arXiv subjects

Taoufik Hmidi

Publications and source records attributed to Taoufik Hmidi.

At least 19 recordsLinked to original sources

arXiv2026

Point vortex dynamics in quasi-periodic channels: transporting trajectories and ergodic distribution of equilibria

We study the dynamics of a single point vortex in an unbounded planar channel whose interfaces are quasi-periodic in the longitudinal direction. The motion is governed by the Robin function. We introduce a hull formulation that lifts the quasi-periodic geometry to a periodic problem on a finite-dimensional torus and yields an exact quasi-periodic representation of the Robin function. Under a natural transversality condition, we construct global quasi-periodic invariant graphs describing transporting vortex trajectories and show that a full neighborhood of each boundary component is foliated by such graphs. Under a Diophantine condition on the spatial frequencies, the dynamics along each graph can be straightened to a constant drift, so that the vortex motion is quasi-periodic modulo translation. A central contribution concerns the distribution of vortex equilibria. In the quasi-periodic setting, where no fundamental spatial cell exists, we identify critical points of the Robin function with crossings of a hypersurface by a Kronecker flow on the hull torus. Exploiting unique ergodicity, we establish a general zero-counting theorem, allowing finite-order tangencies, which yields an explicit geometric flux formula for the asymptotic density of critical points and their limiting phase distribution. The result is nonperturbative once the critical hull is constructed. Explicit periodic and quasi-periodic models reveal bifurcations and phase transitions in the critical-point distribution, while numerical computations provide quantitative validation of the analytical results.

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arXiv2026

Rotating vortex patches: rigidity, bifurcation, and unified structures

This monograph presents a systematic account of several classical and recent developments in the theory of rotating vortex patches, or V-states, for the two-dimensional Euler equations and related active scalar models. Its purpose is both expository and structural: we revisit some of the foundational results of the subject, provide detailed proofs and alternative formulations, and present more recent results within a common analytical framework. We begin with Euler vortex patches and the contour dynamics formulation of rigidly rotating solutions. We then develop two complementary approaches to the V-state equation, based on Cauchy integrals and conformal mappings, and discuss their connections with potential theory and Faber polynomials. These tools are used to revisit classical examples, including Rankine vortices and Kirchhoff ellipses, as well as rigidity and classification results. A substantial part is devoted to Burbea construction of noncircular V-states via bifurcation theory. The final part develops a unified approach to rotating patches for a broad class of incompressible active scalar equations. Structural properties of the interaction kernel, in particular complete monotonicity and the resulting spectral factorization, yield a common bifurcation framework encompassing the Euler, generalized surface quasi-geostrophic, quasi-geostrophic shallow-water, and related models.

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arXiv2026

Time-periodic vortices near translating symmetric dipole patches

We prove the existence of time-periodic solutions of the two-dimensional incompressible Euler equations bifurcating from a translating vortex pair. The reference configuration consists of two symmetric vortex patches of equal strength and opposite sign traveling at constant speed. In a regime of large separation between the vortices, the dynamics may be viewed as a small perturbation of an integrable system. Working in a co-moving frame and using the contour dynamics formulation, we reduce the problem to a nonlinear transport equation for the vortex boundaries. The linearized operator exhibits degeneracies associated with symmetries and transport effects. By combining a Lyapunov-Schmidt reduction, Nash-Moser scheme, and spectral analysis with sharp asymptotic expansions of the eigenvalues in order to overcome the degeneracy, we construct families of non-rigid, time-periodic vortex patch solutions for a large Cantor set of parameters. The analysis reveals that translating dipoles possess a surprisingly rich nearby dynamics, far beyond the classical rigid paradigm usually associated with vortex patch motion. More generally, the approach developed in this work is flexible and robust, and is expected to extend to a broader class of nonlocal PDEs from Fluid Mechanics with degenerate behaviours.

