Hyperbolic systems of balance laws via vanishing viscosity

Christoforou, Cleopatra

doi:10.1016/j.jde.2005.03.010

Cited by 34 publications

(63 citation statements)

References 15 publications

(24 reference statements)

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“…In connection with vanishing viscosity approximations, uniform BV bounds for systems of balance laws with dissipative sources were established in [24]. Viscous approximations to the initial-boundary value problem, with suitable boundary conditions, have been studied by Ancona and Bianchini [2].…”

Section: Extensions and Open Problemsmentioning

confidence: 99%

Hyperbolic Conservation Laws: An Illustrated Tutorial

Bressan

2012

Lecture Notes in Mathematics

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These notes provide an introduction to the theory of hyperbolic systems of conservation laws in one space dimension. The various chapters cover the following topics: 1. Meaning of a conservation equation and definition of weak solutions. 2. Hyperbolic systems. Explicit solutions in the linear, constant coefficients case. Nonlinear effects: loss of regularity and wave interactions. 3. Shock waves: Rankine-Hugoniot equations and admissibility conditions. 4. Genuinely nonlinear and linearly degenerate characteristic fields. Centered rarefaction waves. The general solution of the Riemann problem. Wave interaction estimates. 5. Weak solutions to the Cauchy problem, with initial data having small total variation. Approximations generated by the front-tracking method and by the Glimm scheme. 6. Continuous dependence of solutions w.r.t. the initial data, in the L 1 distance. 7. Characterization of solutions which are limits of front tracking approximations. Uniqueness of entropy-admissible weak solutions. 8. Vanishing viscosity approximations. 9.Extensions and open problems. The survey is concluded with an Appendix, reviewing some basic analytical tools used in the previous sections.Throughout the exposition, technical details are mostly left out. The main goal of these notes is to convey basic ideas, also with the aid of a large number of figures.

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Section: Extensions and Open Problemsmentioning

confidence: 99%

Hyperbolic Conservation Laws: An Illustrated Tutorial

Bressan

2012

Lecture Notes in Mathematics

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“…Even in the general case of systems which are neither genuinely nonlinear nor linearly degenerate, global solutions have been constructed by the Glimm scheme [18,21,23,27], by front tracking approximations [4,5], and by vanishing viscosity approximations [7]. In some special cases, existence and uniqueness of global solutions in the presence of a source term were proved in [16,22,12] and in [2,15,1,13,14], respectively.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Toward the uniform bounds on h and p, we consider the weights 12) for some suitable constants κ 1 and κ 2 . Recall that W is defined in (4.4).…”

Section: The Lmentioning

confidence: 99%

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Global Existence of Large BV Solutions in a Model of Granular Flow

Amadori

Shen

2009

Communications in Partial Differential Equations

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In this paper we analyze a set of equations proposed by Hadeler and Kuttler [20], describing the flow of granular matter in terms of the heights of a standing layer and of a moving layer. By a suitable change of variables, the system can be written as a 2 × 2 hyperbolic system of balance laws, which we study in the one-dimensional case. The system is linearly degenerate along two straight lines in the phase plane, and therefore is weakly linearly degenerate at the point of the intersection. The source term is quadratic, consisting of product of two quantities, which are transported with strictly different speeds. Assuming that the initial height of the moving layer is sufficiently small, we prove the global existence of entropy-weak solutions to the Cauchy problem, for a class of initial data with bounded but possibly large total variation.

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“…by Chistoforou [7]. The dissipativeness assumptions imposed to (1.6) are the same as the ones suggested by Dafermos-Hsiao [14], who employed Glimm's scheme [16].…”

Section: Introductionmentioning

confidence: 94%

Systems of Hyperbolic Conservation Laws With Memory

Christoforou

2007

J. Hyper. Differential Equations

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Abstract. Global weak solutions of bounded variation to systems of balance laws with non-local source are constructed by the method of vanishing viscosity. Suitable dissipativeness assumptions are imposed on the source terms to assure convergence of the method. Under these hypotheses, the total variation remains uniformly bounded and integrable in time, the vanishing viscosity solutions are uniformly stable in L 1 with respect to the initial data and converge to equilibrium as t → ∞. The motivation to study these systems is the observation that conservations laws with fading memory can be written in such form under appropriate conditions on the flux.

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Hyperbolic systems of balance laws via vanishing viscosity

Cited by 34 publications

References 15 publications

Hyperbolic Conservation Laws: An Illustrated Tutorial

Hyperbolic Conservation Laws: An Illustrated Tutorial

Global Existence of Large BV Solutions in a Model of Granular Flow

Systems of Hyperbolic Conservation Laws With Memory

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