arXiv subjects

Isabelle Gallagher

Publications and source records attributed to Isabelle Gallagher.

At least 37 records · Page 2Linked to original sources

arXiv2020

Fluctuation theory in the Boltzmann--Grad limit

We develop a rigorous theory of hard-sphere dynamics in the kinetic regime, away from thermal equilibrium. In the low density limit, the empirical density obeys a law of large numbers and the dynamics is governed by the Boltzmann equation. Deviations from this behaviour are described by dynamical correlations, which can be fully characterized for short times. This provides both a fluctuating Boltzmann equation and large deviation asymptotics.

math.AP↗

arXiv2019

A microscopic view on the Fourier law

The Fourier law of heat conduction describes heat diffusion in macroscopic systems. This physical law has been experimentally tested for a large class of physical systems. A natural question is to know whether it can be derived from the microscopic models using the fundamental laws of mechanics.

math.AP↗

arXiv2019

On the convergence of smooth solutions from Boltzmann to Navier-Stokes

In this work, we are interested in the link between strong solutions of the Boltzmann and the Navier-Stokes equations. To justify this connection, our main idea is to use information on the limit system (for instance the fact that the Navier-Stokes equations are globally wellposed in two space dimensions or when the data are small). In particular we prove that the life span of the solutions to the rescaled Boltzmann equation is bounded from below by that of the Navier-Stokes system. We deal with general initial data in the whole space in dimensions 2 and 3, and also with well-prepared data in the case of periodic boundary conditions.

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arXiv2018

Solutions of Navier--Stokes--Maxwell systems in large energy spaces

Large weak solutions to Navier--Stokes--Maxwell systems are not known to exist in their corresponding energy space in full generality. Here, we mainly focus on the three-dimensional setting of a classical incompressible Navier--Stokes--Maxwell system and --- in an effort to build solutions in the largest possible functional spaces --- prove that global solutions exist under the assumption that the initial velocity and electromagnetic fields have finite energy, and that the initial electromagnetic field is small in $\dot H^s\left({\mathbb R}^3\right)$ with $s\in \left[\frac 12,\frac 32\right)$. We also apply our method to improve known results in two dimensions by providing uniform estimates as the speed of light tends to infinity. The method of proof relies on refined energy estimates and a Grönwall-like argument, along with a new maximal estimate on the heat flow in Besov spaces. The latter parabolic estimate allows us to bypass the use of the so-called Chemin--Lerner spaces altogether, which is crucial and could be of independent interest.

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arXiv2018

Deterministics descriptions of the turbulence in the Navier-Stokes equations

This PhD thesis is devoted to deterministic study of the turbulence in the Navier- Stokes equations. The thesis is divided in four independent chapters.The first chapter involves a rigorous discussion about the energy's dissipation law, proposed by theory of the turbulence K41, in the deterministic setting of the homogeneous and incompressible Navier-Stokes equations, with a stationary external force (the force only depends of the spatial variable) and on the whole space R3. The energy's dissipation law, also called the Kolmogorov's dissipation law, characterizes the energy's dissipation rate (in the form of heat) of a turbulent fluid and this law was developed by A.N. Kolmogorov in 1941. However, its deduction (which uses mainly tools of statistics) is not fully understood until our days and then an active research area consists in studying this law in the rigorous framework of the Navier-Stokes equations which describe in a mathematical way the fluids motion and in particular the movement of turbulent fluids. In this setting, the purpose of this chapter is to highlight the fact that if we consider the Navier-Stokes equations on R3 then certain physical quantities, necessary for the study of the Kolmogorov's dissipation law, have no a rigorous definition and then to give a sense to these quantities we suggest to consider the Navier-Stokes equations with an additional damping term. In the framework of these damped equations, we obtain some estimates for the energy's dissipation rate according to the Kolmogorov's dissipation law.In the second chapter we are interested in study the stationary solutions of the damped Navier- Stokes introduced in the previous chapter. These stationary solutions are a particular type of solutions which do not depend of the temporal variable and their study is motivated by the fact that we always consider the Navier-Stokes equations with a stationary external force. In this chapter we study two properties of the stationary solutions : the first property concerns the stability of these solutions where we prove that if we have a control on the external force then all non stationary solution (with depends of both spatial and temporal variables) converges toward a stationary solution. The second property concerns the decay in spatial variable of the stationary solutions. These properties of stationary solutions are a consequence of the damping term introduced in the Navier-Stokes equations.In the third chapter we still study the stationary solutions of Navier-Stokes equations but now we consider the classical equations (without any additional damping term). The purpose of this chapter is to study an other problem related to the deterministic description of the turbulence : the frequency decay of the stationary solutions. Indeed, according to the K41 theory, if the fluid is in a laminar setting then the stationary solutions of the Navier-Stokes equations must exhibit a exponential frequency decay which starts at lows frequencies. But, if the fluid is in a turbulent setting then this exponential frequency decay must be observed only at highs frequencies. In this chapter, using some Fourier analysis tools, we give a precise description of this exponential frequency decay in the laminar and in the turbulent setting.In the fourth and last chapter we return to the stationary solutions of the classical Navier-Stokes equations and we study the uniqueness of these solutions in the particular case without any external force. Following some ideas of G. Seregin, we study the uniqueness of these solutions first in the framework of Lebesgue spaces of and then in the a general framework of Morrey spaces.

