dafermos
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Total Variation ◽  
Space Dimension ◽  
Hyperbolic Systems ◽  
Relaxation Processes ◽  
Balance Laws ◽  
Type Condition ◽  
Systems Of Balance Laws ◽  
The Cauchy Problem

For hyperbolic systems of balance laws governing relaxation processes, in one space dimension, with source incurring nonnegative entropy production and satisfying a Kawashima-type condition, it is shown that when the initial data have small total variation on (-∞, ∞) and decay rapidly to zero, as |x| → ∞, then the Cauchy problem possesses a unique admissible BV solution, in the large, with total variation decaying to zero, as t → ∞.

HYPERBOLIC SYSTEMS OF BALANCE LAWS WITH WEAK DISSIPATION

2006 ◽  
Vol 03 (03) ◽  
pp. 505-527 ◽  
Author(s):  
C. M. DAFERMOS
Keyword(s):  
Cauchy Problem ◽  
Hyperbolic Systems ◽  
Balance Laws ◽  
Rich Family ◽  
Systems Of Balance Laws ◽  
The Cauchy Problem ◽  
Weak Dissipation

Global BV solutions are constructed to the Cauchy problem for strictly hyperbolic systems of balance laws endowed with a rich family of entropies and source that is merely weakly dissipative, of the type induced by relaxation mechanisms.

HYPERBOLIC SYSTEMS OF BALANCE LAWS WITH WEAK DISSIPATION II

2013 ◽  
Vol 10 (01) ◽  
pp. 173-179 ◽  
Author(s):  
CONSTANTINE M. DAFERMOS
Keyword(s):  
Cauchy Problem ◽  
Entropy Production ◽  
Hyperbolic Systems ◽  
Positive Definite ◽  
Balance Laws ◽  
Systems Of Balance Laws ◽  
The Cauchy Problem ◽  
Weak Dissipation

By extending the analysis in [C. M. Dafermos, Hyperbolic systems of balance laws with weak dissipation, J. Hyperbolic Differ. Equ.3 (2006) 505–527], this note constructs global BV solutions to the Cauchy problem for strictly hyperbolic systems of balance laws endowed with a convex entropy, under the assumption that the entropy production is positive definite.

scholarly journals The Cauchy Problem for the Incompressible 2D-MHD with Power Law-Type Nonlinear Viscous Fluid

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Jae-Myoung Kim
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Global Existence ◽  
Viscous Fluid ◽  
Power Law ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Fluid Flows ◽  
The Cauchy Problem ◽  
Global Existence And Uniqueness

We investigate a motion of the incompressible 2D-MHD with power law-type nonlinear viscous fluid. In this paper, we establish the global existence and uniqueness of a weak solution u , b depending on a number q in ℝ 2 . Moreover, the energy norm of the weak solutions to the fluid flows has decay rate 1 + t − 1 / 2 .

scholarly journals Non-convex entropies for conservation laws with involutions

2013 ◽  
Vol 371 (2005) ◽  
pp. 20120344 ◽  
Author(s):  
Constantine M. Dafermos
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
State Space ◽  
Weak Solutions ◽  
Hyperbolic Systems ◽  
Entropy Inequality ◽  
Classical Solutions ◽  
The Cauchy Problem ◽  
Systems Of Conservation Laws ◽  
Well Posed

The paper discusses systems of conservation laws endowed with involutions and contingent entropies. Under the assumption that the contingent entropy function is convex merely in the direction of a cone in state space, associated with the involution, it is shown that the Cauchy problem is locally well posed in the class of classical solutions, and that classical solutions are unique and stable even within the broader class of weak solutions that satisfy an entropy inequality. This is on a par with the classical theory of solutions to hyperbolic systems of conservation laws endowed with a convex entropy. The equations of elastodynamics provide the prototypical example for the above setting.

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