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arXiv2026

Singular traveling waves for the Euler-Poisson system

We consider the Euler-Poisson system for ions where the electrons are given by a Maxwell-Boltzmann distribution, and we investigate the existence of one-dimensional periodic traveling waves. More precisely, we first establish the existence of a smooth global branch of bifurcation emanating from a constant equilibrium. We then construct a singular traveling wave emerging as the limiting profile at the end of the global curve of bifurcation. Our analysis accommodates a wide class of pressure laws and provides a comprehensive characterization of both smooth and singular traveling waves. A central difficulty in this model arises from the exponential nonlinearity, induced by the nonlocal Poisson-Boltzmann equation, which prevents any explicit representation of the electron field in terms of the ion density. This poses significant obstacles compared to previous studies on related models, where such explicit formulas were crucial for global bifurcation arguments.

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arXiv2026

Time-periodic leapfrogging vortex rings in the 3D Euler equations

We prove the existence of time-periodic leapfrogging vortex rings for the three-dimensional incompressible Euler equations, thereby providing a rigorous realization of a phenomenon first conjectured by Helmholtz (1858). In the leapfrogging motion, two coaxial vortex rings periodically exchange positions, a striking behavior repeatedly observed in experiments and numerical simulations, yet lacking complete mathematical justification. Our construction relies on a desingularization of two interacting vortex filaments within the contour dynamics formulation, which yields a Hamiltonian description of nearly concentric vortex rings. The main difficulty stems from a singular small-divisor problem arising in the linearized transport dynamics, where the effective time scale degenerates with the ring thickness parameter. To overcome this obstruction, we develop a degenerate KAM-type analysis combined with pseudo-differential operator techniques to control the linearized dynamics around symmetric configurations. Combining these tools with a Nash-Moser iteration scheme, we construct families of nontrivial time-periodic solutions in an almost uniformly translating frame. This establishes the first rigorous construction of classical leapfrogging motion for axisymmetric Euler flows without swirl, with no restriction on the time interval of existence.

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arXiv2025

Doubly Connected V-States in Geophysical Models: A General Framework

In this paper, we prove the existence of doubly connected V-states (rotating patches) close to an annulus for active scalar equations with completely monotone kernels. This provides a unified framework for various results related to geophysical flows. This allows us to recover existing results on this topic while also extending to new models, such as the gSQG and QGSW equations in radial domains and 2D Euler equation in annular domains.

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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

Rigorous derivation of the leapfrogging motion for planar Euler equations

The main goal of this paper is to explore the leapfrogging phenomenon in the inviscid planar flows. We show for 2d Euler equations that under suitable constraints, four concentrated vortex patches leapfrog for all time. When observed from a translating frame of reference, the evolution of these vortex patches can be described as a non-rigid time periodic motion. Our proof hinges upon two key components. First, we desingularize the symmetric four point vortex configuration, which leapfrogs in accordance with Love's result \cite{Love1893}, by concentrated vortex patches. Second, we borrow some tools from KAM theory to effectively tackle the small divisor problem and deal with the degeneracy in the time direction. Our approach is robust and flexible and solves a long-standing problem that has remained unresolved for many decades.

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arXiv2023

Unified theory on V-states structures for active scalar equations

This paper revolves around the existence of V-states close to Rankine vortices for active scalar equations with completely monotone kernels. This allows to unify various results on this topic related to geophysical flows. A key ingredient is a new factorization formula for the spectrum using a universal function which is independent of the model. This function admits several interesting properties allowing to track the spectrum distribution.

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arXiv2023

Time periodic solutions close to localized radial monotone profiles for the 2D Euler equations

In this paper, we address for the 2D Euler equations the existence of rigid time periodic solutions close to stationary radial vortices of type $f_0(|x|){\bf 1}_{\mathbb{D}}(x)$, with $\mathbb{D}$ the unit disc and $f_0$ being a strictly monotonic profile with constant sign. We distinguish two scenarios according to the sign of the profile: defocusing and focusing. In the first regime, we have scarcity of the bifurcating curves associated with lower symmetry. However in the focusing case we get a countable family of bifurcating solutions associated with large symmetry. The approach developed in this work is new and flexible, and the explicit expression of the radial profile is no longer required as in [41] with the quadratic shape. The alternative for that is a refined study of the associated spectral problem based on Sturm-Liouville differential equation with a variable potential that changes the sign depending on the shape of the profile and the location of the time period. Deep hidden structure on positive definiteness of some intermediate integral operators are also discovered and used in a crucial way. Notice that a special study will be performed for the linear problem associated with the first mode founded on Prüfer transformation and Kneser's Theorem on the non-oscillation phenomenon.

