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].

Cauchy problem for an extended model of combustion

1992 ◽  
Vol 120 (3-4) ◽  
pp. 349-360 ◽  
Author(s):  
Yun-guang Lu
Keyword(s):  
Cauchy Problem ◽  
Existence Results ◽  
Young Measures ◽  
Weak Continuity ◽  
Finite Limit ◽  
Extended Model ◽  
Lipschitz Continuous ◽  
Simplified Method ◽  
The Cauchy Problem ◽  
Definition Of

SynopsisThis paper considers the Cauchy problem for an extended model of combustion (u + qz)t + f(u)x = 0, zt + kg(u)z = 0 with Lp bounded initial data, where g(u) is a piecewise Lipschitz continuous function and its discontinuous points have no finite limit point. The integral representation gives a definition of a weak solution in Lp space. Some existence results are obtained based on a simplified method of compensated compactness in which the weak continuity theorem of 2 * 2 determinants plays a more important role, but the idea of Young measures has been avoided.

Asymptotic behaviour of solutions of hyperbolic conservation laws ut + (um)x = 0 as m → ∞ with inconsistent initial values

1989 ◽  
Vol 113 (1-2) ◽  
pp. 61-71 ◽  
Author(s):  
Xiangsheng Xu
Keyword(s):  
Singular Perturbation ◽  
Asymptotic Behaviour ◽  
Conservation Laws ◽  
Positive Measure ◽  
Hyperbolic Conservation Laws ◽  
Singular Perturbation Problem ◽  
Stationary Equation ◽  
Hyperbolic Conservation ◽  
Perturbation Problem ◽  
Image Position

SynopsisWe study the behaviour of solutions u = um of ut, + (um)x = 0 for t > 0, x ∊ R, u(x, 0) = u0(x), u0 ≧0, u0 ∊ L1(R) as m → ∞. This is a singular perturbation problem about m = ∞ if u0 > 1 on a set of positive measure. It is shown that the limit exists and satisfies the stationary equation

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