scholarly journals A NUMERICAL SCHEME WITH A MESH ON CHARACTERISTICS FOR THE CAUCHY PROBLEM FOR ONE-DIMENSIONAL HYPERBOLIC CONSERVATION LAWS

2009 ◽  
Vol 24 (3) ◽  
pp. 459-466
Author(s):  
Dae-Ki Yoon ◽  
Hong-Joong Kim ◽  
Woon-Jae Hwang
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
Numerical Scheme ◽  
Hyperbolic Conservation Laws ◽  
One Dimensional ◽  
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].

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