arXiv subjects

David Gérard-Varet

Publications and source records attributed to David Gérard-Varet.

At least 19 recordsLinked to original sources

arXiv2026

A variational approach to the derivation of reduced models for bubbly flows

In this paper, we derive reduced models for the motion of gas bubbles in an ambient inviscid liquid, using Hamilton's least action principle. We first explain how to recover from this principle the classical sharp interface model, in which the pressure is continuous across the surfaces of the bubbles. We then show how to reduce the complexity of the model, by simplifying the description of those surfaces. Namely, we impose them to evolve within a subclass of hypersurfaces described by a finite number of parameters (the simplest example being spheres, that is neglecting deviation of the bubbles from sphericity). The difficulty from a mathematical and modeling point of view is to determine the interface conditions that substitute to pressure continuity. We complete the derivation of the reduced models by some well-posedness analysis, in the case of curl-free liquid flow and homogeneous pressure in the bubbles.

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arXiv2025

Nonlinear stability for active suspensions

This paper is devoted to the nonlinear analysis of a kinetic model introduced by Saintillan and Shelley to describe suspensions of active rodlike particles in viscous flows. We investigate the stability of the constant state $Ψ(t,x,p) = \frac{1}{4π}$ corresponding to a distribution of particles that is homogeneous in space (variable $x \in \mathbb{T}^3$) and uniform in orientation (variable $p \in \mathbb{S}^2$). We prove its nonlinear stability under the optimal condition of linearized spectral stability. The main achievement in this work is that the smallness condition on the initial perturbation is independent of the translational diffusion and only depends on the rotational diffusion, which is particularly relevant for dilute suspensions. Upgrading our previous linear study to such nonlinear stability result requires new mathematical ideas, due to the presence of a quasilinear term in $x$ associated with nonlinear convection. This term cannot be treated as a source, because it is not controllable by the rotational diffusion in $p$. Also, it prevents the decoupling of $x$-Fourier modes crucially used in our previous paper. A key feature of our work is an analysis of enhanced dissipation and mixing properties of the advection diffusion operator $\partial_t + (p + u(t,x)) \cdot \nabla_x - νΔ_p$ on $\mathbb{T}^3 \times \mathbb{S}^2$ for a given appropriately small vector field $u$. We hope this linear analysis to be of independent interest, and useful in other contexts with partial or anisotropic diffusions.

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arXiv2025

Weak solutions of a viscous model for fluid-bubbles interaction

We present a system of Navier-Stokes type that describes the dynamics of several spherical bubbles of gas in a liquid. It is derived from a more complete model, where the bubbles are seen as inclusions of gas of homogeneous barotropic pressure with free surfaces. The usual condition of continuity of the stress is relaxed in order to preserve the sphericity of the bubbles through time. We construct weak solutions à la Leray for this relaxed system, up to collision between the bubbles. Although these solutions are reminiscent of weak solutions for fluid-solid interaction systems, accounting for the compression/dilation of the bubbles creates new and significant mathematical difficulties.

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arXiv2025

Correction to Doi type models for suspensions

Starting from microscopic $N$ particle systems, we study the derivation of Doi type models for suspensions of non-spherical particles in Stokes flows. While Doi models accurately describe the effective evolution of the spatial particle density to the first order in the particle volume fraction, this accuracy fails regarding the evolution of the particle orientations. We rigorously attribute this failure to the singular interaction of the particles via a $-3$-homogeneous kernel. In the situation that the particles are initially distributed according to a stationary ergodic point process, we identify the limit of this singular interaction term. It consists of two parts. The first corresponds to a classical term in Doi type models. The second new term depends on the (microscopic) $2$-point correlation of the point process. By including this term, we provide a modification of the Doi model that is accurate to first order in the particle volume fraction.

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arXiv2024

Analysis of a sedimenting suspension near a vertical wall

We consider a sedimenting suspension in a Stokes flow, in the presence of a vertical wall. We study the effect of a particle-depleted fluid layer near the wall on the bulk dynamics of the suspension. We show that this effect can be captured by an appropriate wall law of Navier type. We provide in this way a rigorous justification of the apparent slip observed in many experiments. We also discuss the phenomenon of intrinsic convection predicted in some physics articles.

