arXiv subjects

Gérard Iooss

Publications and source records attributed to Gérard Iooss.

10 recordsLinked to original sources

arXiv2026

Couette-Taylor instabilities in the small gap regime: the very counter-rotating case

In this paper, we study the Couette-Taylor instability of a viscous fluid between two rotating cylinders in the small-gap, slow rescaled rotation rate, high Reynolds number regime, focusing on the very counter-rotating case $μ< μ_c \approx -0.8$ where the primary instability is non-axisymmetric. Starting from the Navier-Stokes equations, we derive a limit system that captures the leading-order dynamics and compute the critical Taylor number $T_c(μ)$ together with the critical axial and azimuthal wavenumbers. Near criticality, the weakly nonlinear behaviour is governed by a system of two coupled complex Ginzburg-Landau equations. All coefficients of this amplitude system including the cubic nonlinear terms are evaluated numerically from the linearised eigenfunctions and the associated adjoint problem. The reduced equations admit helicoidal waves (travelling in both the axial and azimuthal directions) and ribbon waves (standing axially, travelling azimuthally), and their existence and stability criteria are discussed. We also examine more exotic spatially modulated solutions that satisfy a third-order dynamical system, whose complete classification remains an open challenge.

math.AP↗

arXiv2026

Couette Taylor instabilities for counter-rotating cylinders in the small-gap regime

We study the Couette Taylor instabilities for an incompressible viscous fluid between two coaxial cylinders of nearly equal radii, allowing counter-rotation with the ratio of rotation rate $μ\in [-1,1]$. Working in a rotating frame and in a small-gap and small-viscosity regime, we derive the corresponding limiting Navier Stokes system and analyze the linear stability of the Couette flow. In particular, we numerically compute the critical Taylor number for general perturbations and identify a transition near $μ_c \approx -0.8$: for $μ> μ_c$ the most unstable mode is axisymmetric, whereas for $μ< μ_c$ the most unstable mode is non-axisymmetric. Near criticality, slowly varying traveling waves are governed by a time-independent Ginzburg Landau equation. The nonlinear coefficient changes sign near $\hatμ_c \approx -0.65$, yielding a supercritical regime for $μ> \hatμ_c$ and a subcritical regime for $μ_c < μ< \hatμ_c$. In the subcritical range, we classify small-amplitude steady states, including Taylor vortex flows, wavy vortices, a two-parameter family of quasi-periodic flows, and a localized traveling perturbation of the Couette flow.

math.AP↗

arXiv2026

Couette Taylor instabilities in the small-gap regime

The Couette-Taylor instability occurs in a viscous fluid confined between two coaxial rotating cylinders. When the Taylor number surpasses a critical value, the stable Couette flow destabilizes, giving way to steady Taylor vortices. As the Taylor number increases further, these vortices themselves become unstable, transitioning into wavy Taylor vortices. In this article, we focus on the small-gap limit, where the ratio of the cylinder radii approaches unity and the rotation rates of the cylinders are nearly identical. We provide a rigorous proof of the existence of a critical Taylor number $T_c$, at which the Couette flow loses stability. For Taylor numbers just above $T_c$, under fixed axial periodicity, the solutions to the limiting Navier-Stokes system are governed by a Ginzburg-Landau-type partial differential equation. Beyond the classical Taylor vortex flow, we demonstrate that a two-parameter family of solutions emerges at criticality for $T>T_c$. This family includes not only wavy vortices but also a variety of other exotic flow patterns, all of which remain steady in the frame rotating at the average angular velocity of the cylinders.

math.AP↗

arXiv2025

Existence of orthogonal domain walls in B{é}nard-Rayleigh convection

In B{é}nard-Rayleigh convection we consider the pattern defect in orthogonal domain walls connecting a set of convective rolls with another set of rolls orthogonal to the first set. This is understood as an heteroclinic orbit of a reversible system where the x-coordinate plays the role of time. This appears as a perturbation of the heteroclinic orbit proved to exist in a reduced 6-dimensional system studied by a variational method in [3], and studied analytically in [10]. We then prove for a given amplitude, and an imposed symmetry in coordinate y, the existence of a one parameter family of heteroclinic connections between orthogonal sets of rolls, with wave numbers (different in general) which are linked to an adapted ''shift'' of rolls parallel to the wall.

math-ph↗

arXiv2025

Heteroclinic for a 6-dimensional reversible system occuring in orthogonal domain walls in convection

A six-dimensional reversible normal form system occurs in B{é}nard-Rayleigh convection between parallel planes, when we look for domain walls intersecting orthogonally (see Buffoni et al [1]). On the truncated system, we prove analytically the existence, local uniqueness, and analyticity in parameters, of a heteroclinic connection between two equilibria, each corresponding to a system of convective rolls. We prove that the 3-dimensional unstable manifold of one equilibrium, intersects transversally the 3-dimensional stable manifold of the other equilibrium, both manifolds lying on a 5-dimensional invariant manifold. We also study the linearized operator along the heteroclinic, allowing to prove (in [9]) the persistence under reversible perturbation, of the heteroclinic obtained in [1].

