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Tarek M. Elgindi

Publications and source records attributed to Tarek M. Elgindi.

At least 37 records · Page 2Linked to original sources
arXiv2020

Inviscid limit of vorticity distributions in Yudovich class

We prove that given initial data $ω_0\in L^\infty(\mathbb{T}^2)$, forcing $g\in L^\infty(0,T; L^\infty(\mathbb{T}^2))$, and any $T>0$, the solutions $u^ν$ of Navier-Stokes converge strongly in $L^\infty(0,T;W^{1,p}(\mathbb{T}^2))$ for any $p\in [1,\infty)$ to the unique Yudovich weak solution $u$ of the Euler equations. A consequence is that vorticity distribution functions converge to their inviscid counterparts. As a byproduct of the proof, we establish continuity of the Euler solution map for Yudovich solutions in the $L^p$ vorticity topology. The main tool in these proofs is a uniformly controlled loss of regularity property of the linear transport by Yudovich solutions. Our results provide a partial foundation for the Miller--Robert statistical equilibrium theory of vortices as it applies to slightly viscous fluids.

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arXiv2020

Finite-Time Singularity Formation for $C^{1,α}$ Solutions to the Incompressible Euler Equations on $\mathbb{R}^3$

It has been known since work of Lichtenstein [42] and Gunther [29] in the 1920's that the $3D$ incompressible Euler equation is locally well-posed in the class of velocity fields with Hölder continuous gradient and suitable decay at infinity. It is shown here that these local solutions can develop singularities in finite time, even for some of the simplest three-dimensional flows.

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arXiv2020

Anomalous Dissipation in Passive Scalar Transport

We study anomalous dissipation in hydrodynamic turbulence in the context of passive scalars. Our main result produces an incompressible $C^\infty([0,T)\times \mathbb{T}^d)\cap L^1([0,T]; C^{1-}(\mathbb{T}^d))$ velocity field which explicitly exhibits anomalous dissipation. As a consequence, this example also shows non-uniqueness of solutions to the transport equation with an incompressible $L^1([0,T]; C^{1-}(\mathbb{T}^d))$ drift, which is smooth except at one point in time. We also provide three sufficient conditions for anomalous dissipation provided solutions to the inviscid equation become singular in a controlled way. Finally, we discuss connections to the Obukhov-Corrsin monofractal theory of scalar turbulence along with other potential applications.

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arXiv2020

The incompressible Euler equations under octahedral symmetry: singularity formation in a fundamental domain

We consider the 3D incompressible Euler equations in vorticity form in the following fundamental domain for the octahedral symmetry group: $\{ (x_1,x_2,x_3): 0<x_3<x_2<x_1 \}.$ In this domain, we prove local well-posedness for $C^α$ vorticities not necessarily vanishing on the boundary with any $0<α<1$, and establish finite-time singularity formation within the same class for smooth and compactly supported initial data. The solutions can be extended to all of $\mathbb{R}^3$ via a sequence of reflections, and therefore we obtain finite-time singularity formation for the 3D Euler equations in $\mathbb{R}^3$ with bounded and piecewise smooth vorticities.

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arXiv2019

On Singular Vortex Patches, I: Well-posedness Issues

The purpose of this work is to discuss the well-posedness theory of singular vortex patches. Our main results are of two types: well-posedness and ill-posedness. On the well-posedness side, we show that globally $m-$fold symmetric vortex patches with corners emanating from the origin are globally well-posed in natural regularity classes as long as $m\geq 3$. In this case, all of the angles involved solve a \emph{closed} ODE system which dictates the global-in-time dynamics of the corners and only depends on the initial locations and sizes of the corners. {Along the way we obtain a global well-posedness result for a class of symmetric patches with boundary singular at the origin, which includes logarithmic spirals.} On the ill-posedness side, we show that \emph{any} other type of corner singularity in a vortex patch cannot evolve continuously in time except possibly when all corners involved have precisely the angle $\fracπ{2}$ for all time. Even in the case of vortex patches with corners of angle $\fracπ{2}$ or with corners which are only locally $m-$fold symmetric, we prove that they are generically ill-posed. We expect that in these cases of ill-posedness, the vortex patches actually cusp immediately in a self-similar way and we derive some asymptotic models which may be useful in giving a more precise description of the dynamics. In a companion work, we discuss the long-time behavior of symmetric vortex patches with corners and use them to construct patches on $\mathbb{R}^2$ with interesting dynamical behavior such as cusping and spiral formation in infinite time.

