On the Barotropic Compressible Navier–Stokes Equations

Mellet, Antoine; Vasseur, Alexis

doi:10.1080/03605300600857079

Cited by 262 publications

(355 citation statements)

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“…This is an other essential difference with the case of non-density dependent viscosity. To solve this problem, a new estimate is established in Mellet-Vasseur [32], providing a L ∞ (0, T ; L log L(Ω)) control on ρ|u| 2 . This new estimate provides the weak stability of smooth solutions of (1.3).…”

Section: Introductionmentioning

confidence: 99%

“…The method is based on the Bresch and Desjardins entropy conservation [2]. The main contribution of this paper is to derive the Mellet-Vasseur type inequality [32] for the weak solutions, even if it is not verified by the first level of approximation. This provides existence of global solutions in time, for the compressible Navier-Stokes equations, for any γ > 1 in two dimensional space and for 1 < γ < 3 in three dimensional space, with large initial data possibly vanishing on the vacuum.…”

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confidence: 99%

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Existence of global weak solutions for 3D degenerate compressible Navier–Stokes equations

2016

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Abstract. In this paper, we prove the existence of global weak solutions for 3D compressible Navier-Stokes equations with degenerate viscosity. The method is based on the Bresch and Desjardins entropy conservation [2]. The main contribution of this paper is to derive the Mellet-Vasseur type inequality [32] for the weak solutions, even if it is not verified by the first level of approximation. This provides existence of global solutions in time, for the compressible Navier-Stokes equations, for any γ > 1 in two dimensional space and for 1 < γ < 3 in three dimensional space, with large initial data possibly vanishing on the vacuum. This solves an open problem proposed by Lions in [27].

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%

Existence of global weak solutions for 3D degenerate compressible Navier–Stokes equations

2016

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“…Hsiao and Li [26] need the presence of the drag friction −nu|u| in the momentum equation to prove the strong convergence of √ n δ w δ . This convergence was obtained by Mellet and Vasseur in [43] by proving a bound in a space slightly better than L ∞ (0, T ; L 2 (T d )) (however, excluding a Faedo-Galerkin strategy). Bresch and Desjardins [7] impose conditions on the viscosity coefficients allowing for compactness results for negative powers of the particle density.…”

Section: Corollary 12 (Global Existence For the Quantum Navier-stokementioning

confidence: 99%

“…[16,17] for Navier-Stokes equations and [14,23] for Korteweg-type models. Nonconstant viscosity coefficients are admissible in the analysis of [6,9,26,43]. Hsiao and Li [26] need the presence of the drag friction −nu|u| in the momentum equation to prove the strong convergence of √ n δ w δ .…”

Section: Corollary 12 (Global Existence For the Quantum Navier-stokementioning

confidence: 99%

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Diffusive semiconductor moment equations using Fermi–Dirac statistics

Jüngel

Krause

Pietra

2010

Z. Angew. Math. Phys.

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Abstract. The global-in-time existence of weak solutions to the barotropic compressible quantum Navier-Stokes equations in a three-dimensional torus for large data is proved. The model consists of the mass conservation equation and a momentum balance equation, including a nonlinear thirdorder differential operator, with the quantum Bohm potential, and a density-dependent viscosity. The system has been derived by Brull and Méhats [10] from a Wigner equation using a moment method and a Chapman-Enskog expansion around the quantum equilibrium. The main idea of the existence analysis is to reformulate the quantum Navier-Stokes equations by means of a so-called effective velocity involving a density gradient, leading to a viscous quantum Euler system. The advantage of the new formulation is that there exists a new energy estimate which implies bounds on the second derivative of the particle density. The global existence of weak solutions to the viscous quantum Euler model is shown by using the Faedo-Galerkin method and weak compactness techniques. As a consequence, we deduce the existence of solutions to the quantum Navier-Stokes system if the viscosity constant is smaller than the scaled Planck constant.Key words. Compressible Navier-Stokes equations, quantum Bohm potential, density-dependent viscosity, global existence of solutions, viscous quantum hydrodynamic equations, third-order derivative, energy estimates. Gardner [20] from the Wigner equation by a moment method. More recently, dissipative quantum fluid models have been proposed. For instance, the moment method applied to the Wigner-Fokker-Planck equation leads to viscous quantum Euler models [21], and a Chapman-Enskog expansion in the Wigner equation leads under certain assumptions to quantum Navier-Stokes equations [10]. In this paper, we will reveal a connection between these two models by introducing an effective velocity variable, first used in capillary Korteweg-type models [5], and we will prove the global existence of weak solutions to the multidimensional initial-value problems for any finite-energy initial data. AMSIn the following, we describe the two dissipative quantum systems studied in this paper. The barotropic quantum Navier-Stokes equations for the particle density n and the particle velocity u read as

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Finite Energy Method for Compressible Fluids: The Navier‐Stokes‐Korteweg Model

Germain

LeFloch

2015

Comm Pure Appl Math

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This is the first of a series of papers devoted to the initial value problem for the Euler system of compressible fluids and augmented versions containing higher-order terms. We encompass solutions that have finite total energy and enjoy a certain symmetry (for instance, plane symmetry); these solutions may have unbounded amplitude and contain cavitation regions in which the mass density vanishes. In the present paper, we are interested in dispersive shock waves and analyze the zero viscosity-capillarity limit associated with the Navier-Stokes-Korteweg system. Specifically, we establish the existence of finite energy solutions as well as their convergence toward entropy solutions to the Euler system. We encompass a broad class of nonlinear Navier-Stokes-Korteweg constitutive laws, which is determined by two main conditions relating the viscosity and capillarity coefficients, that is, on one hand the strong coercivity condition (as we call it) which provides a favorable sign for the integrated dissipation associated with an effective energy, and on the other hand the tame capillarity condition (as we call it), which restricts pointwise the strength of the capillarity relatively to the viscosity. Rather mild conditions on the growth of the constitutive functions are aso imposed, which are required in order to define finite energy weak solutions to the Navier-Stokes-Korteweg system, even in the presence of cavitation. Our method of proof relies on fine algebraic properties of the Euler system and combines together energy and effective energy estimates, dissipation and effective dissipation estimates, a nonlinear Sobolev inequality, highintegrability properties for the mass density and for the velocity, and compactness properties based on entropies.

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On the Barotropic Compressible Navier–Stokes Equations

Cited by 262 publications

References 14 publications

Existence of global weak solutions for 3D degenerate compressible Navier–Stokes equations

Existence of global weak solutions for 3D degenerate compressible Navier–Stokes equations

Diffusive semiconductor moment equations using Fermi–Dirac statistics

Finite Energy Method for Compressible Fluids: The Navier‐Stokes‐Korteweg Model

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