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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

Emergence of time periodic solutions for the generalized surface quasi-geostrophic equation in the disc

In this paper we address the existence of time periodic solutions for the generalized inviscid SQG equation in the unit disc with homogeneous Dirichlet boundary condition when $α\in (0,1)$. We show the existence of a countable family of bifurcating curves from the radial patches. In contrast with the preceding studies in active scalar equations, the Green function is no longer explicit and we circumvent this issue by a suitable splitting into a singular explicit part (which coincides with the planar one) and a smooth implicit one induced by the boundary of the domain. Another problem is connected to the analysis of the linear frequencies which admit a complicated form through a discrete sum involving Bessel functions and their zeros. We overcome this difficulty by using Sneddon's formula leading to a suitable integral representation of the frequencies.

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arXiv2021

Time periodic solutions for 3D quasi-geostrophic model

This paper aims to study time periodic solutions for 3D inviscid quasi-geostrophic model. We show the existence of non trivial rotating patches by suitable perturbation of stationary solutions given by generic revolution shapes around the vertical axis. The construction of those special solutions are done through bifurcation theory. In general, the spectral problem is very delicate and strongly depends on the shape of the initial stationary solutions. More specifically, the spectral study can be related to an eigenvalue problem of a self-adjoint compact operator. We are able to implement the bifurcation only from the largest eigenvalues of the operator, which are simple. Additional difficulties generated by the singularities of the poles are solved through the use of suitable function spaces with Dirichlet boundary condition type and refined potential theory with anisotropic kernels.

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arXiv2021

KAM theory for active scalar equations

In this paper, we establish the existence of time quasi-periodic solutions to generalized surface quasi-geostrophic equation $({\rm gSQG})_α$ in the patch form close to Rankine vortices. We show that invariant tori survive when the order $α$ of the singular operator belongs to a Cantor set contained in $(0,\frac12)$ with almost full Lebesgue measure. The proof is based on several techniques from KAM theory, pseudo-differential calculus together with Nash-Moser scheme in the spirit of the recent works \cite{Baldi-Berti2018,Berti-Bolle15}. One key novelty here is a refined Egorov type theorem established through a new approach based on the kernel dynamics together with some hidden Töpliz structures.

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arXiv2021

Rigidity aspects of singular patches in stratified flows

We explore the local well-posedness theory for the 2d inviscid Boussinesq system when the vorticity is given by a singular patch. We give a significant improvement of \cite{Hassainia-Hmidi} by replacing their compatibility assumption on the density with a constraint on its platitude degree on the singular set. The second main contribution focuses on the same issue for the partial viscous Boussinesq system. We establish a uniform LWP theory with respect to the vanishing conductivity. This issue is much more delicate than the inviscid case and one should carefully deal with various difficulties related to the diffusion effects which tend to alter some local structures. The weak a priori estimates are not trivial and refined analysis on transport-diffusion equation subject to a logarithmic singular potential is required. Another difficulty stems from some commutators arising in the control of the co-normal regularity that we counterbalance in part by the maximal smoothing effects of transport-diffusion equation advected by a velocity field which scales slightly below the Lipschitz class.

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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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arXiv2020

Ergodicity effects on transport-diffusion equations with localized damping

The main objective of this paper is to study the time decay of transport-diffusion equation with inhomogeneous localized damping in the multi-dimensional torus. The drift is governed by an autonomous Lipschitz vector field and the diffusion by the standard heat equation with small viscosity parameter $ν$. In the first part we deal with the inviscid case and show some results on the time decay of the energy using in a crucial way the ergodicity and the unique ergodicity of the flow generated by the drift. In the second part we analyze the same problem with small viscosity and provide quite similar results on the exponential decay uniformly with respect to the viscosity in some logarithmic time scaling of the \mbox{type $t\in [0,C_0\ln(1/ν)]$}.

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