Global solution for an initial boundary value problem of a quasilinear hyperbolic system

Nishida, Takaaki

doi:10.3792/pja/1195521083

Cited by 189 publications

(191 citation statements)

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“…By setting ρ = v −1 (mass) and U = t (ρ, u), initial data are given by U (x, 0) = t (ρ 0 (x), u 0 (x)), (1.3) where ρ 0 (x) ≥ ρ > 0 and ρ 0 (x), u 0 (x) ∈ BV (R) : the space of functions having bounded total variation in R. If the initial data have small total variation, Glimm [7] says that there exists a global in time weak solution. For the isothermal gas equations (γ = 1), Nishida [10] has proved the existence of global weak solutions for the initial data having arbitrarily large total variation. Nishida-Smoller [11] has shown that global weak solutions exist if (γ − 1) times the total variation of the initial data is sufficiently small, which is a generalisation of [10].…”

Section: Wave-front Tracking For the Equations Of Isentropic Gas Dynamentioning

confidence: 99%

“…For the isothermal gas equations (γ = 1), Nishida [10] has proved the existence of global weak solutions for the initial data having arbitrarily large total variation. Nishida-Smoller [11] has shown that global weak solutions exist if (γ − 1) times the total variation of the initial data is sufficiently small, which is a generalisation of [10]. These authors use Glimm's random choice scheme.…”

Section: Wave-front Tracking For the Equations Of Isentropic Gas Dynamentioning

confidence: 99%

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Wave-front tracking for the equations of isentropic gas dynamics

Asakura

2004

Quart. Appl. Math.

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Abstract. We study the 2 × 2 system of conservation laws of the form, which are the model equations of isentropic gas dynamics. Weak global in time solutions are obtained by Nishida-Smoller (CPAM 1973) provided (γ − 1) times the total variation of the initial data is sufficiently small. The aim of this paper is to give an alternative proof by using the Dafermos-Bressan-Risebro wavefront tracking scheme. We obtain new estimates of the total amount of interactions, which also imply the asymptotic decay of the solution. The main idea is to define appropriate amplitude to the path that is a continuation of shock fronts.

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Section: Wave-front Tracking For the Equations Of Isentropic Gas Dynamentioning

confidence: 99%

Section: Wave-front Tracking For the Equations Of Isentropic Gas Dynamentioning

confidence: 99%

Wave-front tracking for the equations of isentropic gas dynamics

Asakura

2004

Quart. Appl. Math.

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“…They established the existence of entropy solutions, under the assumption that the initial mass density is bounded, is bounded away from zero, and has bounded variation. Their result is based on the Glimm scheme and extends an earlier approach for the non-relativistic version (Nishida, 1968).…”

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confidence: 60%

Entropy solutions of the Euler equations for isothermal relativistic fluids

LeFloch

Yamazaki

2007

IJDSDE

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Abstract. We investigate the initial-value problem for the relativistic Euler equations governing isothermal perfect fluid flows, and generalize an approach introduced by LeFloch and Shelukhin in the non-relativistic setting. We establish the existence of globally defined, bounded measurable, entropy solutions with arbitrary large amplitude. An earlier result by Smoller and Temple for the same system covered solutions with bounded variation that avoid the vacuum state. The new framework proposed here provides entropy solutions in a larger function space and allows for the mass density to vanish and the velocity field to approach the speed of light. The relativistic Euler equations become strongly degenerate in both regimes, as the conservative or the flux variables vanish or blow-up. Our proof is based on the method of compensated compactness for nonlinear systems of conservation laws (Tartar, DiPerna) and takes advantage of a scaling invariance property of the isothermal fluid equations. We also rely on properties of the fundamental kernel that generates the mathematical entropy and entropy flux pairs. This kernel exhibits certain singularities on the boundary of its support and we are led to analyze certain nonconservative products (after Dal Maso, LeFloch, and Murat) consisting of functions of bounded variation by measures.

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“…Le résultat principal est la décroissance de la variation totale de In (n) en temps avec le schéma de Glimm (voir [9] et aussi [17]). Nous utilisons également un argument de domaine invariant pour le problème de Riemann dans le plan (n,u) pour contrôler la norme || u || L~ qui détermine la condition C.F.L.…”

Section: Dérivation Du Modèle Et Principaux Résultatsunclassified

Système Euler-Poisson non linéaire. Existence globale de solutions faibles entropiques

Cordier¹,

Peng²

1998

ESAIM: M2AN

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Global solution for an initial boundary value problem of a quasilinear hyperbolic system

Cited by 189 publications

References 4 publications

Wave-front tracking for the equations of isentropic gas dynamics

Wave-front tracking for the equations of isentropic gas dynamics

Entropy solutions of the Euler equations for isothermal relativistic fluids

Système Euler-Poisson non linéaire. Existence globale de solutions faibles entropiques

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