arXiv subjects

Thierry Gallay

Publications and source records attributed to Thierry Gallay.

At least 19 recordsLinked to original sources

arXiv2026

The three-vortex system: Hopf fibration, symplectic reduction, and near-collisions

We study the dynamics of collisions and near-collisions in the system of three point vortices in the plane. The dynamics is known to be completely integrable and can be reduced to the study of Hamiltonian systems on two-dimensional symplectic leaves. This still allows for interesting behavior of the solutions, various aspects of which have been investigated in a large body of work by many authors. Our focus will be on collisions and the possibility of their regularization through perturbations of suitable parameters of the system. We will use the reduced coordinate given by $\zeta=(z_2-z_1)/(z_3-z_1)$, where $z_1,z_2,z_3$ are the complex numbers representing the positions of the vortices. This choice seems very natural, although we have not come across it in the existing literature on the topic. Some of our results concerning the near-collision behavior appear to be new, and we also prove inequalities between critical energy values that we have not found in the literature. The calculations we do to analyze the collisions and near-collisions also reproduce various classical results that were previously obtained by other authors using different coordinates, and parts of the paper therefore have an expository character.

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arXiv2026

Viscous evolution of a point vortex in a half-plane

As a model for vortex-wall interactions, we consider the two-dimensional incompressible Navier--Stokes equations in the half-plane $R^2_+$ with no-slip boundary condition and point vortices as initial data. We focus on the paradigmatic example of a single vortex in an otherwise stagnant fluid, which is already quite challenging from a mathematical point of view. We prove that this system has a unique global solution for all values of the Reynolds number $|\Gamma|/\nu$, where $\Gamma$ is the circulation of the vortex and $\nu$ the kinematic viscosity of the fluid. The solution we construct has finite energy for positive times and converges to zero in energy norm as $t \to +\infty$. Uniqueness holds under the assumption that the solution is close to a Lamb--Oseen vortex for small times. To our knowledge, all previous results in domains with boundaries assume that the initial vorticity has small or zero atomic part. In our particular situation, we remove the smallness condition by decomposing the solution into a vortex and a boundary layer term, so that we can apply the techniques developed in the whole plane $R^2$ to avoid the difficulties related to the large circulation of the vortex.

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arXiv2025

Fast relaxation of a viscous vortex in an external flow

We study the evolution of a concentrated vortex advected by a smooth, divergence-free velocity field in two space dimensions. In the idealized situation where the initial vorticity is a Dirac mass, we compute an approximation of the solution which accurately describes, in the regime of high Reynolds numbers, the motion of the vortex center and the deformation of the streamlines under the shear stress of the external flow. For ill-prepared initial data, corresponding to a sharply peaked Gaussian vortex, we prove relaxation to the previous solution on a time scale that is much shorter than the diffusive time, due to enhanced dissipation inside the vortex core.

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arXiv2024

The long way of a viscous vortex dipole

We consider the evolution of a viscous vortex dipole in $R^2$ originating from a pair of point vortices with opposite circulations. At high Reynolds number $Re >> 1$, the dipole can travel a very long way, compared to the distance between the vortex centers, before being slowed down and eventually destroyed by diffusion. In this regime we construct an accurate approximation of the solution in the form of a two-parameter asymptotic expansion involving the aspect ratio of the dipole and the inverse Reynolds number. We then show that the exact solution of the Navier-Stokes equations remains close to the approximation on a time interval of length $O(Re^\sigma)$, where $\sigma < 1$ is arbitrary. This improves upon previous results which were essentially restricted to $\sigma = 0$. As an application, we provide a rigorous justification of an existing formula which gives the leading order correction to the translation speed of the dipole due to finite size effects.

