arXiv subjects

Didier Bresch

Publications and source records attributed to Didier Bresch.

32 records · Page 2Linked to original sources

arXiv2019

Global Existence of Entropy-Weak Solutions to the Compressible Navier-Stokes Equations with Non-Linear Density Dependent Viscosities

In this paper, we extend considerably the global existence results of entropy-weak solutions related to compressible Navier-Stokes system with density dependent viscosities obtained, independently (using different strategies), by Vasseur-Yu [Inventiones mathematicae (2016) and arXiv:1501.06803 (2015)] and by Li-Xin [arXiv:1504.06826 (2015)].More precisely we are able to consider a physical symmetric viscous stress tensor $σ=2μ(ρ)\,{\mathbb{D}}(u)+\bigl(λ(ρ){\rm div}u -P(ρ)\bigr)\, {\rm Id}$ where ${\mathbb D}(u) = [\nabla u + \nabla^T u]/2$ with a shear and bulk viscosities (respectively $μ(ρ)$ and $λ(ρ)$) satisfying the BD relation $λ(ρ)=2(μ'(ρ)ρ- μ(ρ))$ and a pressure law $P(ρ)=aρ^γ$ (with $a>0$ a given constant) for any adiabatic constant $γ>1$. The nonlinear shear viscosity $μ(ρ)$ satisfies some lower and upper bounds for low and high densities (our mathematical result includes the case $μ(ρ)= μρ^α$ with $2/3 < α< 4$ and $μ>0$ constant). This provides an answer to a longstanding mathematical question on compressible Navier-Stokes equations with density dependent viscosities as mentioned for instance by F. Rousset in the Bourbaki 69ème année, 2016--2017, no 1135.

math.AP↗

arXiv2018

BD Entropy and Bernis-Friedman Entropy

In this note, we propose in the full generality a link between the BD entropy introduced by D. Bresch andB. Desjardins for the viscous shallow-water equations and the Bernis-Friedman (called BF) dissipative entropyintroduced to study the lubrications equations. Dierent dissipative entropies are obtained playing with the dragterms on the viscous shallow water equations. It helps for instance to prove global existence of nonnegativeweak solutions for the lubrication equations starting from the global existence of nonnegative weak solutions forappropriate viscous shallow-water equations.

math.AP↗

arXiv2018

Finite-energy solutions for compressible two-fluid Stokes system

We prove existence of global in time weak solutions to a compressible two-fluid Stokes system with a single velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. The system appears to be outside the class of problems that can be treated using the classical Lions-Feireisl approach. Adapting the novel compactness tool developed by the first author and P.-E. Jabin in the mono-fluid compressible Navier-Stokes setting, we first prove the weak sequential stability of solutions. Next, we construct weak solutions via unconventional approximation using the Lagrangian formulation, truncations and stability result of trajectories for rough velocity fields.

math.AP↗

arXiv2018

On Navier-Stokes-Korteweg and Euler-Korteweg Systems: Application to Quantum Fluids Models

In this paper, the main objective is to generalize to the Navier-Stokes-Korteweg (with density dependent viscosities satisfying the BD relation) and Euler-Korteweg systems a recent relative entropy [proposed by D. Bresch, P. Noble and J.--P. Vila, (2016)] introduced for the compressible Navier-Stokes equations with a linear density dependent shear viscosity and a zero bulk viscosity. As a concrete application, this helps to justify mathematically the convergence between global weak solutions of the quantum Navier-Stokes system [recently obtained simultaneously by I. Lacroix-Violet and A. Vasseur (2017)] and dissipative solutions of the quantum Euler system when the viscosity coefficient tends to zero: This selects a dissipative solution as the limit of a viscous system. We also get weak-strong uniqueness for the Quantum-Euler and for the Quantum-Navier-Stokes equations. Our results are based on the fact that Euler-Korteweg systems and corresponding Navier--Stokes-Korteweg systems can be reformulated through an augmented system such as the compressible Navier-Stokes system with density dependent viscosities satisfying the BD algebraic relation. This was also observed recently [by D. Bresch, F. Couderc, P. Noble and J.--P. Vila, (2016)] for the Euler-Korteweg system for numerical purposes. As a by-product of our analysis, we show that this augmented formulation helps to define relative entropy estimates for the Euler Korteweg systems in a simplest way compared to recent works [See D. Donatelli, E. Feireisl, P. Marcati (2015) and J. Giesselmann, C. Lattanzio, A.-E. Tzavaras (2017)] with less hypothesis required on the capillary coefficient.