math-ph↗

arXiv2025

Bifurcations of viscous boundary layers in the half space

It is well-established that shear flows are linearly unstable provided the viscosity is small enough, when the horizontal Fourier wave number lies in some interval, between the so-called lower and upper marginally stable curves. In this article, we prove that, under a natural spectral assumption, shear flows undergo a Hopf bifurcation near their upper marginally stable curve. In particular, close to this curve, there exists space periodic traveling waves solutions of the full incompressible Navier-Stokes equations. For the linearized operator, the occurrence of an essential spectrum containing the entire negative real axis causes certain difficulties which are overcome. Moreover, if this Hopf bifurcation is super-critical, these time and space periodic solutions are linearly and nonlinearly asymptotically stable.

math.AP↗

arXiv2022

Patterns and quasipatterns from the superposition of two hexagonal lattices

When two-dimensional pattern-forming problems are posed on a periodic domain, classical techniques (Lyapunov-Schmidt, equivariant bifurcation theory) give considerable information about what periodic patterns are formed in the transition where the featureless state loses stability. When the problem is posed on the whole plane, these periodic patterns are still present. Recent work on the Swift-Hohenberg equation (an archetypal pattern-forming partial differential equation) has proved the existence of quasipatterns, which are not spatially periodic and yet still have long-range order. Quasipatterns may have 8-fold, 10-fold, 12-fold and higher rotational symmetry, which preclude periodicity. There are also quasipatterns with 6-fold rotational symmetry made up from the superposition of two equal-amplitude hexagonal patterns rotated by almost any angle $α$ with respect to each other. Here, we revisit the Swift-Hohenberg equation (with quadratic as well as cubic nonlinearities) and prove existence of several new quasipatterns. The most surprising are hexa-rolls: periodic and quasiperiodic patterns made from the superposition of hexagons and rolls (stripes) oriented in almost any direction with respect to each other and with any relative translation; these bifurcate directly from the featureless solution. In addition, we find quasipatterns made from the superposition of hexagons with unequal amplitude (provided the coefficient of the quadratic nonlinearity is small). We consider the periodic case as well, and extend the class of known solutions, including the superposition of hexagons and rolls. While we have focused on the Swift-Hohenberg equation, our work contributes to the general question of what periodic or quasiperiodic patterns should be found generically in pattern-forming problems on the plane.

nlin.PS↗

arXiv2021

Heteroclinic orbits for a system of amplitude equations for orthogonal domain walls

Using a variational method, we prove the existence of heteroclinic solutions for a 6dimensional system of ordinary differential equations. We derive this system from the classical B{é}nard-Rayleigh problem near the convective instability threshold. The constructed heteroclinic solutions provide first order approximations for domain walls between two orthogonal convective rolls.

math.AP↗

arXiv2012

Existence of quasipatterns solutions of the Swift-Hohenberg equation

We consider the steady Swift - Hohenberg partial differential equation. It is a one-parameter family of PDE on the plane, modeling for example Rayleigh - Bénard convection. For values of the parameter near its critical value, we look for small solutions, quasiperiodic in all directions of the plane and which are invariant under rotations of angle π/q, q\geq 4. We solve an unusual small divisor problem, and prove the existence of solutions for small parameter values.

math.AP↗

arXiv2006

Small divisor problem in the theory of three-dimensional water gravity waves

We consider doubly-periodic travelling waves at the surface of an infinitely deep perfect fluid, only subjected to gravity $g$ and resulting from the nonlinear interaction of two simply periodic travelling waves making an angle $2θ$ between them. \newline Denoting by $μ=gL/c^{2}$ the dimensionless bifurcation parameter ($L$ is the wave length along the direction of the travelling wave and $c$ is the velocity of the wave), bifurcation occurs for $μ=\cos θ$. For non-resonant cases, we first give a large family of formal three-dimensional gravity travelling waves, in the form of an expansion in powers of the amplitudes of two basic travelling waves. "Diamond waves" are a particular case of such waves, when they are symmetric with respect to the direction of propagation.\newline \emph{The main object of the paper is the proof of existence} of such symmetric waves having the above mentioned asymptotic expansion. Due to the \emph{occurence of small divisors}, the main difficulty is the inversion of the linearized operator at a non trivial point, for applying the Nash Moser theorem. This operator is the sum of a second order differentiation along a certain direction, and an integro-differential operator of first order, both depending periodically of coordinates. It is shown that for almost all angles $θ$, the 3-dimensional travelling waves bifurcate for a set of "good" values of the bifurcation parameter having asymptotically a full measure near the bifurcation curve in the parameter plane $(θ,μ).$

math.AP↗