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arXiv2024

Vanishing viscosity limit for axisymmetric vortex rings

For the incompressible Navier-Stokes equations in $R^3$ with low viscosity $ν>0$, we consider the Cauchy problem with initial vorticity $ω_0$ that represents an infinitely thin vortex filament of arbitrary given strength $Γ$ supported on a circle. The vorticity field $ω(x,t)$ of the solution is smooth at any positive time and corresponds to a vortex ring of thickness $\sqrt{νt}$ that is translated along its symmetry axis due to self-induction, an effect anticipated by Helmholtz in 1858 and quantified by Kelvin in 1867. For small viscosities, we show that $ω(x,t)$ is well-approximated on a large time interval by $ω_{lin}(x-a(t),t)$, where $ω_{lin}(\cdot,t)=\exp(νtΔ)ω_0$ is the solution of the heat equation with initial data $ω_0$, and $\dot a(t)$ is the instantaneous velocity given by Kelvin's formula. This gives a rigorous justification of the binormal motion for circular vortex filaments in weakly viscous fluids. The proof relies on the construction of a precise approximate solution, using a perturbative expansion in self-similar variables. To verify the stability of this approximation, one needs to rule out potential instabilities coming from very large advection terms in the linearized operator. This is done by adapting V. I. Arnold's geometric stability methods developed in the inviscid case $ν=0$ to the slightly viscous situation. It turns out that although the geometric structures behind Arnold's approach are no longer preserved by the equation for $ν> 0$, the relevant quadratic forms behave well on larger subspaces than those originally used in Arnold's theory and interact favorably with the viscous terms.

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arXiv2023

Viscous shocks and long-time behavior of scalar conservation laws

We study the long-time behavior of scalar viscous conservation laws via the structure of $ω$-limit sets. We show that $ω$-limit sets always contain constants or shocks by establishing convergence to shocks for arbitrary monotone initial data. In the particular case of Burgers' equation, we review and refine results that parametrize entire solutions in terms of probability measures, and we construct initial data for which the $ω$-limit set is not reduced to the translates of a single shock. Finally we propose several open problems related to the description of long-time dynamics.

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arXiv2023

Enhanced dissipation and Taylor dispersion in higher-dimensional parallel shear flows

We consider the evolution of a passive scalar advected by a parallel shear flow in an infinite cylinder with bounded cross section, in arbitrary space dimension. The essential parameters of the problem are the molecular diffusivity $ν$, which is assumed to be small, and the wave number $k$ in the streamwise direction, which can take arbitrary values. Under generic assumptions on the shear velocity $v$, we obtain optimal decay estimates for large times, both in the enhanced dissipation regime $ν\ll |k|$ and in the Taylor dispersion regime $|k| \ll ν$. Our results can be deduced from resolvent estimates using a quantitative version of the Gearhart-Prüss theorem, or can be established more directly via the hypocoercivity method. Both approaches are explored in the present example, and their relative efficiency is compared.

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arXiv2021

Arnold's variational principle and its application to the stability of planar vortices

We consider variational principles related to V. I. Arnold's stability criteria for steady-state solutions of the two-dimensional incompressible Euler equation. Our goal is to investigate under which conditions the quadratic forms defined by the second variation of the associated functionals can be used in the stability analysis, both for the Euler evolution and for the the Navier-Stokes equation at low viscosity. In particular, we revisit the classical example of Oseen's vortex, providing a new stability proof with stronger geometric flavor. Our analysis involves a fairly detailed functional-analytic study of the inviscid case, which may be of independent interest, and a careful investigation of the influence of the viscous term in the particular example of the Gaussian vortex.

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arXiv2021

Diffusive relaxation to equilibria for an extended reaction-diffusion system on the real line

We study the long-time behavior of the solutions of a two-component reaction-diffusion system on the real line, which describes the basic chemical reaction $A <=> 2 B$. Assuming that the initial densities of the species $A, B$ are bounded and nonnegative, we prove that the solution converges uniformly on compact sets to the manifold $E$ of all spatially homogeneous chemical equilibria. The result holds even if the species diffuse at very different rates, but the proof is substantially simpler for equal diffusivities. In the spirit of our previous work on extended dissipative systems [18], our approach relies on localized energy estimates, and provides an explicit bound for the time needed to reach a neighborhood of the manifold $E$ starting from arbitrary initial data. The solutions we consider typically do not converge to a single equilibrium as $t \to +\infty$, but they are always quasiconvergent in the sense that their omega-limit sets consist of chemical equilibria.

