arXiv subjects

David Gerard-Varet

Publications and source records attributed to David Gerard-Varet.

At least 19 recordsLinked to original sources

arXiv2025

Stability analysis for active Brownian particle models

We carry out a comprehensive linear stability analysis of active Brownian particle systems around a constant homogeneous state. These scalar models, being important prototypes for the continuous description of active matter, are Fokker-Planck type equations in position-orientation and are known to exhibit motility-induced phase separation. We fully characterize the linear stability and instability regimes, with an explicit threshold depending on the effective speed of the particles. In this way, we rigorously confirm a conjecture on phase separation originating in the physics and applied literature. Our sharp and quantitative (in)stability results are valid both in the non-diffusive case and in the case of small angular diffusion. In the stable non-diffusive regime, we uncover a mixing mechanism reminiscent of Landau damping for the Vlasov equation, albeit with significantly weaker decay. This decay is non-integrable in time and gives rise to substantial mathematical difficulties; in particular, it prevents the use of classical perturbative arguments to treat the case of small angular diffusion.

math.AP↗

arXiv2025

Diffusion-free boundary conditions for the Navier-Stokes equations

We provide a mathematical analysis of the `diffusion-free' boundary conditions recently introduced by Lin and Kerswell for the numerical treatment of inertial waves in a fluid contained in a rotating sphere. We consider here the full setting of the nonlinear Navier-Stokes equation in a general bounded domain $Ω$ of $\mathbb{R}^d$, $d=2$ or $3$. We show that diffusion-free boundary conditions $$ Δu \cdot τ\vert_{\partial Ω} = 0, \quad u \cdot n\vert_{\partial Ω} = 0 \quad \text{ when } d=2, $$ $$ Δu \times n\vert_{\partial Ω} = 0, \quad u \cdot n\vert_{\partial Ω} = 0 \quad \text{ when } d=3, $$ allow for a satisfactory well-posedness theory of the full Navier-Stokes equations (global in time for $d=2$, local for $d=3$). Moreover, we perform a boundary layer analysis in the limit of vanishing viscosity $ν\rightarrow 0$. We establish that the amplitude of the boundary layer flow is in this case of order $ν$, i.e. much lower than in the case of standard Dirichlet or even stress-free conditions. This confirms analytically that this choice of boundary conditions may be used to reduce diffusive effects in numerical studies relying on the Navier-Stokes equation to approach nearly inviscid solutions.

math.AP↗

arXiv2022

Improved well-posedness for the Triple-Deck and related models via concavity

We establish linearized well-posedness of the Triple-Deck system in Gevrey-$\frac32$ regularity in the tangential variable, under concavity assumptions on the background flow. Due to the recent result \cite{DietertGV}, one cannot expect a generic improvement of the result of \cite{IyerVicol} to a weaker regularity class than real analyticity. Our approach exploits two ingredients, through an analysis of space-time modes on the Fourier-Laplace side: i) stability estimates at the vorticity level, that involve the concavity assumption and a subtle iterative scheme adapted from \cite{GVMM} ii) smoothing properties of the Benjamin-Ono like equation satisfied by the Triple-Deck flow at infinity. Interestingly, our treatment of the vorticity equation also adapts to the so-called hydrostatic Navier-Stokes equations: we show for this system a similar Gevrey-$\frac32$ linear well-posedness result for concave data, improving at the linear level the recent work \cite{MR4149066}.

math.AP↗

arXiv2021

A simple justification of effective models for conducting or fluid media with dilute spherical inclusions

We present a gentle approach to the justification of effective media approximations, for PDE's set outside the union of $n \gg 1$ spheres with low volume fraction. To illustrate our approach, we consider three classical examples: the derivation of the so-called strange term, made popular by Cioranescu and Murat, the derivation of the Brinkman term in the Stokes equation, and a scalar analogue of the effective viscosity problem. Under some separation assumption on the spheres, valid for periodic and random distributions of the centers, we recover effective models as $n$ goes to infinity by simple arguments.