math.AP↗

arXiv2016

Aging and linear response in the Hébraud-Lequeux model for amorphous rheology

We analyse the aging dynamics of the Hébraud-Lequeux model, a self-consistent stochastic model for the evolution of local stress in an amorphous material. We show that the model exhibits initial-condition dependent freezing: the stress diffusion constant decays with time as $D\sim 1/t^2$ during aging so that the cumulative amount of memory that can be erased, which is given by the time integral of $D(t)$, is finite. Accordingly the shear stress relaxation function, which we determine in the long-time regime, only decays to a plateau and becomes progressively elastic as the system ages. The frequency-dependent shear modulus exhibits a corresponding overall decay of the dissipative part with system age, while the characteristic relaxation times scale linearly with age as expected.

cond-mat.soft↗

arXiv2016

Global Existence of Weak Solutions for Compresssible Navier--Stokes Equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor

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).

math.AP↗

arXiv2015

Relative entropy for compressible Navier-Stokes equations with density dependent viscosities and applications

Recently, A. Vasseur and C. Yu have proved the existence of global entropy-weak solutions to the compressible Navier-Stokes equations with viscosities $ν(\varrho)=μ\varrho$ and $λ(\varrho)=0$ and a pressure law under the form $p(\varrho)=a\varrho^γ$ with $a>0$ and $γ>1$ constants. In this note, we propose a non-trivial relative entropy for such system in a periodic box and give some applications. This extends, in some sense, results with constant viscosities initiated by E. Feiersl, B.J. Jin and A. Novotny. We present some mathematical results related to the weak-strong uniqueness, convergence to a dissipative solution of compressible or incompressible Euler equations. As a by-product, this mathematically justifies the convergence of solutions of a viscous shallow water system to solutions of the inviscid shall-water system.

math.AP↗

arXiv2015

New Extended Formulations of Euler-Korteweg Equations Based on a Generalization of the Quantum Bohm Identity

In this note, we propose an original extended formulation of Euler-Korteweg systems based on a generalization of the quantum Bohm potential identity. This new formulation allows to propose a useful construction of a numerical scheme with entropy stability property under a hyperbolic CFL condition. We also comment the use of the identity for compressible Navier-Stokes equations with degenerate viscosities.

math.NA↗

arXiv2015

Two-velocity hydrodynamics in fluid mechanics: Part I Well posedness for zero Mach number systems

In this paper we prove global in time existence of weak solutions to zero Mach number systems arising in fluid mechanics. Relaxing a certain algebraic constraint between the viscosity and the conductivity introduced in [D. Bresch, E.H. Essoufi, and M. Sy, J. Math. Fluid Mech. 2007] gives a more complete answer to an open question formulated in [P.-L. Lions, Oxford 1998]. A new mathematical entropy shows clearly the existence of two-velocity hydrodynamics with a fixed mixture ratio. As an application of our result we first discuss a model of gaseous mixture extending the results of [P. Embid, Comm. Partial Diff. Eqs. 1987] to the global weak solutions framework. Second, we present the ghost effect system studied by [C.D. Levermore, W. Sun, K. Trivisa, SIAM J. Math. Anal. 2012] and discuss a contribution of the density-dependent heat-conductivity coefficient to the issue of existence of weak solutions.

math.AP↗

arXiv2015

Two-velocity hydrodynamics in fluid mechanics: Part II Existence of global $κ$-entropy solutions to compressible Navier-Stokes systems with degenerate viscosities

This paper addresses the issue of global existence of so-called $κ$-entropy solutions to the Navier-Stokes equations for viscous compressible and barotropic fluids with degenerate viscosities. We consider the three dimensional space domain with periodic boundary conditions. Our solutions satisfy the weak formulation of the mass and momentum conservation equations and also a generalization of the BD-entropy identity called: $κ$-entropy. This new entropy involves a mixture parameter $κ\in (0,1)$ between the two velocities $\bf{u}$ and $\bf{u} + 2\nablaφ(\varrho)$ (the latter was introduced by the first two authors in [C. R. Acad. Sci. Paris 2004]), where $\bf{u}$ is the velocity field and $φ$ is a function of the density $\varrho$ defined by $φ'(s)=μ'(s)/s$. As a byproduct of the existence proof, we show that two-velocity hydrodynamics (in the spirit of S.C. Shugrin 1994) is a possible formulation of a model of barotropic compressible flow with degenerate viscosities. It is used in construction of approximate solutions based on a conservative augmented approximate scheme.

