Equations of First Order and Conservation Laws

1995 ◽  
pp. 343-410
Author(s):  
Emmanuele DiBenedetto
Keyword(s):  
Conservation Laws ◽  
First Order

First‐order equivalent Lagrangians and conservation laws

1984 ◽  
Vol 25 (6) ◽  
pp. 1776-1779 ◽  
Author(s):  
Sergio Hojman ◽  
Javier Gómez
Keyword(s):  
Conservation Laws ◽  
First Order

scholarly journals Lagrangian Representations for Linear and Nonlinear Transport

2017 ◽  
Vol 63 (3) ◽  
pp. 418-436
Author(s):  
Stefano Bianchini ◽  
Paolo Bonicatto ◽  
Elio Marconi
Keyword(s):  
Conservation Laws ◽  
Continuity Equation ◽  
Space Dimension ◽  
Nonlinear Transport ◽  
Scalar Conservation Laws ◽  
First Order ◽  
Entropy Dissipation ◽  
Linear And Nonlinear ◽  
Scalar Conservation ◽  
The One

In this note we present a unifying approach for two classes of first order partial differential equations: we introduce the notion of Lagrangian representation in the settings of continuity equation and scalar conservation laws. This yields, on the one hand, the uniqueness of weak solutions to transport equation driven by a two dimensional BV nearly incompressible vector field. On the other hand, it is proved that the entropy dissipation measure for scalar conservation laws in one space dimension is concentrated on countably many Lipschitz curves.

scholarly journals A Bhatnagar–Gross–Krook approximation to scalar conservation laws with discontinuous flux

2010 ◽  
Vol 140 (5) ◽  
pp. 953-972 ◽  
Author(s):  
F. Berthelin ◽  
J. Vovelle
Keyword(s):  
Conservation Laws ◽  
Entropy Solution ◽  
Limit Problem ◽  
Scalar Conservation Laws ◽  
First Order ◽  
The Cauchy Problem ◽  
Discontinuous Flux ◽  
Scalar Conservation ◽  
Bgk Approximation ◽  
Well Posed

AbstractWe study the Bhatnagar–Gross–Krook (BGK) approximation to first-order scalar conservation laws with a flux which is discontinuous in the space variable. We show that the Cauchy problem for the BGK approximation is well posed and that, as the relaxation parameter tends to 0, it converges to the (entropy) solution of the limit problem.

ABOUT UNIQUENESS OF WEAK SOLUTIONS TO FIRST ORDER QUASI-LINEAR EQUATIONS

2002 ◽  
Vol 12 (11) ◽  
pp. 1599-1615 ◽  
Author(s):  
J. NIETO ◽  
J. SOLER ◽  
F. POUPAUD
Keyword(s):  
Conservation Laws ◽  
Weak Solutions ◽  
Entropy Solution ◽  
Linear Equations ◽  
Particle Method ◽  
Entropy Condition ◽  
First Order

In this paper we give a criterion to discriminate the entropy solution to quasi-linear equations of first order among weak solutions. This uniqueness statement is a generalization of Oleinik's criterion, which makes reference to the measure of the increasing character of weak solutions. The link between Oleinik's criterion and the entropy condition due to Kruzhkov is also clarified. An application of this analysis to the convergence of the particle method for conservation laws is also given by using the Filippov characteristics.

scholarly journals Conservation laws in anisotropic elasticity I. Basic framework

1993 ◽  
Vol 443 (1917) ◽  
pp. 139-151 ◽  
Keyword(s):  
Conservation Laws ◽  
General Procedure ◽  
Anisotropic Elasticity ◽  
Two Dimensional ◽  
Real Form ◽  
Elastic Materials ◽  
Complete Class ◽  
First Order ◽  
Cauchy's Theorem ◽  
Anisotropic Elastic

A complete class of first order conservation laws for two dimensional deformations in general anisotropic elastic materials is derived. The derivations are based on Stroh’s formalism for anisotropic elasticity. The general procedure proposed by P. J. Olver for the construction of conservation integrals is followed. It is shown that the conservation laws are intimately connected with Cauchy’s theorem for complex analytic functions. Real-form conservation laws that are valid for degenerate or non-degenerate materials are given.

Conservation laws for plane steady potential barotropic flow

2013 ◽  
Vol 24 (6) ◽  
pp. 789-801 ◽  
Author(s):  
YU. A. CHIRKUNOV ◽  
S. B. MEDVEDEV
Keyword(s):  
Conservation Laws ◽  
Velocity Potential ◽  
Zero Order ◽  
System Of Equations ◽  
Nonlinear System Of Equations ◽  
Nonlinear Conservation Laws ◽  
First Order ◽  
Barotropic Flow ◽  
Chaplygin System ◽  
Steady Potential

It is shown that the set of conservation laws for the nonlinear system of equations describing plane steady potential barotropic flow of gas is given by the set of conservation laws for the linear Chaplygin system. All the conservation laws of zero order for the Chaplygin system are found. These include both known and new nonlinear conservation laws. It is found that the number of conservation laws of the first order is not more than three, assuming that the laws do not depend on the velocity potential and are not non-obvious ones. The components of these conservation laws are quadratic with respect to the stream function and its derivatives. All the Chaplygin functions are found, for which the Chaplygin system has three non-obvious conservation laws of the first order that are independent of velocity potential. All such non-obvious first-order conservation laws are found.

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