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arXiv2021

Propagation fronts in a simplified model of tumor growth with degenerate cross-dependent self-diffusivity

Motivated by tumor growth in Cancer Biology, we provide a complete analysis of existence and non-existence of invasive fronts for the reduced Gatenby--Gawlinski model \[ \partial_t U = U\{f(U)-dV\}, \qquad \partial_t V = \partial_x \{f(U)\,\partial_x V\} + r V f(V), \] where $f(u) = 1-u$ and the parameters $d,r$ are positive. Denoting by $(\mathcal{U},\mathcal{V})$ the traveling wave profile and by $(\mathcal{U}_\pm,\mathcal{V}_\pm)$ its asymptotic states at $\pm\infty$, we investigate existence in the regimes i) $d > 1$ (homogeneous invasion) : $(\mathcal{U}_-,\mathcal{V}_-) = (0,1)$, $(\mathcal{U}_+,\mathcal{V}_+) = (1,0)$; ii) $d < 1$ (heterogeneous invasion) : $(\mathcal{U}_-,\mathcal{V}_-) = (1-d,1)$, $(\mathcal{U}_+,\mathcal{V}_+) = (1,0)$. In both cases, we prove that a propagating front exists whenever the speed parameter $c$ is strictly positive. We also derive an accurate approximation of the front profile in the singular limit $c \to 0$.

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arXiv2020

Asymptotic self-similarity in diffusion equations with nonconstant radial limits at infinity

We study the long-time behavior of localized solutions to linear or semilinear parabolic equations in the whole space $\mathbb{R}^n$, where $n \ge 2$, assuming that the diffusion matrix depends on the space variable $x$ and has a finite limit along any ray as $|x| \to \infty$. Under suitable smallness conditions in the nonlinear case, we prove convergence to a self-similar solution whose profile is entirely determined by the asymptotic diffusion matrix. Examples are given which show that the profile can be a rather general Gaussian-like function, and that the approach to the self-similar solution can be arbitrarily slow depending on the continuity and coercivity properties of the asymptotic matrix. The proof of our results relies on appropriate energy estimates for the diffusion equation in self-similar variables. The new ingredient consists in estimating not only the difference $w$ between the solution and the self-similar profile, but also an antiderivative $W$ obtained by solving a linear elliptic problem which involves $w$ as a source term. Hence, a good part of our analysis is devoted to the study of linear elliptic equations whose coefficients are homogeneous of degree zero.

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arXiv2019

On the linear stability of vortex columns in the energy space

We investigate the linear stability of inviscid columnar vortices with respect to finite energy perturbations. For a large class of vortex profiles, we show that the linearized evolution group has a sub-exponential growth in time, which means that the associated growth bound is equal to zero. This implies in particular that the spectrum of the linearized operator is entirely contained in the imaginary axis. This contribution complements the results of a previous work, where spectral stability was established for the linearized operator in the enstrophy space.

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arXiv2019

Spectral stability of inviscid columnar vortices

Columnar vortices are stationary solutions of the three-dimensional Euler equations with axial symmetry, where the velocity field only depends on the distance to the axis and has no component in the axial direction. Stability of such flows was first investigated by Lord Kelvin in 1880, but despite a long history the only analytical results available so far provide necessary conditions for instability under either planar or axisymmetric perturbations. The purpose of this paper is to show that columnar vortices are spectrally stable with respect to three-dimensional perturbations with no particular symmetry. Our result applies to a large family of velocity profiles, including the most common models in atmospheric flows and engineering applications. The proof is based on a homotopy argument, which allows us to concentrate in the spectral analysis of the linearized operator to a small neighborhood of the imaginary axis, where unstable eigenvalues can be excluded using integral identities and a careful study of the so-called critical layers.

