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Tarek M. Elgindi

Publications and source records attributed to Tarek M. Elgindi.

At least 19 recordsLinked to original sources
arXiv2026

On the flexibility of 2D Euler steady states

We consider steady states of the incompressible Euler equation on two-dimensional domains. For non-radial analytic steady states on bounded simply connected domains, it was shown previously that there must be a global functional relationship between the stream function and the vorticity. We show that this does not extend to smooth functions, even under further structural assumptions such as the Morse condition or Arnold's stability criterion. More precisely, we show that a broad class of steady states with multiple critical points can be perturbed to smooth steady states for which the vorticity is not a single-valued function of the stream function. We also establish an analogous flexibility result near the cellular flow on the flat torus, which is a degenerate case. As a consequence of our constructions, there are "branches" of smooth steady states that are isolated from analytic ones. In some cases, the resulting isolated branches can even consist entirely of linearly stable steady states.

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arXiv2026

A Classification Theorem for Steady Euler Flows

Fix a bounded, analytic, and simply connected domain $Ω\subset\mathbb{R}^2.$ We show that all analytic steady states of the Euler equations with stream function $ψ$ are either radial or solve a semi-linear elliptic equation of the form $Δψ= F(ψ)$ with Dirichlet boundary conditions. In particular, if $Ω$ is not a ball, then there exists a one to one correspondence between analytic steady states of the Euler equations and analytic solutions of equations of the form $Δψ= F(ψ)$ with Dirichlet boundary conditions.

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arXiv2025

Dynamics of Ideal Fluid Flows

We will discuss various aspects of the incompressible Euler equation. We will discuss, in particular, problems related to the least action principle, the existence of special solutions, the problem of solvability, singularity formation, and asymptotic behavior.

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arXiv2025

On the existence of fibered three-dimensional perfect fluid equilibria without continuous Euclidean symmetry

Following Lortz, we construct a family of smooth steady states of the ideal, incompressible Euler equation in three dimensions that possess no continuous Euclidean symmetry. As in Lortz, they do possess a planar reflection symmetry and, as such, all the orbits of the velocity are closed. Different from Lortz, our construction has a discrete m-fold symmetry and is foliated by invariant cylindrical level sets of a non-degenerate Bernoulli pressure. Such examples narrow the scope of validity of Grad's conjecture that the only solutions with a continuous symmetry can be fibered by pressure surfaces.

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arXiv2025

Cusp Formation in Vortex Patches

We prove instantaneous cusp formation for any initial vortex patch with acute corners. This was conjectured to occur in the numerical literature.

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arXiv2024

Twisting in Hamiltonian Flows and Perfect Fluids

We introduce a notion of stability for non-autonomous Hamiltonian flows on two-dimensional annular surfaces. This notion of stability is designed to capture the sustained twisting of particle trajectories. The main Theorem is applied to establish a number of results that reveal a form of irreversibility in the Euler equations governing the motion of an incompressible and inviscid fluid. In particular, we show that nearby general stable steady states (i) all fluid flows exhibit indefinite twisting (ii) vorticity generically exhibits gradient growth and wandering. We also give examples of infinite time gradient growth for smooth solutions to the SQG equation and of smooth vortex patches that entangle and develop unbounded perimeter in infinite time.

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arXiv2024

Finite-time singularity formation for scalar stretching equations

We consider equations of the type: \[\partial_t ω= ωR(ω),\] for general linear operators $R$ in any spatial dimension. We prove that such equations almost always exhibit finite-time singularities for smooth and localized solutions. Singularities can even form in settings where solutions dissipate an energy. Such equations arise naturally as models in various physical settings such as inviscid and complex fluids.

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arXiv2023

Wellposedness and singularity formation beyond the Yudovich class

We introduce a local-in-time existence and uniqueness class for solutions to the 2d Euler equation with unbounded vorticity. Furthermore, we show that solutions belonging to this class can develop stronger singularities in finite time, meaning that they experience finite time blow up and exit the wellposedness class. Such solutions may be continued as weak solutions (potentially non-uniquely) after the singularity. While the general dynamics of 2d Euler solutions beyond the Yudovich class will certainly not be so tame, studying such solutions gives a way to study singular phenomena in a more controlled setting.

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arXiv2023

From Instability to Singularity Formation in Incompressible Fluids

We establish finite-time singularity formation for $C^{1,α}$ solutions to the Boussinesq system that are compactly supported on $\mathbb{R}^2$ and infinitely smooth except in the radial direction at the origin. The solutions are smooth in the angular variable at the blow-up point, which was a fundamental obstruction in previous works. This is done by exploiting a second-order effect, related to the classical Rayleigh--Bénard instability, that overcomes the regularizing effect of transport. A similar result is established for the 3d Euler system based on the Taylor--Couette instability.

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arXiv2023

Norm Growth, Non-uniqueness, and Anomalous Dissipation in Passive Scalars

We construct a divergence-free velocity field $u:[0,T] \times \mathbb{T}^2 \to \mathbb{R}^2$ satisfying $$u \in C^\infty([0,T];C^α(\mathbb{T}^2)) \quad \forall α\in [0,1)$$ such that the corresponding drift-diffusion equation exhibits anomalous dissipation for every smooth initial data. We also show that, given any $α_0 < 1$, the flow can be modified such that it is uniformly bounded only in $C^{α_0}(\mathbb{T}^2)$ and the regularity of solutions satisfy sharp (time-integrated) bounds predicted by the Obukhov-Corrsin theory. The proof is based on a general principle implying $H^1$ growth for all solutions to the transport equation, which may be of independent interest.

