Global BV Entropy Solutions and Uniqueness for Hyperbolic Systems of Balance Laws

Amadori, Debora; Gosse, Laurent; Guerra, Graziano

doi:10.1007/s002050200198

Cited by 76 publications

(148 citation statements)

References 31 publications

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“…Note that the integral estimates 6 and 7 in Theorem 1.2 ensure that the solution constructed here coincide with those in [1]. Similarly, the characterization 4 of the tangent vector imply that the present solutions coincide with those in [2].…”

Section: Local Inhomogeneous Sourcementioning

confidence: 59%

Hyperbolic Balance Laws with a Non Local Source

Colombo

Guerra

2007

Communications in Partial Differential Equations

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Section: Local Inhomogeneous Sourcementioning

confidence: 59%

Hyperbolic Balance Laws with a Non Local Source

Colombo

Guerra

2007

Communications in Partial Differential Equations

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“…Extensions of these well-posedness results to the initial-boundary value problem and to balance laws with source terms can be found in [31] and in [1], respectively.…”

Section: Extensions and Open Problemsmentioning

confidence: 90%

“…. uniformly distributed over the interval [0,1]. This means that, for every λ ∈ [0, 1], the percentage of points θ i , 1 ≤ i ≤ N which fall inside [0, λ] should approach λ as N → ∞, i.e.…”

Section: The Glimm Schemementioning

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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“…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%

“…6 Concluding Remarks (1). In this paper we proved the global existence of large BV solutions to a 2 × 2 system of balance laws in one space dimension, arising from a model of granular flow.…”

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confidence: 79%

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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Global BV Entropy Solutions and Uniqueness for Hyperbolic Systems of Balance Laws

Cited by 76 publications

References 31 publications

Hyperbolic Balance Laws with a Non Local Source

Hyperbolic Balance Laws with a Non Local Source

Hyperbolic Conservation Laws: An Illustrated Tutorial

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

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