arXiv subjects

Emeric Roulley

Publications and source records attributed to Emeric Roulley.

16 recordsLinked to original sources

arXiv2026

On stationary Quasi-Geostrophic Shallow-Water flows

In this paper, we prove the existence of $\mathbf{m}$-fold doubly-connected stationary vortex patches for the quasi-geostrophic shallow-water equations. The solutions are obtained through a bifurcation analysis based on the Crandall-Rabinowitz theorem, with either the inner radius of an annulus or the Rossby deformation length serving as the bifurcation parameter. A central feature of the work is the highly nontrivial analysis of modified Bessel functions arising in the spectral study of the linearized operator. The proof requires delicate and extensive manipulations of these special functions, including precise asymptotic expansions, differentiation formulas, recurrence identities, monotonicity properties and the analysis of singular quantities governing the bifurcation mechanism. These ingredients are essential for characterizing the bifurcation points and establishing the transversality conditions. Finally, we investigate the radial symmetry of stationary and uniformly rotating simply-connected vortex patch solutions, therefore motivating the previous bifurcation results.

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arXiv2026

Quasi-periodic traveling electron layers

We consider the one dimensional space-periodic Vlasov-Poisson equations and construct, close to symmetric flat velocity strips, small amplitude traveling quasi-periodic electron-layers, namely strip-shaped patches of electrons in the phase space. These solutions are found for most values of the strip area. The proof uses a Nash-Moser construction together with reducibility arguments based on pseudo-differential homogeneous expansions and KAM reductions. Thanks to a suitable linear transformation of the unknowns, we reveal a connection between the electron patch problem and the classical, physically relevant, electronic Euler-Poisson system with cubic pressure law. As a direct, non trivial consequence, we obtain small-amplitude quasi-periodic traveling waves for this model as well. To the best of our knowledge, this work provides the first rigorous example of construction of quasi-periodic solutions for kinetic models.

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arXiv2026

Confinement results near point vortices on the rotating sphere

We study the Euler equation on the rotating sphere in the case where the absolute vorticity is initially sharply concentrated around several points. We follow the literature already concerning vorticity confinement for the planar Euler equations, and obtain similar results on the rotating sphere, with new challenges due to the geometry. More precisely, we show the improbability of collisions for point-vortices, logarithmic in time absolute vorticity confinement for general configurations, the optimality of this last result in general, and the existence of configurations with power-law long confinement. We take this opportunity to write a unified, self-contained, and improved version of all the proofs, previously scattered across multiple papers on the planar case, with detailed exposition for pedagogical clarity.

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arXiv2026

Ions-electrons-states for the two-component Vlasov-Poisson equation

We establish both local and global bifurcation results for traveling periodic solutions of the one-dimensional two-species Vlasov-Poisson equation. These solutions consist of strip-like regions of ions and electrons in phase space that propagate coherently and emerge from spatially homogeneous, velocity-dependent equilibrium layers. Depending on the geometry of the underlying equilibrium and on the selected Fourier mode, the bifurcation diagram exhibits either two or four solution branches. In all cases, the bifurcation is of pitchfork type; in symmetric configurations, the local structure near the equilibrium has a hyperbolic geometry. We further show that these locally constructed branches extend globally. This work extends the previous study by the same author of the purely electronic case, where the ions were modeled as an immobile neutralizing background. Allowing both species to evolve dynamically leads to a more intricate, higher-dimensional analysis. Finally, by means of an affine change of variables, we reveal a connection with the one-dimensional two-component Euler-Poisson system, which in turn enables the construction of traveling periodic waves of both small and large amplitude for that model as well.

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arXiv2025

Filamentation near monotone zonal vortex caps

We study the Euler equations on a rotating unit sphere, focusing on the dynamics of vortex caps. Leveraging the $L^1$-stability of monotone, longitude-independent profiles, we demonstrate that certain ill-prepared initial data within the vortex cap class exhibit an instability characterized by the growth of the interface perimeter. These configurations are nearly equivalent in area to a zonal vortex cap but are perturbed by a localized latitudinal bump. By comparing the longitudinal flows at points along the zonal interface and within the bump region, we track the induced stretching and capture the underlying instability mechanism.