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arXiv2019

On the Stability of Self-similar Blow-up for $C^{1,α}$ Solutions to the Incompressible Euler Equations on $\mathbb{R}^3$

We study the stability of recently constructed self-similar blow-up solutions to the incompressible Euler equation. A consequence of our work is the existence of finite-energy $C^{1,α}$ solutions that become singular in finite time in a locally self-similar manner. As a corollary, we also observe that the Beale-Kato-Majda criterion cannot be improved in the class of $C^{1,α}$ solutions.

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arXiv2019

On Singular Vortex Patches, II: Long-time dynamics

In a companion paper, we gave a detailed account of the well-posedness theory for singular vortex patches. Here, we discuss the long-time dynamics of some of the classes of vortex patches we showed to be globally well-posed in the companion work. In particular, we give examples of time-periodic behavior, cusp formation in infinite time at an exponential rate, and spiral formation in infinite time.

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arXiv2019

Enhanced dissipation in the Navier-Stokes equations near the Poiseuille flow

We consider solutions to the 2d Navier-Stokes equations on $\mathbb{T}\times\mathbb{R}$ close to the Poiseuille flow, with small viscosity $ν>0$. Our first result concerns a semigroup estimate for the linearized problem. Here we show that the $x$-dependent modes of linear solutions decay on a time-scale proportional to $ν^{-1/2}|\logν|$. This effect is often referred to as \emph{enhanced dissipation} or \emph{metastability} since it gives a much faster decay than the regular dissipative time-scale $ν^{-1}$ (this is also the time-scale on which the $x$-independent mode naturally decays). We achieve this using an adaptation of the method of hypocoercivity. Our second result concerns the full nonlinear equations. We show that when the perturbation from the Poiseuille flow is initially of size at most $ν^{3/4+}$, then it remains so for all time. Moreover, the enhanced dissipation also persists in this scenario, so that the $x$-dependent modes of the solution are dissipated on a time scale of order $ν^{-1/2}|\logν|$. This transition threshold is established by a bootstrap argument using the semigroup estimate and a careful analysis of the nonlinear term in order to deal with the unboundedness of the domain and the Poiseuille flow itself.

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arXiv2018

Universal Mixers in All Dimensions

We construct universal mixers, incompressible flows that mix arbitrarily well general solutions to the corresponding transport equation, in all dimensions. This mixing is exponential in time (i.e., essentially optimal) for any initial condition with at least some regularity, and we also show that a uniform mixing rate for all initial conditions cannot be achieved. The flows are uniformly-in-time bounded in spaces $W^{s,p}$ for a range of $(s,p)$ that includes $s > 1$ and $p>2$.

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arXiv2018

Symmetries and Critical Phenomena in Fluids

We show that given an initial vorticity which is bounded and $m$-fold rotationally symmetric for $m \ge 3$, there is a unique global solution to the 2D Euler equation on the whole plane. That is, in the well-known $L^1 \cap L^\infty$ theory of Yudovich, the $L^1$ assumption can be dropped upon having an appropriate symmetry condition. This contains a class of radially homogeneous solutions to the 2D Euler equation, which gives rise to a new 1D fluid model. We discuss several interesting properties of this 1D system. Using this framework, we construct time quasi-periodic solutions to the 2D Euler equation exhibiting pendulum-like behavior. We also exhibit solutions with compact support for which angular derivatives grow linearly in time or exponential in time (the latter being in the presence of a boundary). A similar study can be done for the surface quasi-geostrophic (SQG) equation. Using the same symmetry condition, we prove local existence and uniqueness of solutions which are merely Lipschitz continuous near the origin - though, without the symmetry, Lipschitz initial data is expected to lose its Lipschitz continuity immediately. Once more, a special class of radially homogeneous solutions is considered and we extract a 1D model which bears great resemblance to the so-called De Gregorio model. We then show that finite-time singularity formation for the 1D model implies finite-time singularity formation in the class of Lipschitz solutions to the SQG equation which are compact support. While the study of special infinite energy solutions to fluid models is classical, this appears to be the first case where they can be embedded into a natural existence/uniqueness class for the equation. Moreover, these special solutions approximate finite-energy solutions for long time and have a direct bearing on the global regularity problem for finite-energy solutions.