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arXiv2023

Orientation mixing in active suspensions

We study a popular kinetic model introduced by Saintillan and Shelley for the dynamics of suspensions of active elongated particles where the particles are described by a distribution in space and orientation. The uniform distribution of particles is the stationary state of incoherence which is known to exhibit a phase transition. We perform an extensive study of the linearised evolution around the incoherent state. We show (i) in the non-diffusive regime corresponding to spectral (neutral) stability that the suspensions experiences a mixing phenomenon similar to Landau damping and we provide optimal pointwise in time decay rates in weak topology. Further, we show (ii) in the case of small rotational diffusion $ν$ that the mixing estimates persist up to time scale $ν^{-1/2}$ until the exponential decay at enhanced dissipation rate $ν^{1/2}$ takes over.The interesting feature is that the usual velocity variable of kinetic models is replaced by an orientation variable on the sphere. The associated \emph{orientation mixing} leads to limited algebraic decay for macroscopic quantities. For the proof, we start with a general pointwise decay results for Volterra equations that may be of independent interest. While, in the non-diffusive case, explicit formulas on the sphere allow to conclude the desired decay, much more work is required in the diffusive case: here we prove mixing estimates for the advection-diffusion equation on the sphere by combining an optimized hypocoercive approach with the vector field method. One main point in this context is to identify good commuting vector fields for the advection-diffusion operator on the sphere. Our results in this direction may be useful to other models in collective dynamics, where an orientation variable is involved.

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arXiv2021

Homogenization of stiff inclusions through network approximation

We investigate the homogenization of inclusions of infinite conductivity, randomly stationary distributed inside a homogeneous conducting medium. A now classical result by Zhikov shows that, under a logarithmic moment bound on the inter-particle distance, an effective model with finite homogeneous conductivity exists. Relying on ideas from network approximation, we provide a relaxed criterion ensuring homogenization. Several examples not covered by the previous theory are discussed.

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arXiv2021

On the ill-posedness of the triple deck model

We analyze the stability properties of the so-called triple deck model, a classical refinement of the Prandtl equation to describe boundary layer separation. Combining the methodology introduced in [2], based on complex analysis tools, and stability estimates inspired from [3], we exhibit unstable linearizations of the triple deck equation. The growth rates of the corresponding unstable eigenmodes scale linearly with the tangential frequency. This shows that the recent result of Iyer and Vicol [11] of local well-posedness for analytic data is essentially optimal.

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arXiv2020

Mild assumptions for the derivation of Einstein's effective viscosity formula

We provide a rigorous derivation of Einstein's formula for the effective viscosity of dilute suspensions of $n$ rigid balls, $n \gg 1$, set in a volume of size $1$. So far, most justifications were carried under a strong assumption on the minimal distance between the balls: $d_{min} \ge c n^{-\frac{1}{3}}$, $c > 0$. We relax this assumption into a set of two much weaker conditions: one expresses essentially that the balls do not overlap, while the other one gives a control of the number of balls that are close to one another. In particular, our analysis covers the case of suspensions modelled by standard Poisson processes with almost minimal hardcore condition.

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arXiv2020

On the correction to Einstein's formula for the effective viscosity

This paper is a follow-up of article [6], on the derivation of accurate effective models for viscous dilute suspensions. The goal is to identify an effective Stokes equation providing a $o(λ^2)$ approximation of the exact fluid-particle system, with $λ$ the solid volume fraction of the particles. This means that we look for an improvement of Einstein's formula for the effective viscosity in the form $μ_{eff}(x) = μ+ \frac{5}{2} μρ(x) λ+ μ_2(x) λ^2$. Under a separation assumption on the particles, we proved in [6] that if a $o(λ)^2$ Stokes effective approximation exists, the correction $μ_2$ is necessarily given by a mean field limit, that can then be studied and computed under further assumptions on the particle configurations. Roughly, we go here from the conditional result of [6] to an unconditional result: we show that such a $o(λ^2)$ Stokes approximation indeed exists, as soon as the mean field limit exists. This includes the case of periodic and random stationary particle configurations.

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arXiv2018

High frequency analysis of the unsteady Interactive Boundary Layer model

The present paper is about a famous extension of the Prandtl equation, the so-called Interactive Boundary Layer model (IBL). This model has been used intensively in the numerics of steady boundary layer flows, and compares favorably to the Prandtl one, especially past separation. We consider here the unsteady version of the IBL, and study its linear well-posedness, namely the linear stability of shear flow solutions to high frequencyperturbations. We show that the IBL model exhibits strong unrealistic instabilities, that are in particular distinct from the Tollmien-Schlichting waves. We also exhibit similar instabilities for a Prescribed Displacement Thickness model (PDT), which is one of the building blocks of numerical implementations of the IBL model.

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arXiv2017

The vanishing viscosity limit for 2D Navier-Stokes in a rough domain

We study the high Reynolds number limit of a viscous fluid in the presence of a rough boundary. We consider the two-dimensional incompressible Navier-Stokes equations with Navier slip boundary condition, in a domain whose boundaries exhibit fast oscillations in the form $x_2 = \varepsilon^{1+α} η(x_1/\varepsilon)$, $α> 0$. Under suitable conditions on the oscillating parameter $\varepsilon$ and the viscosity $ν$, we show that solutions of the Navier-Stokes system converge to solutions of the Euler system in the vanishing limit of both $ν$ and $\varepsilon$. The main issue is that the curvature of the boundary is unbounded as $\varepsilon \rightarrow 0$, which precludes the use of standard methods to obtain the inviscid limit. Our approach is to first construct an accurate boundary layer approximation to the Euler solution in the rough domain, and then to derive stability estimates for this approximation under the Navier-Stokes evolution.

