Compactness for nonlinear continuity equations

Belgacem, Fethi Ben; Jabin, Pierre‐Emmanuel

doi:10.1016/j.jfa.2012.10.005

Cited by 23 publications

(28 citation statements)

References 40 publications

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“…This is achieved through a complete revisiting of the classical compactness theory by obtaining quantitative regularity estimates. The idea is inspired by estimates obtained for nonlinear continuity equations in [7], though with a different method than the one introduced here. Those estimates correspond to critical spaces, also developed and used for instance in works by J. Bourgain, H. Brézis and P. Mironescu and by A.C. Ponce, see [12] and [60].…”

Section: Introductionmentioning

confidence: 99%

Global existence of weak solutions for compressible Navier--Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor

2018

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We prove global existence of appropriate weak solutions for the compressible Navier-Stokes equations for more general stress tensor than those covered by P.-L. Lions and E. Feireisl's theory. More precisely we focus on more general pressure laws which are not thermodynamically stable; we are also able to handle some anisotropy in the viscous stress tensor. To give answers to these two longstanding problems, we revisit the classical compactness theory on the density by obtaining precise quantitative regularity estimates: This requires a more precise analysis of the structure of the equations combined to a novel approach to the compactness of the continuity equation. These two cases open the theory to important physical applications, for instance to describe solar events (virial pressure law), geophysical flows (eddy viscosity) or biological situations (anisotropy).

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Section: Introductionmentioning

confidence: 99%

Global existence of weak solutions for compressible Navier--Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor

2018

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“…We present in this section our main existence result concerning System (1). As usual for global existence of weak solutions to nonlinear PDEs, one has to prove stability estimates for sequences of approximate solutions and construct such approximate sequences.…”

Section: Statements Of the Resultsmentioning

confidence: 99%

Global Weak Solutions of PDEs for Compressible Media: A Compactness Criterion to Cover New Physical Situations

Bresch

Jabin

2017

Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics

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This short paper is an introduction of the memoir recently written by the two authors (see [D.Bresch., P.-E. Jabin, arXiv:1507.04629, (2015]) which concerns the resolution of two longstanding problems: Global existence of weak solutions for compressible Navier-Stokes equations with thermodynamically unstable pressure and with anisotropic stress tensor. We focus here on a Stokes-like system which can for instance model flows in a compressible tissue in biology or in a compressible porous media in petroleum engineering. This allows to explain, on a simpler but still relevant and important system, the tools recently introduced by the authors and to discuss the important results that have been obtained on the compressible Navier-Stokes equations. It is finally a real pleasure to dedicate this paper to G. MÉTIVIER for his 65's Birthday.

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“…The proof of Th. 2 mostly follows the steps of [9], the main improvement being the more precise Proposition 7. First of all by Kruzkov's doubling of variables, see [35], any entropy solution u to (1.1) satisfies in the sense of distributions that ∂ t |u(t, x) − u(t, y)| + div x (a(t, x) F (u(t, x), u(t, y)) + div y (a(t, y) F (u(t, x), u(t, y)) + G(u(t, x), u(t, y)) div x a(t, x) + G(u(t, y), u(t, x)) div y a(t, y) ≤ 0, Note that up to adding a constant in f , we may assume that f (0) = 0 thus normalizingḠ s.t.Ḡ(0, 0) = 0.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Convergence of Numerical Approximations to Non-linear Continuity Equations with Rough Force Fields

Belgacem

Jabin

2019

Arch Rational Mech Anal

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We prove quantitative regularity estimates for the solutions to nonlinear continuity equations and their discretized numerical approximations on Cartesian grids when advected by a rough force field. This allow us to recover the known optimal regularity for linear transport equations but also to obtain the convergence of a wide range of numerical schemes. Our proof is based on a novel commutator estimates, quantifying and extending to the non-linear case the classical commutator approach of the theory of renormalized solutions.

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Compactness for nonlinear continuity equations

Cited by 23 publications

References 40 publications

Global existence of weak solutions for compressible Navier--Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor

Global existence of weak solutions for compressible Navier--Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor

Global Weak Solutions of PDEs for Compressible Media: A Compactness Criterion to Cover New Physical Situations

Convergence of Numerical Approximations to Non-linear Continuity Equations with Rough Force Fields

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