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arXiv2018

A non linear estimate on the life span of solutions of the three dimensional Navier-Stokes equations

The purpose of this article is to establish bounds from below for the life span of regular solutions to the incompressible Navier-Stokes system, whichinvolve norms not only of the initial data, but also of nonlinear functions of the initial data. We provide examples showing that those bounds are significant improvements to the one provided by the classical fixed point argument. One of the important ingredients is the use of a scale-invariant energy estimate.

math.AP↗

arXiv2018

Universal dynamics for the defocusing logarithmic Schrodinger equation

We consider the nonlinear Schrodinger equation with a logarithmic nonlinearity in a dispersive regime. We show that the presence of the nonlinearity affects the large time behavior of the solution: the dispersion is faster than usual by a logarithmic factor in time and the positive Sobolev norms of the solution grow logarithmically in time. Moreover, after rescaling in space by the dispersion rate, the modulus of the solution converges to a universal Gaussian profile. These properties are suggested by explicit computations in the case of Gaussian initial data, and remain when an extra power-like nonlinearity is present in the equation. One of the key steps of the proof consists in using the Madelung transform to reduce the equation to a variant of the isothermal compressible Euler equation, whose large time behavior turns out to be governed by a parabolic equation involving a Fokker-Planck operator.

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arXiv2017

Derivation of an ornstein-uhlenbeck process for a massive particle in a rarified gas of particles

We consider the statistical motion of a convex rigid body in a gas of N smaller (spherical) atoms close to thermodynamic equilibrium. Because the rigid body is much bigger and heavier, it undergoes a lot of collisions leading to small deflections. We prove that its velocity is described, in a suitable limit, by an Ornstein-Uhlenbeck process. The strategy of proof relies on Lanford's arguments [17] together with the pruning procedure from [3] to reach diffusive times, much larger than the mean free time. Furthermore, we need to introduce a modified dynamics to avoid pathological collisions of atoms with the rigid body: these collisions, due to the geometry of the rigid body, require developing a new type of trajectory analysis.

math.AP↗

arXiv2017

On stationary two-dimensional flows around a fast rotating disk

We study the two-dimensional stationary Navier-Stokes equations describing flows around a rotating disk. The existence of unique solutions is established for any rotating speed, and qualitative effects of a large rotation are described precisely by exhibiting a boundary layer structure and an axisymmetrization of the flow.

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arXiv2017

Asymptotics of fast rotating density-dependent incompressible fluids in two space dimensions

In the present paper we study the fast rotation limit for viscous incompressible fluids with variable density, whose motion is influenced by the Coriolis force. We restrict our analysis to two dimensional flows. In the case when the initial density is a small perturbation of a constant state, we recover in the limit the convergence to the homogeneous incompressible Navier-Stokes equations (up to an additional term, due to density fluctuations). For general non-homogeneous fluids, the limit equations are instead linear, and the limit dynamics is described in terms of the vorticity and the density oscillation function: we lack enough regularity on the latter to prove convergence on the momentum equation itself. The proof of both results relies on a compensated compactness argument, which enables one to treat also the possible presence of vacuum.

math.AP↗

arXiv2016

One-sided convergence in the Boltzmann-Grad limit

We review various contributions on the fundamental work of Lanford deriving the Boltzmann equation from hard-sphere dynamics in the low density limit. We focus especially on the assumptions made on the initial data and on how they encode irreversibility. The impossibility to reverse time in the Boltzmann equation (expressed for instance by Boltzmann's H-theorem) is related to the lack of convergence of higher order marginals on some singular sets. Explicit counterexamples single out the microscopic sets where the initial data should converge in order to produce the Boltzmann dynamics.

math.PR↗

arXiv2016

From hard sphere dynamics to the Stokes-Fourier equations: An $L^2$ analysis of the Boltzmann-Grad limit

We derive the linear acoustic and Stokes-Fourier equations as the limiting dynamics of a system of N hard spheres of diameter $ε$ in two space dimensions, when N $\rightarrow$ $\infty$, $ε$ $\rightarrow$ 0, N $ε$ = $α$ $\rightarrow$ $\infty$, using the linearized Boltzmann equation as an intermediate step. Our proof is based on Lanford's strategy [18], and on the pruning procedure developed in [5] to improve the convergence time to all kinetic times with a quantitative control which allows us to reach also hydrodynamic time scales. The main novelty here is that uniform L 2 a pri-ori estimates combined with a subtle symmetry argument provide a weak version of chaos, in the form of a cumulant expansion describing the asymptotic decorrelation between the particles. A refined geometric analysis of recollisions is also required in order to discard the possibility of multiple recollisions.