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arXiv2023

Remark on the Stability of Energy Maximizers for the 2D Euler equation on $\mathbb{T}^2$

It is well-known that the first energy shell, \[\mathcal{S}_1^{c_0}:=\{α\cos(x+μ)+β\cos(y+λ): α^2+β^2=c_0\,\, \&\,\, (μ,λ)\in\mathbb{R}^2\}\] of solutions to the 2d Euler equation is Lyapunov stable on $\mathbb{T}^2$. This is simply a consequence of the conservation of energy and enstrophy. Using the idea of Wirosoetisno and Shepherd \cite{WS}, which is to take advantage of conservation of a properly chosen Casimir, we give a simple and quantitative proof of the $L^2$ stability of single modes up to translation. In other words, each \[\mathcal{S}_1^{α,β}:=\{α\cos(x+μ)+β\cos(y+λ): (μ,λ)\in\mathbb{R}^2\}\] is Lyapunov stable. Interestingly, our estimates indicate that the extremal cases $α=0,$ $β=0$, and $α=\pmβ$ may be markedly less stable than the others.

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arXiv2023

Optimal enhanced dissipation and mixing for a time-periodic, Lipschitz velocity field on $\mathbb{T}^2$

We consider the advection-diffusion equation on $\mathbb{T}^2$ with a Lipschitz and time-periodic velocity field that alternates between two piecewise linear shear flows. We prove enhanced dissipation on the timescale $|\log ν|$, where $ν$ is the diffusivity parameter. This is the optimal decay rate as $ν\to 0$ for uniformly-in-time Lipschitz velocity fields. We also establish exponential mixing for the $ν= 0$ problem.

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arXiv2022

Singularity formation in the incompressible Euler equation in finite and infinite time

Some classical and recent results on the Euler equations governing perfect (incompressible and inviscid) fluid motion are collected and reviewed, with some small novelties scattered throughout. The perspective and emphasis will be given through the lens of infinite-dimensional dynamical systems, and various open problems are listed and discussed.

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arXiv2022

Strong Ill-Posedness in $L^\infty$ for the Riesz Transform Problem

We prove strong ill-posedness in $L^{\infty}$ for linear perturbations of the 2d Euler equations of the form: \[\partial_t ω+ u\cdot\nablaω= R(ω),\] where $R$ is any non-trivial second order Riesz transform. Namely, we prove that there exist smooth solutions that are initially small in $L^{\infty}$ but become arbitrarily large in short time. Previous works in this direction relied on the strong ill-posedness of the linear problem, viewing the transport term perturbatively, which only led to mild growth. In this work we derive a nonlinear model taking all of the leading order effects into account to determine the precise pointwise growth of solutions for short time. Interestingly, the Euler transport term does counteract the linear growth so that the full nonlinear equation grows an order of magnitude less than the linear one. In particular, the (sharp) growth rate we establish is consistent with the global regularity of smooth solutions.

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arXiv2022

Propagation of singularities by Osgood vector fields and for 2D inviscid incompressible fluids

We show that certain singular structures (Hölderian cusps and mild divergences) are transported by the flow of homeomorphisms generated by an Osgood velocity field. The structure of these singularities is related to the modulus of continuity of the velocity and the results are shown to be sharp in the sense that slightly more singular structures cannot generally be propagated. For the 2D Euler equation, we prove that certain singular structures are preserved by the motion, e.g. a system of $\log\log_+(1/|x|)$ vortices (and those that are slightly less singular) travel with the fluid in a nonlinear fashion, up to bounded perturbations. We also give stability results for weak Euler solutions away from their singular set.

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arXiv2020

Stationary Structures near the Kolmogorov and Poiseuille Flows in the 2d Euler Equations

We study the behavior of solutions to the incompressible $2d$ Euler equations near two canonical shear flows with critical points, the Kolmogorov and Poiseuille flows, with consequences for the associated Navier-Stokes problems. We exhibit a large family of new, non-trivial stationary states of analytic regularity, that are arbitrarily close to the Kolmogorov flow on the square torus $\mathbb{T}^2$. This situation contrasts strongly with the setting of some monotone shear flows, such as the Couette flow: in both cases the linearized problem exhibits an "inviscid damping" mechanism that leads to relaxation of perturbations of the base flows back to nearby shear flows. While this effect persists nonlinearly for suitably small and regular perturbations of some monotone shear flows, for the Kolmogorov flow our result shows that this is not possible. Our construction of these stationary states builds on a degeneracy in the global structure of the Kolmogorov flow on $\mathbb{T}^2$. In this regard both the Kolmogorov flow on a rectangular torus and the Poiseuille flow in a channel are very different, and we show that the only stationary states near them must indeed be shears, even in relatively low regularity $H^3$ resp. $H^{5+}$. In addition, we show that this behavior is mirrored closely in the related Navier-Stokes settings: the linearized problems near the Poiseuille and Kolmogorov flows both exhibit an enhanced rate of dissipation. Previous work by us and others shows that this effect survives in the full, nonlinear problem near the Poiseuille flow and near the Kolmogorov flow on rectangular tori, provided that the perturbations lie below a certain threshold. However, we show here that the corresponding result cannot hold near the Kolmogorov flow on $\mathbb{T}^2$.

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