scholarly journals Global solution of the cauchy problem for a class of 2 × 2 nonstrictly hyperbolic conservation laws

1982 ◽  
Vol 3 (3) ◽  
pp. 335-375 ◽  
Author(s):  
Blake Temple
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
Global Solution ◽  
Hyperbolic Conservation Laws ◽  
Hyperbolic Conservation ◽  
The Cauchy Problem

Cauchy problem for hyperbolic conservation laws with a relaxation term

1996 ◽  
Vol 126 (4) ◽  
pp. 821-828 ◽  
Author(s):  
Christian Klingenberg ◽  
Yun-guang Lu
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Original System ◽  
Young Measures ◽  
Weak Continuity ◽  
Hyperbolic Conservation ◽  
Relaxation Term ◽  
The Cauchy Problem ◽  
Image Position

This paper considers the Cauchy problem for hyperbolic conservation laws arising in chromatography:with bounded measurable initial data, where the relaxation term g(δ, u, v) converges to zero as the parameter δ > 0 tends to zero. We show that a solution of the equilibrium equationis given by the limit of the solutions of the viscous approximationof the original system as the dissipation ε and the relaxation δ go to zero related by δ = O(ε). The proof of convergence is obtained by a simplified method of compensated compactness [2], avoiding Young measures by using the weak continuity theorem (3.3) of two by two determinants.

scholarly journals Well-posedness theory for stochastically forced conservation laws on Riemannian manifolds

2019 ◽  
Vol 16 (03) ◽  
pp. 519-593
Author(s):  
L. Galimberti ◽  
K. H. Karlsen
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Generalized Solutions ◽  
Well Posedness ◽  
Scalar Conservation Laws ◽  
Stochastic Forcing ◽  
Rigidity Result ◽  
Hyperbolic Conservation ◽  
The Cauchy Problem ◽  
Scalar Conservation

We investigate a class of scalar conservation laws on manifolds driven by multiplicative Gaussian (Itô) noise. The Cauchy problem defined on a Riemanian manifold is shown to be well-posed. We prove existence of generalized kinetic solutions using the vanishing viscosity method. A rigidity result àla Perthame is derived, which implies that generalized solutions are kinetic solutions and that kinetic solutions are uniquely determined by their initial data ([Formula: see text] contraction principle). Deprived of noise, the equations we consider coincide with those analyzed by Ben-Artzi and LeFloch [Well-posedness theory for geometry-compatible hyperbolic conservation laws on manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire 24(6) (2007) 989–1008], who worked with Kružkov–DiPerna solutions. In the Euclidian case, the stochastic equations agree with those examined by Debussche and Vovelle [Scalar conservation laws with stochastic forcing, J. Funct. Anal. 259(4) (2010) 1014–1042].

ON SCALAR HYPERBOLIC CONSERVATION LAWS WITH A DISCONTINUOUS FLUX

2011 ◽  
Vol 21 (01) ◽  
pp. 89-113 ◽  
Author(s):  
MIROSLAV BULÍČEK ◽  
PIOTR GWIAZDA ◽  
JOSEF MÁLEK ◽  
AGNIESZKA ŚWIERCZEWSKA-GWIAZDA
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Entropy Solution ◽  
Spatial Dimension ◽  
Entropy Measure ◽  
Hyperbolic Conservation ◽  
Jump Discontinuities ◽  
New Concepts ◽  
The Cauchy Problem ◽  
Weak Entropy Solution

We study the Cauchy problem for scalar hyperbolic conservation laws with a flux that can have jump discontinuities. We introduce new concepts of entropy weak and measure-valued solution that are consistent with the standard ones if the flux is continuous. Having various definitions of solutions to the problem, we then answer the question what kind of properties the flux should possess in order to establish the existence and/or uniqueness of solution of a particular type. In any space dimension we establish the existence of measure-valued entropy solution for a flux having countable jump discontinuities. Under the additional assumption on the Hölder continuity of the flux at zero, we prove the uniqueness of entropy measure-valued solution, and as a consequence, we establish the existence and uniqueness of weak entropy solution. If we restrict ourselves to one spatial dimension, we prove the existence of weak solution to the problem where the flux has merely monotone jumps; in such a setting we do not require any continuity of the flux at zero.

Sign in / Sign up

Export Citation Format

Share Document