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arXiv2018

On the relation between enhanced dissipation time-scales and mixing rates

We study diffusion and mixing in different linear fluid dynamics models, mainly related to incompressible flows. In this setting, mixing is a purely advective effect which causes a transfer of energy to high frequencies. When diffusion is present, mixing enhances the dissipative forces. This phenomenon is referred to as enhanced dissipation, namely the identification of a time-scale faster than the purely diffusive one. We establish a precise connection between quantitative mixing rates in terms of decay of negative Sobolev norms and enhanced dissipation time-scales. The proofs are based on a contradiction argument that takes advantage of the cascading mechanism due to mixing, an estimate of the distance between the inviscid and viscous dynamics, and of an optimization step in the frequency cut-off. Thanks to the generality and robustness of our approach, we are able to apply our abstract results to a number of problems. For instance, we prove that contact Anosov flows obey logarithmically fast dissipation time-scales. To the best of our knowledge, this is the first example of a flow that induces an enhanced dissipation time-scale faster than polynomial. Other applications include passive scalar evolution in both planar and radial settings and fractional diffusion.

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arXiv2018

Finite-time Singularity Formation for Strong Solutions to the Boussinesq System

The global regularity problem for the Boussinesq system is a well known open problem in mathematical fluid dynamics. As a follow up to our work \cite{EJSI}, we give examples of finite-energy and Lipschitz continuous velocity field and density $(u_0,ρ_0)$ which are $C^\infty$-smooth away from the origin and belong to a natural local well-posedness class for the Boussinesq equation whose corresponding local solution becomes singular in finite time. That is, while the sup norm of the gradient of the velocity field and the density remain finite on the time interval $t\in [0,1)$, both quantities become infinite as $t\rightarrow 1$. The key is to use scale-invariant solutions similar to those introduced in \cite{EJSI}. The proof consists of three parts: local well-posedness for the Boussinesq equation in critical spaces, the analysis of certain special infinite-energy solutions belonging to those critical spaces, and finally a cut-off argument to ensure finiteness of energy. All of this is done on spatial domains $\{(x_1,x_2): x_1 \ge γ|x_2|\}$ for any $γ> 0$ so that we can get arbitrarily close to the half-space case. We remark that the $2D$ Euler equation is globally well-posed in all of the situations we look at, so that the singularity is not coming from the domain or the lack of smoothness on the data but from the vorticity amplification due to the presence of a density gradient. It is conceivable that our methods can be adapted to produce finite-energy $C^\infty$ solutions on $\mathbb{R}^2_+$ which become singular in finite time.

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arXiv2018

Finite-time Singularity formation for Strong Solutions to the axi-symmetric $3D$ Euler Equations

For all $ε>0$, we prove the existence of finite-energy strong solutions to the axi-symmetric $3D$ Euler equations on the domains $ \{(x,y,z)\in\mathbb{R}^3: (1+ε|z|)^2\leq x^2+y^2\}$ which become singular in finite time. We further show that solutions with 0 swirl are necessarily globally regular. The proof of singularity formation relies on the use of approximate solutions at exactly the critical regularity level which satisfy a $1D$ system which has solutions which blow-up in finite time. The construction bears similarity to our previous result on the Boussinesq system \cite{EJB} though a number of modifications must be made due to anisotropy and since our domains are not scale-invariant. This seems to be the first construction of singularity formation for finite-energy strong solutions to the actual $3D$ Euler system.

