Twenty-eight years with “Hyperbolic conservation laws with relaxation”

MASCIA, Corrado

doi:10.1016/s0252-9602(15)30023-0

Cited by 6 publications

(5 citation statements)

References 88 publications

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“…Similar results have been obtained for convection-diffusion systems under the diffusive scaling [29,10,27,2,25,22]. Originally, they presented continuous velocities, see [34], but later on discrete velocities BGK models inspired by the relaxation method have been introduced, see [31] for a survey. In the spirit of the relaxation approximations, the main advantage of discrete velocities BGK models is to deal with semilinear systems, see [32,12,23].…”

Section: Introductionsupporting

confidence: 70%

Strong convergence of a vector-BGK model to the incompressible Navier-Stokes equations via the relative entropy method

Bianchini

2019

Journal de Mathématiques Pures et Appliquées

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The aim of this paper is to prove the strong convergence of the solutions to a vector-BGK model under the diffusive scaling to the incompressible Navier-Stokes equations on the two-dimensional torus. This result holds in any interval of time [0, T ], with T > 0. We also provide the global in time uniform boundedness of the solutions to the approximating system. Our argument is based on the use of local in time H s -estimates for the model, established in a previous work, combined with the L 2 -relative entropy estimate and the interpolation properties of the Sobolev spaces.

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Section: Introductionsupporting

confidence: 70%

Strong convergence of a vector-BGK model to the incompressible Navier-Stokes equations via the relative entropy method

Bianchini

2019

Journal de Mathématiques Pures et Appliquées

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“…In this kind of model and natural generalisations, the question of the stability of constant flows and small travelling waves have been studied extensively using energy methods. See for instance [13] and reference therein, as well as [4], [10] for the stability of constant flows, [12], [19] for the stability of travelling waves, and [2], [18] for generalisations.…”

Section: Previous Stability Results On the Constant Flowmentioning

confidence: 99%

Localisation of perturbations of a constant state in a traffic flow model

Ghoul¹,

Masmoudi²,

Pacherie³

2022

Preprint

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We consider, in the Aw-Rascle-Zhang traffic flow model, the problem of the asymptotic stability of constant flows. By using a perturbative approach, we show the stability in a larger space of perturbation than previous results. Furthermore, we are able to compute where the perturbation is mainly localised in space for a given time, based on the localisation of the perturbation initially. These new ideas can be applied to various other models of hyperbolic conservation laws with relaxations.

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“…This model has been studied in [18,6,10], and the hyperbolic relaxation limit has been investigated. A complete review on hyperbolic conservation laws with relaxation, and a focus on the Jin-Xin system is presented in [16]. By means of the Chapman-Enskog expansion, local attractivity of diffusion waves for the Jin-Xin model was established in [6].…”

mentioning

confidence: 99%

“…Notice that from the theory on hyperbolic systems, [14], the Cauchy problem for (16) with initial data w 0 in H m (R), m ≥ 2, has a unique local smooth solution w ε for each fixed ε > 0. We denote by T ε the maximum time of existence of this local solution and, hereafter, we consider the time interval [0, T * ], with T * ∈ [0, T ε ) for every ε.…”

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confidence: 99%

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Uniform Asymptotic and Convergence Estimates for the Jin--Xin Model Under the Diffusion Scaling

Bianchini¹

2018

SIAM J. Math. Anal.

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We provide sharp decay estimates in time in the context of Sobolev spaces, for smooth solutions to the one dimensional Jin-Xin model under the diffusion scaling, which are uniform with respect to the singular parameter of the scaling. This provides convergence to the limit nonlinear parabolic equation both for large time, and for the vanishing singular parameter. The analysis is performed by means of two main ingredients. First, a crucial change of variables highlights the dissipative property of the Jin-Xin system, and allows to observe a faster decay of the dissipative variable with respect to the conservative one, which is essential in order to close the estimates. Next, the analysis relies on a deep investigation on the Green function of the linearized Jin-Xin model, depending on the singular parameter, combined with the Duhamel formula in order to handle the nonlinear terms.

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Twenty-eight years with “Hyperbolic conservation laws with relaxation”

Cited by 6 publications

References 88 publications

Strong convergence of a vector-BGK model to the incompressible Navier-Stokes equations via the relative entropy method

Strong convergence of a vector-BGK model to the incompressible Navier-Stokes equations via the relative entropy method

Localisation of perturbations of a constant state in a traffic flow model

Uniform Asymptotic and Convergence Estimates for the Jin--Xin Model Under the Diffusion Scaling

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