Regularizing effect for conservation laws with a Lipschitz convex flux

Guelmame, Billel; Junca, Stéphane; Clamond, Didier

doi:10.4310/cms.2019.v17.n8.a6

Cited by 10 publications

(16 citation statements)

References 31 publications

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“…In this section, we will build some examples to show the optimality of smoothing effect in BV s for all time. This regularity has been obtained in [13,30,17,38,39]. The optimality for all time is new.…”

Section: Introductionmentioning

confidence: 81%

Optimal regularity for all time for entropy solutions of conservation laws in $$BV^s$$

Ghoshal

Guelmame

Jana

et al. 2020

Nonlinear Differ. Equ. Appl.

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This paper deals with the optimal regularity for entropy solutions of conservation laws. For this purpose, we use two key ingredients: (a) fine structure of entropy solutions and (b) fractional BV spaces. We show that optimality of the regularizing effect for the initial value problem from L ∞ to fractional Sobolev space and fractional BV spaces is valid for all time. Previously, such optimality was proven only for a finite time, before the nonlinear interaction of waves. Here for some well-chosen examples, the sharp regularity is obtained after the interaction of waves. Moreover, we prove sharp smoothing in BV s for a convex scalar conservation law with a linear source term. Next, we provide an upper bound of the maximal smoothing effect for nonlinear scalar multi-dimensional conservation laws and some hyperbolic systems in one or multi-dimension.

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Section: Introductionmentioning

confidence: 81%

Optimal regularity for all time for entropy solutions of conservation laws in $$BV^s$$

Ghoshal

Guelmame

Jana

et al. 2020

Nonlinear Differ. Equ. Appl.

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“…For a convex flux with a power law degeneracy, the authors in [8] show an optimal regularizing effects on the solution u inasmuch as the solution u becomes instantaneously BV s loc with s depending on the power law of the flux. For a general nonlinear convex flux locally Lipschitz, the solution belongs for positive time in a generalized BV space related to the nonlinearity of the flux, see [27]. These results have been extended for a nonlinear non convex flux, at least C 3 , in a generalized BV space, and for a more regular flux with polynomial degeneracy in the optimal BV s space by Marconi in [36,37].…”

Section: Examplesmentioning

confidence: 99%

Fractional $BV$ solutions for $2\times 2$ systems of conservation laws with a linearly degenerate field

Haspot,

Junca

2020

Preprint

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The class of 2 × 2 nonlinear hyperbolic systems with one genuinely nonlinear field and one linearly degenerate field are considered. Existence of global weak solutions for small initial data in fractional BV spaces BV s is proved. The exponent s is related to the usual fractional Sobolev derivative. Riemann invariants w and z corresponding respectively to the genuinely nonlinear component and to the linearly degenerate component play different key roles in this work. We obtain the existence of a global weak solution provided that the initial data written in Riemann coordinates (w 0 , z 0 ) are small in BV s × L ∞ , 1/3 ≤ s < 1. The restriction on the exponent s is due to a fundamental result of P.D. Lax, the variation of the Riemann invariant z on the Lax shock curve depends in a cubic way of the variation of the other Riemann invariant w.

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“…and initial discontinuity at the point x x, ξ(ω) := x − (ξ(ω) − 1 /2) is analyzed. According to [25], the pointwise entropy solution is…”

Section: Applications To Hyperbolic Conservation Lawsmentioning

confidence: 99%

“…Scalar conservation law with Lipschitz continuous flux. We consider the example[25, Ex. 3.1], where the scalar conservation law(4.1) …”

mentioning

confidence: 99%

Haar-type stochastic Galerkin formulations for hyperbolic systems with Lipschitz continuous flux function

Gerster¹,

Sikstel²,

Visconti³

2022

Preprint

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This work is devoted to the Galerkin projection of highly nonlinear random quantities. The dependency on a random input is described by Haar-type wavelet systems. The classical Haar sequence has been used by Pettersson, Iaccarino, Nordström (2014) for a hyperbolic stochastic Galerkin formulation of the one-dimensional Euler equations. This work generalizes their approach to several multi-dimensional systems with Lipschitz continuous and non-polynomial flux functions. Theoretical results are illustrated numerically by a genuinely multidimensional CWENO reconstruction.

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Regularizing effect for conservation laws with a Lipschitz convex flux

Cited by 10 publications

References 31 publications

Optimal regularity for all time for entropy solutions of conservation laws in $$BV^s$$

Optimal regularity for all time for entropy solutions of conservation laws in $$BV^s$$

Fractional $BV$ solutions for $2\times 2$ systems of conservation laws with a linearly degenerate field

Haar-type stochastic Galerkin formulations for hyperbolic systems with Lipschitz continuous flux function

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