math.AP↗

arXiv2021

Derivation of Batchelor-Green formula for random suspensions

This paper is dedicated to the effective viscosity of suspensions without inertia, at low solid volume fraction $ϕ$. The goal is to derive rigorously a $o(ϕ^2)$ formula for the effective viscosity. In previous works, such formula was given for rigid spheres satisfying the strong separation assumption $ d_{min} \ge c ϕ^{-\frac13} r$, where $d_{min}$ is the minimal distance between the spheres and $r$ their radius. It was then applied to both periodic and random configurations with separation, to yield explicit values for the $O(ϕ^2)$ coefficient. We consider here complementary (and certainly more realistic) random configurations, satisfying soft assumptions of separation and long range decorrelation. We justify in this setting the famous Batchelor-Green formula. Our result applies for instance to hardcore Poisson point process with almost minimal hardcore assumption $d_{min} > (2+ε) r$, $ε> 0$.

math.AP↗

arXiv2020

Analysis of the viscosity of dilute suspensions beyond Einstein's formula

We provide a mathematical analysis of the effective viscosity of suspensions of spherical particles in a Stokes flow, at low solid volume fraction $ϕ$. Our objective is to go beyond the Einstein's approximation $μ_{eff}=(1+\frac{5}{2}ϕ)μ$. Assuming a lower bound on the minimal distance between the $N$ particles, we are able to identify the $O(ϕ^2)$ correction to the effective viscosity, which involves pairwise particle interactions. Applying the methodology developped over the last years on Coulomb gases, we are able to tackle the limit $N \rightarrow +\infty$ of the $O(ϕ^2)$-correction, and provide explicit formula for this limit when the particles centers can be described by either periodic or stationary ergodic point processes.

math.AP↗

arXiv2020

Optimal Prandtl expansion around concave boundary layer

We provide an optimal Gevrey stability result for general boundary layer expansions, under a mild concavity condition on the boundary layer profile. Our result generalizes (and even improves in the non strictly concave case) the one obtained in [Gerard-Varet et al, Duke Math. J. 167 (2018)], restricted to expansions of shear flow type.

math.AP↗

arXiv2018

Well-posedness of the Prandtl equation without any structural assumption

We show the local in time well-posedness of the Prandtl equation for data with Gevrey $2$ regularity in $x$ and $H^1$ regularity in $y$. The main novelty of our result is that we do not make any assumption on the structure of the initial data: no monotonicity or hypothesis on the critical points. Moreover, our general result is optimal in terms of regularity, in view of the ill-posedness result of [9].

math.AP↗

arXiv2018

Sobolev stability of Prandtl expansions for the steady Navier-Stokes equations

We show the $H^1$ stability of shear flows of Prandtl type: $U^ν= (U_s(y/\sqrtν),0)$, in the steady two-dimensional Navier-Stokes equations, under the natural assumptions that $U_s(Y) > 0$ for $Y > 0$, $U_s(0) = 0$, and $U_s'(0) > 0$. Our result is in sharp contrast with the unsteady ones, in which at most Gevrey stability can be obtained, even under global monotonicity and concavity hypotheses. It provides the first positive answer to the inviscid limit problem in Sobolev regularity for a non-trivial class of steady Navier-Stokes flows with no-slip boundary condition.

math.AP↗

arXiv2018

Well-posedness of the hydrostatic Navier-Stokes equations

We address the local well-posedness of the hydrostatic Navier-Stokes equations. These equations, sometimes called reduced Navier-Stokes/Prandtl, appear as a formal limit of the Navier-Stokes system in thin domains, under certain constraints on the aspect ratio and the Reynolds number. It is known that without any structural assumption on the initial data, real-analyticity is both necessary and sufficient for the local well-posedness of the system. In this paper we prove that for convex initial data, local well-posedness holds under simple Gevrey regularity.

math.AP↗

arXiv2016

Formal Derivation and Stability Analysis of Boundary Layer Models in MHD

We provide a systematic derivation of boundary layer models in magnetohydrodynamics (MHD), through an asymptotic analysis of the incompressible MHD system. We recover classical linear models, related to the famous Hartmann and Shercliff layers, as well as nonlinear ones, that we call magnetic Prandtl models. We perform their linear stability analysis, emphasizing the stabilizing effect of the magnetic field.

