arXiv subjects

Michele Coti Zelati

Publications and source records attributed to Michele Coti Zelati.

51 records · Page 3Linked to original sources

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

A sufficient condition for the Kolmogorov 4/5 law for stationary martingale solutions to the 3D Navier-Stokes equations

We prove that statistically stationary martingale solutions of the 3D Navier-Stokes equations on $\mathbb{T}^3$ subjected to white-in-time (colored-in-space) forcing satisfy the Kolmogorov 4/5 law (in an averaged sense and over a suitable inertial range) using only the assumption that the kinetic energy is $o(ν^{-1})$ as $ν\rightarrow 0$ (where $ν$ is the inverse Reynolds number). This plays the role of a weak anomalous dissipation. No energy balance or additional regularity is assumed (aside from that satisfied by all martingale solutions from the energy inequality). If the force is statistically homogeneous, then any homogeneous martingale solution satisfies the spherically averaged 4/5 law pointwise in space. An additional hypothesis of approximate isotropy in the inertial range gives the traditional version of the Kolmogorov law. We demonstrate a necessary condition by proving that energy balance and an additional quantitative regularity estimate as $ν\rightarrow 0$ imply that the 4/5 law (or any similar scaling law) cannot hold.

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arXiv2018

Invariant measures for the stochastic one-dimensional compressible Navier-Stokes equations

We investigate the long-time behavior of solutions to a stochastically forced one-dimensional Navier-Stokes system, describing the motion of a compressible viscous fluid, in the case of linear pressure law. We prove existence of an invariant measure for the Markov process generated by strong solutions. We overcome the difficulties of working with non-Feller Markov semigroups on non-complete metric spaces by generalizing the classical Krylov-Bogoliubov method, and by providing suitable polynomial and exponential moment bounds on the solution, together with pathwise estimates.

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arXiv2018

On degenerate circular and shear flows: the point vortex and power law circular flows

We consider the problem of asymptotic stability and linear inviscid damping for perturbations of a point vortex and similar degenerate circular flows. Here, key challenges include the lack of strict monotonicity and the necessity of working in weighted Sobolev spaces whose weights degenerate as the radius tends to zero or infinity. Prototypical examples are given by circular flows with power law singularities or zeros as $r\downarrow 0$ or $r \uparrow \infty$.

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arXiv2017

Vortex axisymmetrization, inviscid damping, and vorticity depletion in the linearized 2D Euler equations

Coherent vortices are often observed to persist for long times in turbulent 2D flows even at very high Reynolds numbers and are observed in experiments and computer simulations to potentially be asymptotically stable in a weak sense for the 2D Euler equations. We consider the incompressible 2D Euler equations linearized around a radially symmetric, strictly monotone decreasing vorticity distribution. For sufficiently regular data, we prove the inviscid damping of the $θ$-dependent radial and angular velocity fields with the optimal rates $\|u^r(t)\| \lesssim \langle t \rangle^{-1}$ and $\|u^θ(t)\| \lesssim \langle t \rangle^{-2}$ in the appropriate radially weighted $L^2$ spaces. We moreover prove that the vorticity weakly converges back to radial symmetry as $t \rightarrow \infty$, a phenomenon known as vortex axisymmetrization in the physics literature, and characterize the dynamics in higher Sobolev spaces. Furthermore, we prove that the $θ$-dependent angular Fourier modes in the vorticity are ejected from the origin as $t \to \infty$, resulting in faster inviscid damping rates than those possible with passive scalar evolution. This non-local effect is called vorticity depletion. Our work appears to be the first to find vorticity depletion relevant for the dynamics of vortices.

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arXiv2017

Long time behavior and critical limit of subcritical SQG equations in scale-invariant Sobolev spaces

We consider the subcritical SQG equation in its natural scale invariant Sobolev space and prove the existence of a global attractor of optimal regularity. The proof is based on a new energy estimate in Sobolev spaces to bootstrap the regularity to the optimal level, derived by means of nonlinear lower bounds on the fractional laplacian. This estimate appears to be new in the literature, and allows a sharp use of the subcritical nature of the $L^\infty$ bounds for this problem. As a byproduct, we obtain attractors for weak solutions as well. Moreover, we study the critical limit of the attractors and prove their stability and upper-semicontinuity with respect to the strength of the diffusion.

math.AP↗

arXiv2017

Enhanced dissipation, hypoellipticity, and anomalous small noise inviscid limits in shear flows