math.AP↗

arXiv2010

Mathematical modeling of powder-snow avalanche flows

Powder-snow avalanches are violent natural disasters which represent a major risk for infrastructures and populations in mountain regions. In this study we present a novel model for the simulation of avalanches in the aerosol regime. The second scope of this study is to get more insight into the interaction process between an avalanche and a rigid obstacle. An incompressible model of two miscible fluids can be successfully employed in this type of problems. We allow for mass diffusion between two phases according to the Fick's law. The governing equations are discretized with a contemporary fully implicit finite volume scheme. The solver is able to deal with arbitrary density ratios. Several numerical results are presented. Volume fraction, velocity and pressure fields are presented and discussed. Finally we point out how this methodology can be used for practical problems.

physics.geo-ph↗

arXiv2010

Mathematical derivation of viscous shallow-water equations with zero surface tension

The purpose of this paper is to derive rigorously the so called viscous shallow water equations given for instance page 958-959 in [A. Oron, S.H. Davis, S.G. Bankoff, Rev. Mod. Phys, 69 (1997), 931?980]. Such a system of equations is similar to compressible Navier-Stokes equations for a barotropic fluid with a non-constant viscosity. To do that, we consider a layer of incompressible and Newtonian fluid which is relatively thin, assuming no surface tension at the free surface. The motion of the fluid is described by 3d Navier-Stokes equations with constant viscosity and free surface. We prove that for a set of suitable initial data (asymptotically close to "shallow water initial data"), the Cauchy problem for these equations is well-posed, and the solution converges to the solution of viscous shallow water equations. More precisely, we build the solution of the full problem as a perturbation of the strong solution to the viscous shallow water equations. The method of proof is based on a Lagrangian change of variable that fixes the fluid domain and we have to prove the well-posedness in thin domains: we have to pay a special attention to constants in classical Sobolev inequalities and regularity in Stokes problem.

math.AP↗

arXiv2009

A priori convergence estimates for a rough Poisson-Dirichlet problem with natural vertical boundary conditions

Stents are medical devices designed to modify blood flow in aneurysm sacs, in order to prevent their rupture. Some of them can be considered as a locally periodic rough boundary. In order to approximate blood flow in arteries and vessels of the cardio-vascular system containing stents, we use multi-scale techniques to construct boundary layers and wall laws. Simplifying the flow we turn to consider a 2-dimensional Poisson problem that conserves essential features related to the rough boundary. Then, we investigate convergence of boundary layer approximations and the corresponding wall laws in the case of Neumann type boundary conditions at the inlet and outlet parts of the domain. The difficulty comes from the fact that correctors, for the boundary layers near the rough surface, may introduce error terms on the other portions of the boundary. In order to correct these spurious oscillations, we introduce a vertical boundary layer. Trough a careful study of its behavior, we prove rigorously decay estimates. We then construct complete boundary layers that respect the macroscopic boundary conditions. We also derive error estimates in terms of the roughness size epsilon either for the full boundary layer approximation and for the corresponding averaged wall law.

math.AP↗

arXiv2007

High Order Multi-Scale Wall-Laws, Part I : The Periodic Case

In this work we present new wall-laws boundary conditions including microscopic oscillations. We consider a newtonian flow in domains with periodic rough boundaries that we simplify considering a Laplace operator with periodic inflow and outflow boundary conditions. Following the previous approaches, see [A. Mikelic, W. Jäger, J. Diff. Eqs, 170, 96-122, (2001) ] and [Y. Achdou, O. Pironneau, F. Valentin, J. Comput. Phys, 147, 1, 187-218, (1998)], we construct high order boundary layer approximations and rigorously justify their rates of convergence with respect to epsilon (the roughness' thickness). We establish mathematically a poor convergence rate for averaged second-order wall-laws as it was illustrated numerically for instance in [Y. Achdou, O. Pironneau, F. Valentin, J. Comput. Phys, 147, 1, 187-218, (1998)]. In comparison, we establish exponential error estimates in the case of explicit multi-scale ansatz. This motivates our study to derive implicit first order multi-scale wall-laws and to show that its rate of convergence is at least of order epsilon to the three halves. We provide a numerical assessment of the claims as well as a counter-example that evidences the impossibility of an averaged second order wall-law. Our paper may be seen as the first stone to derive efficient high order wall-laws boundary conditions.

math.AP↗