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arXiv2016

Well-posedness of linearized Taylor equations in magnetohydrodynamics

This paper is a first step in the study of the so-called Taylor model, introduced by J.B. Taylor in \cite{Taylor}. This system of nonlinear PDE's is derived from the viscous incompressible MHD equations, through an asymptotics relevant to the Earth's magnetic field. We consider here a simple class of linearizations of the Taylor model, for which we show well-posedness.

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arXiv2016

Dispersive estimates for the Schrödinger operator on step 2 stratified Lie groups

The present paper is dedicated to the proof of dispersive estimates on stratified Lie groups of step 2, for the linear Schrödinger equation involving a sublaplacian. It turns out that the propagator behaves like a wave operator on a space of the same dimension p as the center of the group, and like a Schrödinger operator on a space of the same dimension k as the radical of the canonical skew-symmetric form, which suggests a decay with exponant -(k+p-1)/2. In this article, we identify a property of the canonical skew-symmetric form under which we establish optimal dispersive estimates with this rate. The relevance of this property is discussed through several examples.

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arXiv2016

Blow-up of critical Besov norms at a potential Navier-Stokes singularity

We prove that if an initial datum to the incompressible Navier-Stokes equations in any critical Besov space $\dot B^{-1+\frac 3p}_{p,q}(\mathbb{R}^3)$, with $3 0$, then the norm of the solution in that Besov space becomes unbounded at time $T$. This result, which treats all critical Besov spaces where local existence is known, generalizes the result of Escauriaza, Seregin and Sverak (Uspekhi Mat. Nauk 58(2(350)):3-44, 2003) concerning suitable weak solutions blowing up in $L^3(\mathbb{R}^3)$. Our proof uses profile decompositions and is based on our previous work (Math. Ann. 355(4):1527--1559, 2013) which provided an alternative proof of the $L^3(\mathbb{R}^3)$ result. For very large values of $p$, an iterative method, which may be of independent interest, enables us to use some techniques from the $L^3(\mathbb{R}^3)$ setting. (To appear in Communications in Mathematical Physics.)

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arXiv2015

The Brownian motion as the limit of a deterministic system of hard-spheres

We provide a rigorous derivation of the brownian motion as the limit of a deterministic system of hard-spheres as the number of particles $N$ goes to infinity and their diameter $\varepsilon$ simultaneously goes to $0$, in the fast relaxation limit $α= N\varepsilon^{d-1}\to \infty $ (with a suitable diffusive scaling of the observation time). As suggested by Hilbert in his sixth problem, we rely on a kinetic formulation as an intermediate level of description between the microscopic and the fluid descriptions: we use indeed the linear Boltzmann equation to describe one tagged particle in a gas close to global equilibrium. Our proof is based on the fundamental ideas of Lanford. The main novelty here is the detailed study of the branching process, leading to explicit estimates on pathological collision trees.

math.AP↗

arXiv2015

Limite de diffusion linéaire pour un système déterministe de sphères dures

We provide a rigorous derivation of the brownian motion as the hydrodynamic limit of a deterministic system of hard-spheres as the number of particles $N$ goes to infinity and their diameter $\varepsilon$ simultaneously goes to $0,$ in the fast relaxation limit $N\varepsilon^{d-1}\to \infty $ (with a suitable scaling of the observation time and length). As suggested by Hilbert in his sixth problem, we use Boltzmann's kinetic theory as an intermediate level of description for the gas close to global equilibrium. Our proof relies on the fundamental ideas of Lanford. The main novelty is the detailed study of the branching process, leading to explicit estimates on pathological collision trees.

math.AP↗

arXiv2013

Stability by rescaled weak convergence for the Navier-Stokes equations

We prove a weak stability result for the three-dimensional homogeneous incompressible Navier-Stokes system. More precisely, we investigate the following problem : if a sequence $(u_{0, n})_{n\in \N}$ of initial data, bounded in some scaling invariant space, converges weakly to an initial data $u_0$ which generates a global regular solution, does $u_{0, n}$ generate a global regular solution ? A positive answer in general to this question would imply global regularity for any data, through the following examples $u_{0,n} = n \vf_0(n\cdot)$ or $u_{0,n} = \vf_0(\cdot-x_n)$ with $|x_n|\to \infty$. We therefore introduce a new concept of weak convergence (rescaled weak convergence) under which we are able to give a positive answer. The proof relies on profile decompositions in anisotropic spaces and their propagation by the Navier-Stokes equations.

math.AP↗