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arXiv2019

Stability of Vortices in Ideal Fluids : the Legacy of Kelvin and Rayleigh

The mathematical theory of hydrodynamic stability started in the middle of the 19th century with the study of model examples, such as parallel flows, vortex rings, and surfaces of discontinuity. We focus here on the equally interesting case of columnar vortices, which are axisymmetric stationary flows where the velocity field only depends on the distance to the symmetry axis and has no component in the axial direction. The stability of such flows was first investigated by Kelvin in 1880 for some particular velocity profiles, and the problem benefited from important contributions by Rayleigh in 1880 and 1917. Despite further progress in the 20th century, notably by Howard and Gupta (1962), the only rigorous results so far are necessary conditions for instability under either two-dimensional or axisymmetric perturbations. This note is a non-technical introduction to a recent work in collaboration with D. Smets, where we prove under mild assumptions that columnar vortices are spectrally stable with respect to general three-dimensional perturbations, and that the linearized evolution group has a subexponential growth as $|t| \to \infty$.

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arXiv2017

Enhanced dissipation and axisymmetrization of two-dimensional viscous vortices

This paper is devoted to the stability analysis of the Lamb-Oseen vortex in the regime of high circulation Reynolds numbers. When strongly localized perturbations are applied, it is shown that the vortex relaxes to axisymmetry in a time proportional to $Re^{2/3}$, which is substantially shorter than the diffusion time scale given by the viscosity. This enhanced dissipation effect is due to the differential rotation inside the vortex core. Our result relies on a recent work by Li, Wei, and Zhang, where optimal resolvent estimates for the linearized operator at Oseen's vortex are established. A comparison is made with the predictions that can be found in the physical literature, and with the rigorous results that were obtained for shear flows using different techniques.

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arXiv2017

On nonlinear stabilization of linearly unstable maps

We examine the phenomenon of nonlinear stabilization, exhibiting a variety of related examples and counterexamples. For Gâteaux differentiable maps, we discuss a mechanism of nonlinear stabilization, in finite and infinite dimensions, which applies in particular to hyperbolic partial differential equations, and, for Fréchet differentiable maps with linearized operators that are normal, we give a sharp criterion for nonlinear exponential instability at the linear rate. These results highlight the fundamental open question whether Fréchet differentiability is sufficient for linear exponential instability to imply nonlinear exponential instability, at possibly slower rate.

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arXiv2016

Existence and stability of viscous vortices

Vorticity plays a prominent role in the dynamics of incompressible viscous flows. In two-dimensional freely decaying turbulence, after a short transient period, evolution is essentially driven by interactions of viscous vortices, the archetype of which is the self-similar Lamb-Oseen vortex. In three dimensions, amplification of vorticity due to stretching can counterbalance viscous dissipation and produce stable tubular vortices. This phenomenon is illustrated in a famous model originally proposed by Burgers, where a straight vortex tube is produced by a linear uniaxial strain field. In real flows vortex lines are usually not straight, and can even form closed curves, as in the case of axisymmetric vortex rings which are very common in nature and in laboratory experiments. The aim of this chapter is to review a few rigorous results concerning existence and stability of viscous vortices in simple geometries.

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arXiv2016

Uniqueness of axisymmetric viscous flows originating from circular vortex filaments

The incompressible Navier-Stokes equations in R^3 are shown to admit a unique axisymmetric solution without swirl if the initial vorticity is a circular vortex filament with arbitrarily large circulation Reynolds number. The emphasis is on uniqueness, as existence has already been established in [10]. The main difficulty which has to be overcome is that the nonlinear regime for such flows is outside of applicability of standard perturbation theory, even for short times. The solutions we consider are archetypal examples of viscous vortex rings, and can be thought of as axisymmetric analogues of the self-similar Lamb-Oseen vortices in two-dimensional flows. Our method provides the leading term in a fixed-viscosity short-time asymptotic expansion of the solution, and may in principle be extended so as to give a rigorous justification, in the axisymmetric situation, of higher-order formal asymptotic expansions that can be found in the literature [7].

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