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arXiv2017

Finite-time Singularity Formation for Strong Solutions to the $3D$ Euler Equations, I

In this paper and the companion paper [EJE2], we establish finite-time singularity formation for finite-energy strong solutions to the axi-symmetric $3D$ Euler equations in the domain $\{(x,y,z)\in\mathbb{R}^3:z^2\leq c(x^2+y^2)\}$ for some $c>0$. In the spirit of our previous works, [EJSI] and [EJB], we do this by first studying scale-invariant solutions which satisfy a one dimensional PDE system and proving that they may become singular in finite time for properly chosen initial data. We then prove local well-posedness for the $3D$ Euler equations in a natural regularity class which includes scale-invariant solutions. While these solutions have uniformly bounded vorticity from time zero until right before the blow-up time, they do not have finite energy. To remedy this, we cut off the scale-invariant data to ensure finite energy and prove that the corresponding local solution must also become singular in finite time. This paper focuses only on the analysis of the scale-invariant solutions themselves and the proof that they can become singular in finite time. The local well-posedness theorem and the cut-off argument are very close to those in [JSI] and [EJB] and are left for the companion paper [EJE2]. It is important to remark that while the fluid domain is the exterior of a cone, we prove global regularity for the axi-symmetric $3D$ Euler equations without swirl in the exact same regularity classes and in the same domain. It is quite possible that the methods we use can be adapted to establish finite-time singularity formation for $C^\infty$ finite-energy solutions to the $3D$ Euler equations on $\mathbb{R}^3_+$.

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arXiv2017

Finite-time Singularity Formation for Strong Solutions to the $3D$ Euler equations, II

This work is a companion to [EJE1] and its purpose is threefold: first, we will establish local well-posedness for the axi-symmetric $3D$ Euler equation in the domains $\{(x_1,x_2,x_3) \in \mathbb{R}^3 : x_3^2 \le \mathfrak{c}(x_1^2 + x_2^2) \}$ for $\mathfrak{c}$ sufficiently small in a scale of critical spaces. Second, we will prove that if the vorticity at $t=0$ can be decomposed into a scale-invariant part and a smoother part vanishing at $x=0$, then this decomposition remains valid for $t>0$ so long as the solution exists. This will then immediately imply singularity formation for finite-energy solutions in those critical spaces using that there are scale-invariant solutions which break down in finite time (this was proved in the companion paper [EJE1]). Third, we establish global regularity in the same domain and in the same scale of critical spaces for $0-$swirl solutions to the system. This implies that the finite-time singularity is not coming from the domain or the critical regularity of the data (though, these are important for the present construction), they are coming from a genuine vorticity growth mechanism due to the presence of the swirl. This work and the companion work seem to be the first case when this mechanism has been effectively used to establish finite-time singularity formation for the actual $3D$ Euler system.

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arXiv2017

Ill-posedness results in critical spaces for some equations arising in hydrodynamics

Many questions related to well-posedness/ill-posedness in critical spaces for hydrodynamic equations have been open for many years. In this article we give a new approach to studying norm inflation (in some critical spaces) for a wide class of equations arising in hydrodynamics. As an application, we prove strong ill-posedness of the $n$-dimensional Euler equations in the class $C^1\cap L^2 (Ω)$ and also in $C^k \cap L^2(Ω)$ where $Ω$ can be the whole space, a smooth bounded domain, or the torus. We also apply our method to the Oldroyd B, surface quasi-geostrophic, and Boussinesq systems.

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arXiv2017

On the Effects of Advection and Vortex Stretching

We prove finite-time singularity formation for De Gregorio's model of the three-dimensional vorticity equation in the class of $L^p\cap C^α(\mathbb{R})$ vorticities for some $α>0$ and $p<\infty$. We also prove finite-time singularity formation from smooth initial data for the Okamoto-Sakajo-Wunsch models in a new range of parameter values. As a consequence, we have finite-time singularity for certain infinite-energy solutions of the surface quasi-geostrophic equation which are $C^α$-regular. One of the difficulties in the models we consider is that there are competing \emph{nonlocal} stabilizing effects (advection) and destabilizing effects (vortex stretching) which are of the same size in terms of scaling. Hence, it is difficult to establish the domination of one effect over the other without having strong control of the solution. We conjecture that strong solutions to the De Gregorio model exhibit the following behavior: for each $0<α<1$ there exists an initial $ω_0\in C^α(\mathbb{R})$ which is compactly supported for which the solution becomes singular in finite-time; on the other hand, solutions to De Gregorio's equation are global whenever $ω_0\in L^p\cap C^{1}(\mathbb{R})$ for some $p<\infty$. Such a dichotomy seems to be a genuinely non-linear effect which cannot be explained merely by scaling considerations since $C^α$ spaces are scaling subcritical for each $α>0$.

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