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arXiv2025

Long-time dynamics for the Kelvin-Helmholtz equations close to circular vortex sheets

We consider the Kelvin-Helmholtz system describing the evolution of a vortex-sheet near the circular stationary solution. Answering previous numerical conjectures in the 90s physics literature, we prove an almost global existence result for small-amplitude solutions. We first establish the existence of a linear stability threshold for the Weber number, which represents the ratio between the square of the background velocity jump and the surface tension. Then, we prove that for almost all values of the Weber number below this threshold any small solution lives for almost all times, remaining close to the equilibrium. Our analysis reveals a remarkable stabilization phenomenon: the presence of both non-zero background velocity jump and capillarity effects enables to prevent nonlinear instability phenomena, despite the inherently unstable nature of the classical Kelvin-Helmholtz problem. This long-time existence would not be achievable in a setting where capillarity alone provides linear stabilization, without the richer modulation induced by the velocity jump. Our proof exploits the Hamiltonian nature of the equations. Specifically, we employ Hamiltonian Birkhoff normal form techniques for quasi-linear systems together with a general approach for paralinearization of non-linear singular integral operators. This approach allows us to control resonances and quasi-resonances at arbitrary order, ensuring the desired long-time stability result.

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arXiv2025

Dynamics of vortex cap solutions on the rotating unit sphere

In this work, we analytically study the existence of periodic vortex cap solutions for the homogeneous and incompressible Euler equations on the rotating unit 2-sphere, which was numerically conjectured by Dritschel-Polvani and Kim-Sakajo-Sohn. Such solutions are piecewise constant vorticity distributions, subject to the Gauss constraint and rotating uniformly around the vertical axis. The proof is based on the bifurcation from zonal solutions given by spherical caps. For the one--interface case, the bifurcation eigenvalues correspond to Burbea's frequencies obtained in the planar case but shifted by the rotation speed of the sphere. The two--interfaces case (also called band type or strip type solutions) is more delicate. Though, for any fixed large enough symmetry, and under some non-degeneracy conditions to avoid spectral collisions, we achieve the existence of at most two branches of bifurcation.

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arXiv2024

Steady vortex sheets in presence of surface tension

We prove a bifurcation result of uniformly-rotating/stationary non-trivial vortex sheets near the circular distribution for a model of two irrotational fluids with same density taking into account surface tension effects. As bifurcation parameters, we play with either the speed of rotation, the surface tension coefficient or the mean vorticity.

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arXiv2024

Desingularization of time-periodic vortex motion in bounded domains via KAM tools

We examine the Euler equations within a simply-connected bounded domain. The dynamics of a single point vortex are governed by a Hamiltonian system, with most of its energy levels corresponding to time-periodic motion. We show that for the single point vortex, under certain non-degeneracy conditions, it is possible to desingularize most of these trajectories into time-periodic concentrated vortex patches. We provide concrete examples of these non-degeneracy conditions, which are satisfied by a broad class of domains, including convex ones. The proof uses Nash-Moser scheme and KAM techniques, in the spirit of the recent work of Hassainia-Hmidi-Masmoudi on the leapfrogging motion, combined with complex geometry tools. Additionally, we employ a vortex duplication mechanism to generate synchronized time-periodic motion of multiple vortices. This approach can be, for instance, applied to desingularize the motion of two symmetric dipoles (with four vortices) in a disc or a rectangle. To our knowledge, this is the first result showing the existence of non-rigid time-periodic motion for Euler equations in generic simply-connected bounded domain. This answers an open problem that has been pointed in the literature, for example by Bartsch-Sacchet.