math.AP↗

arXiv2016

Gevrey Stability of Prandtl Expansions for 2D Navier-Stokes

We investigate the stability of boundary layer solutions of the two-dimensional incompressible Navier-Stokes equations. We consider shear flow solutions of Prandtl type : $$ u^ν(t,x,y) \, = \, \big (U^E(t,y) + U^{BL}(t,\frac{y}{\sqrtν})\,, \, 0 \big )\, , \quad 0<ν\ll 1\,. $$ We show that if $U^{BL}$ is monotonic and concave in $Y = y /\sqrtν$ then $u^ν$ is stable over some time interval $(0,T)$, $T$ independent of $ν$, under perturbations with Gevrey regularity in $x$ and Sobolev regularity in $y$. We improve in this way the classical stability results of Sammartino and Caflisch in analytic class (both in $x$ and $y$). Moreover, in the case where $U^{BL}$ is steady and strictly concave, our Gevrey exponent for stability is optimal. The proof relies on new and sharp resolvent estimates for the linearized Orr-Sommerfeld operator.

math.AP↗

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.

math.AP↗

arXiv2013

Recent progress in the theory of homogenization with oscillating Dirichlet data

This note is a summary of the recent paper [9]. Here, we study the homogenization of elliptic systems with Dirichlet boundary condition, when both the coefficients and the boundary datum are oscillating. In particular, in the paper [9], we showed that, the solutions converge in L2 with a power rate, and we identified the homogenized limit system and the homogenized boundary data. Due to a boundary layer phenomenon, this homogenized system depends in a non trivial way on the boundary. The analysis in [9] answers a longstanding open problem, raised for instance in [4]

math.AP↗

arXiv2010

Remarks on the ill-posedness of the Prandtl equation

In the lines of a recent paper by Gerard-Varet and Dormy, we establish various ill-posedness results for the Prandtl equation. By considering perturbations of stationary shear flows, we show that for some linearizations of the Prandtl equation and some $C^\infty$ initial data, local in time $C^\infty$ solutions do not exist. At the nonlinear level, we prove that if a flow exists in the Sobolev setting, it cannot be Lipschitz continuous. Besides ill-posedness in time, we also establish some ill-posedness in space, that casts some light on the results obtained by Oleinik for monotonic data.

math.AP↗

arXiv2009

On the ill-posedness of the Prandtl equation

The concern of this paper is the Cauchy problem for the Prandtl equation. This problem is known to be well-posed for analytic data, or for data with monotonicity properties. We prove here that it is linearly ill-posed in Sobolev type spaces. The key of the analysis is the construction, at high tangential frequencies, of unstable quasimodes for the linearization around solutions with non-degenerate critical points. Interestingly, the strong instability is due to vicosity, which is coherent with well-posedness results obtained for the inviscid version of the equation. A numerical study of this instability is also provided.

math.AP↗

arXiv2008

Time scales separation for dynamo action

The study of dynamo action in astrophysical objects classically involves two timescales: the slow diffusive one and the fast advective one. We investigate the possibility of field amplification on an intermediate timescale associated with time dependent modulations of the flow. We consider a simple steady configuration for which dynamo action is not realised. We study the effect of time dependent perturbations of the flow. We show that some vanishing low frequency perturbations can yield exponential growth of the magnetic field on the typical time scale of oscillation. The dynamo mechanism relies here on a parametric instability associated with transient amplification by shear flows. Consequences on natural dynamos are discussed.

physics.class-ph↗

arXiv2007

The Navier wall law at a boundary with random roughness

We consider the Navier-Stokes equation in a domain with irregular boundaries. The irregularity is modeled by a spatially homogeneous random process, with typical size $\eps \ll 1$. In a parent paper, we derived a homogenized boundary condition of Navier type as $\eps \to 0$. We show here that for a large class of boundaries, this Navier condition provides a $O(\eps^{3/2} |\ln \eps|^{1/2})$ approximation in $L^2$, instead of $O(\eps^{3/2})$ for periodic irregularities. Our result relies on the study of an auxiliary boundary layer system. Decay properties of this boundary layer are deduced from a central limit theorem for dependent variables.

math.AP↗