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arXiv2016

Landau damping to partially locked states in the Kuramoto model

In the Kuramoto model of globally coupled oscillators, partially locked states (PLS) are stationary solutions that incorporate the emergence of partial synchrony when the interaction strength increases. While PLS have long been considered, existing results on their stability are limited to neutral stability of the linearized dynamics in strong topology, or to specific invariant subspaces (obtained via the so-called Ott-Antonsen (OA) ansatz) with specific frequency distributions for the oscillators. In the mean field limit, the Kuramoto model shows various ingredients of the Landau damping mechanism in the Vlasov equation. This analogy has been a source of inspiration for stability proofs of regular Kuramoto equilibria. Besides, the major mathematical issue with PLS asymptotic stability is that these states consist of heterogeneous and singular measures. Here, we establish an explicit criterion for their spectral stability and we prove their local asymptotic stability in weak topology, for a large class of analytic frequency marginals. The proof strongly relies on a suitable functional space that contains (Fourier transforms of) singular measures, and for which the linearized dynamics is well under control. For illustration, the stability criterion is evaluated in some standard examples. We show in particular that no loss of generality results in assuming the OA ansatz. To our best knowledge, our result provides the first proof of Landau damping to heterogeneous and irregular equilibria, in absence of dissipation.

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arXiv2015

Nonlinear boundary layers for rotating fluids

We investigate the behavior of rotating incompressible flows near a non-flat horizontal bottom. In the flat case, the velocity profile is given explicitly by a simple linear ODE. When bottom variations are taken into account, it is governed by a nonlinear PDE system, with far less obvious mathematical properties. We establish the well-posedness of this system and the asymptotic behavior of the solution away from the boundary. In the course of the proof, we investigate in particular the action of pseudo-differential operators in non-localized Sobolev spaces. Our results extend the older paper [18], restricted to periodic variations of the bottom. It ponders on the recent linear analysis carried in [14].

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arXiv2015

Boundary Layer for a Non-Newtonian Flow over a Rough Surface

We analyze the effect of a rough surface on shear thinning and shear thickening fluids, modeled by power law stress tensors. The roughness is modeled by a small wavelength and small amplitude oscillation, parametrized by eps. We study the boundary layer corrector near the rough surface, and provide error estimates in terms of eps for the associated wall laws. With regards to previous studies, Mathematical difficulties arise from the nonlinearity of the corrector equations.

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arXiv2015

On shape optimization problems involving the fractional laplacian

Our concern is the computation of optimal shapes in problems involving $\(-Δ)^{1/2}$. We focus on the energy $J(Ω)$ associated to the solution $u\_Ω$ of the basic Dirichlet problem $(-Δ)^{1/2} u\_Ω= 1$ in $Ω$, $ u = 0$ in $Ω^c$. We show that regular minimizers $Ω$ of this energy under a volume constraint are disks. Our proof goes through the explicit computation of the shape derivative (that seems to be completely new in the fractional context), and a refined adaptation of the moving plane method.

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arXiv2015

Computation of the effective slip of rough hydrophobic surfaces via homogenization

We present a quantitative analysis of the effect of rough hydrophobic surfaces on viscous newtonian flows. We use a model introduced by Ybert and coauthors in which the rough surface is replaced by a flat plane with alternating small areas of slip and no-slip. We investigate the averaged slip generated at the boundary, depending on the ratio between these areas. This problem reduces to the homogenization of a non-local system, involving the Dirichlet to Neumann map of the Stokes operator, in a domain with small holes. Pondering on works by Allaire, we compute accurate scaling laws of the averaged slip for various types of roughness (riblets, patches). Numerical computations complete and confirm the analysis.

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arXiv2014

Landau damping in the Kuramoto model

We consider the Kuramoto model of globally coupled phase oscillators in its continuum limit, with individual frequencies drawn from a distribution with density of class $C^n$ ($n\geq 4$). A criterion for linear stability of the uniform stationary state is established which, for basic examples of frequency distributions, is equivalent to the standard condition on the coupling strength in the literature. We prove that, under this criterion, the Kuramoto order parameter, when evolved under the full nonlinear dynamics, asymptotically vanishes (with polynomial rate $n$) for every trajectory issued from sufficiently small $C^n$ perturbation. The proof uses techniques from the Analysis of PDEs and closely follows recent proofs of the nonlinear Landau damping in the Vlasov equation and Vlasov-HMF model.

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