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arXiv2024

Uniformly rotating vortices for the lake equation

We investigate the existence of time-periodic vortex patch solutions, in both simply and doubly-connected cases, for the two-dimensional lake equation where the depth function of the lake is assumed to be non-degenerate and radial. The proofs employ bifurcation techniques, where the most challenging steps are related to the regularity study of some nonlinear functionals and the spectral analysis of their linearized operators around Rankine type vortices. The main difficulties stem from the roughness and the implicit form of the Green function connecting the fluid vorticity and its stream function. We handle in part these issues by exploring the asymptotic structure of the solutions to the associated elliptic problem. As to the distribution of the spectrum, it is tackled by a fixed-point argument through a perturbative approach.

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arXiv2023

Local and global bifurcation of electron-states

We study the bifurcation of traveling periodic electron layers, that we call electron-states, from symmetric and asymmetric flat velocity strips in the phase space, for the one dimensional Vlasov-Poisson equation with space periodic condition. The boundaries of the constructed solutions are real-analytic and in uniform translation at the same speed in the space direction. These structures are obtained applying Crandall-Rabinowitz's Theorem using either the velocity or geometrical quantities related to the size of the strip as bifurcation parameters. In the first case, we can prove for any fixed symmetry, the emergence of a pair of branches and the local bifurcation diagram has a hyperbolic structure. In the symmetric situation, we find, for any large enough symmetry, one branch whose orientation close to the stationary solution depends on the sign of the prescribed speed of translation. As for the asymmetric case, we find either a countable or a finite number of bifurcation curves according to some constraints related to the prescribed speed of translation. The pitchfork (subcritical or supercritical) bifurcation is also described in this case. Finally, we briefly discuss the global continuation of these branches.

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arXiv2023

Invariant KAM tori around annular vortex patches for 2D Euler equations

We construct time quasi-periodic vortex patch solutions with one hole for the planar Euler equations. These structures are captured close to any annulus provided that its modulus belongs to a massive Borel set. The proof is based on Nash-Moser scheme and KAM theory applied with a Hamiltonian system governing the radial deformations of the patch. Compared to the scalar case, some technical issues emerge due to the interaction between the interfaces. One of them is related to a new small divisor problem in the second order Melnikov non-resonances condition coming from the transport equations advected with different velocities.

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arXiv2022

Periodic and quasi-periodic Euler-$α$ flows close to Rankine vortices

In the present contribution, we first prove the existence of $\mathbf{m}$-fold simply-connected V-states close to the unit disc for Euler-$α$ equations. These solutions are implicitly obtained as bifurcation curves from the circular patches. We also prove the existence of quasi-periodic in time vortex patches close to the Rankine vortices provided that the scale parameter $α$ belongs to a suitable Cantor-like set of almost full Lebesgue measure. The techniques used to prove this result are borrowed from the Berti-Bolle theory in the context of KAM for PDEs.

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arXiv2022

Boundary effects on the emergence of quasi-periodic solutions for Euler equations

In this paper, we highlight the importance of the boundary effects on the construction of quasi-periodic vortex patches solutions close to Rankine vortices and whose existence is not known in the whole space due to the resonances of the linear frequencies. Availing of the lack of invariance by radial dilation of Euler equations in the unit disc and using a Nash-Moser implicit function iterative scheme we show the existence of such structures when the radius of the Rankine vortex belongs to a suitable massive Cantor-like set with almost full Lebesgue measure.

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arXiv2022

Vortex rigid motion in quasi-geostrophic shallow-water equations

In this paper, we prove the existence of relative equilibria with holes for quasi-geostrophic shallow-water equations. More precisely, using bifurcation techniques, we establish for any m large enough the existence of two branches of m-fold doubly-connected V-states bifurcating from any annulus of arbitrary size.

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arXiv2021

Time quasi-periodic vortex patches for quasi-geostrophic shallow-water equations

In this paper, we shall implement KAM theory in order to construct a large class of time quasi-periodic solutions for an active scalar model arising in fluid dynamics. More precisely, the construction of invariant tori is performed for quasi-geostrophic shallow-water equations when the {\it Rossby deformation length} belongs to a massive Cantor set. As a consequence, we construct pulsating vortex patches whose boundary is localized in a thin annulus for any time.

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