scholarly journals On the Cauchy Problem for a Class of Hyperbolic Systems of Conservation Laws

1994 ◽  
Vol 112 (1) ◽  
pp. 170-178 ◽  
Author(s):  
H. Freistuhler
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
Hyperbolic Systems ◽  
The Cauchy Problem ◽  
Systems Of Conservation Laws

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.

scholarly journals On uniqueness of solutions to conservation laws verifying a single entropy condition

2019 ◽  
Vol 16 (01) ◽  
pp. 157-191 ◽  
Author(s):  
Sam G. Krupa ◽  
Alexis F. Vasseur
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Systems ◽  
Quasilinear Equation ◽  
Scalar Conservation Laws ◽  
Uniqueness Of Solutions ◽  
The Cauchy Problem ◽  
Systems Of Conservation Laws ◽  
Uniqueness Of The Solution ◽  
Scalar Conservation ◽  
Jacobi Equations

For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (see [E. Yu. Panov, Uniqueness of the solution of the Cauchy problem for a first order quasilinear equation with one admissible strictly convex entropy, Mat. Z. 55(5) (1994) 116–129 (in Russian), Math. Notes 55(5) (1994) 517–525]. This single entropy result was proven again by De Lellis, Otto and Westdickenberg about 10 years later [Minimal entropy conditions for Burgers equation, Quart. Appl. Math. 62(4) (2004) 687–700]. These two proofs both rely on the special connection between Hamilton–Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In this paper, we prove the single entropy result for scalar conservation laws without using Hamilton–Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case.

ASYMPTOTIC STABILITY OF NON-PLANAR RIEMANN SOLUTIONS FOR MULTI-D SYSTEMS OF CONSERVATION LAWS WITH SYMMETRIC NONLINEARITIES

2004 ◽  
Vol 01 (03) ◽  
pp. 567-579 ◽  
Author(s):  
HERMANO FRID
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Asymptotic Stability ◽  
Conservation Laws ◽  
Entropy Solution ◽  
Smooth Functions ◽  
Riemann Solutions ◽  
The Cauchy Problem ◽  
Systems Of Conservation Laws ◽  
Self Similar

We study the asymptotic behavior of entropy solutions of the Cauchy problem for multi-dimensional systems of conservation laws of the form [Formula: see text], where the gα are real smooth functions defined in [0,+∞), and when the initial data are perturbations of two-state nonplanar Riemann data. Specifically, if R0(x) is such Riemann data and ψ∈L∞(ℝd)n satisfies ψ(Tx)→0 in [Formula: see text], as T→∞, then an entropy solution, u(x,t), of the Cauchy problem with u(x,0)=R0(x)+ψ(x) satisfies u(ξt,t)→R(ξ) in [Formula: see text], as t→∞, where R(x/t) turns out to be the unique self-similar entropy solution of the corresponding Riemann problem.

scholarly journals Nonlinear conservation laws for the Schrödinger boundary value problems of second order

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Ming Ren ◽  
Shiwei Yun ◽  
Zhenping Li
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Maximum Modulus ◽  
Second Order ◽  
Nonlinear Conservation Laws ◽  
The Cauchy Problem ◽  
Modulus Method ◽  
Semilinear Schrödinger Equation

AbstractIn this paper, we apply a reliable combination of maximum modulus method with respect to the Schrödinger operator and Phragmén–Lindelöf method to investigate nonlinear conservation laws for the Schrödinger boundary value problems of second order. As an application, we prove the global existence to the solution for the Cauchy problem of the semilinear Schrödinger equation. The results reveal that this method is effective and simple.

scholarly journals On the uniqueness of solutions to hyperbolic systems of conservation laws

2021 ◽  
Vol 291 ◽  
pp. 110-153
Author(s):  
Shyam Sundar Ghoshal ◽  
Animesh Jana ◽  
Konstantinos Koumatos
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Systems ◽  
Uniqueness Of Solutions ◽  
Systems Of Conservation Laws
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