scholarly journals Nonclassical Shocks and the Cauchy Problem for Nonconvex Conservation Laws

1999 ◽  
Vol 151 (2) ◽  
pp. 345-372 ◽  
Author(s):  
D. Amadori ◽  
P. Baiti ◽  
P.G. LeFloch ◽  
B. Piccoli
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
The Cauchy 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.

Transport-collapse scheme for heterogeneous scalar conservation laws

2018 ◽  
Vol 15 (01) ◽  
pp. 119-132
Author(s):  
Darko Mitrović ◽  
Andrej Novak
Keyword(s):  
Cauchy Problem ◽  
Kinetic Equation ◽  
Conservation Laws ◽  
Admissible Solution ◽  
Scalar Conservation Laws ◽  
Numerical Examples ◽  
The Cauchy Problem ◽  
Scalar Conservation

We extend Brenier’s transport collapse scheme on the Cauchy problem for heterogeneous scalar conservation laws i.e. for the conservation laws with spacetime-dependent coefficients. The method is based on averaging out the solution to the corresponding kinetic equation, and it necessarily converges toward the entropy admissible solution. We also provide numerical examples.

Global BV solutions and relaxation limit for a system of conservation laws

2001 ◽  
Vol 131 (1) ◽  
pp. 1-26 ◽  
Author(s):  
Debora Amadori ◽  
Graziano Guerra
Keyword(s):  
Cauchy Problem ◽  
Nonlinear System ◽  
Conservation Laws ◽  
Bounded Variation ◽  
Equilibrium Curve ◽  
Equilibrium Equations ◽  
System Of Conservation Laws ◽  
The Cauchy Problem ◽  
Image Position

We consider the Cauchy problem for the (strictly hyperbolic, genuinely nonlinear) system of conservation laws with relaxation Assume there exists an equilibrium curve A(u), such that r(u,A(u)) = 0. Under some assumptions on σ and r, we prove the existence of global (in time) solutions of bounded variation, uε, υε, for ε > 0 fixed.As ε → 0, we prove the convergence of a subsequence of uε, υε to some u, υ that satisfy the equilibrium equations

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