We analyze the decay and instant regularization properties of the evolution semigroups generated by two-dimensional drift-diffusion equations in which the scalar is advected by a shear flow and dissipated by full or partial diffusion. We consider both the space-periodic $\mathbb{T}^2$ setting and the case of a bounded channel $\mathbb{T}\times [0,1]$ with no-flux boundary conditions. In the infinite Péclet number limit (diffusivity $ν\to 0$), our work quantifies the enhanced dissipation effect due to the shear. We also obtain hypoelliptic regularization, showing that solutions are instantly Gevrey regular even with only partial diffusion. The proofs rely on localized spectral gap inequalities and ideas from hypocoercivity with an augmented energy functional with weights replaced by pseudo-differential operators (of a rather simple form). As an application, we study small noise inviscid limits of invariant measures of stochastic perturbations of passive scalars, and show that the classical Freidlin scaling between noise and diffusion can be modified. In particular, although statistically stationary solutions blow up in $H^1$ in the limit $ν\to0$, we show that viscous invariant measures still converge to a unique inviscid measure.

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arXiv2016

Invariant measures for passive scalars in the small noise inviscid limit

We consider a class of invariant measures for a passive scalar $f$ driven by an incompressible velocity field $\boldsymbol{u}$, on a $d$-dimensional periodic domain, satisfying $$ \partial_t f + \boldsymbol{u} \cdot \nabla f = 0, \qquad f(0)=f_0. $$ The measures are obtained as limits of stochastic viscous perturbations. We prove that the span of the $H^1$ eigenfunctions of the operator $\boldsymbol{u}\cdot\nabla$ contains the support of these measures. We also analyze several explicit examples: when $\boldsymbol{u}$ is a shear flow or a relaxation enhancing flow (a generalization of weakly mixing), we can characterize the limiting measure uniquely and compute its covariance structure. We also consider the case of two-dimensional cellular flows, for which further regularity properties of the functions in the support of the measure can be deduced. The main results are proved with the use of spectral theory results, in particular the RAGE theorem, which are used to characterize large classes of orbits of the inviscid problem that are growing in $H^1$.

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arXiv2015

Smooth attractors for weak solutions of the SQG equation with critical dissipation

We consider the evolution of weak vanishing viscosity solutions to the critically dissipative surface quasi-geostrophic equation. Due to the possible non-uniqueness of solutions, we rephrase the problem as a set-valued dynamical system and prove the existence of a global attractor of optimal Sobolev regularity. To achieve this, we derive a new Sobolev estimate involving Hölder norms, which complement the existing estimates based on commutator analysis.

math.AP↗

arXiv2015

Uniformly attracting limit sets for the critically dissipative SQG equation

We consider the global attractor of the critical SQG semigroup $S(t)$ on the scale-invariant space $H^1(\mathbb{T}^2)$. It was shown in~\cite{CTV13} that this attractor is finite dimensional, and that it attracts uniformly bounded sets in $H^{1+δ}(\mathbb{T}^2)$ for any $δ>0$, leaving open the question of uniform attraction in $H^1(\mathbb{T}^2)$. In this paper we prove the uniform attraction in $H^1(\mathbb{T}^2)$, by combining ideas from DeGiorgi iteration and nonlinear maximum principles.

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arXiv2014

Singular limits of Voigt models in fluid dynamics

We investigate the long-term behavior, as a certain regularization parameter vanishes, of the three-dimensional Navier-Stokes-Voigt model of a viscoelastic incompressible fluid. We prove the existence of global and exponential attractors of optimal regularity. We then derive explicit upper bounds for the dimension of these attractors in terms of the three-dimensional Grashof number and the regularization parameter. Finally, we also prove convergence of the (strong) global attractor of the 3D Navier-Stokes-Voigt model to the (weak) global attractor of the 3D Navier-Stokes equation. Our analysis improves and extends recent results obtained by Kalantarov and Titi in [31].

math.DS↗

arXiv2014

On the global regularity for the supercritical SQG equation

We consider the initial value problem for the fractionally dissipative quasi-geostrophic equation \[ \partial_t θ+ \mathcal{R}^\perp θ\cdot \nabla θ+ Λ^γθ= 0, \qquad θ(\cdot,0) =θ_0 \] on $\mathbb{T}^2 = [0,1]^2$, with $γ\in (0,1)$. The coefficient in front of the dissipative term $Λ^γ= (-Δ)^{γ/2}$ is normalized to $1$. We show that given a smooth initial datum with $\|θ_0\|_{L^2}^{γ/2} \|θ_0\|_{\dot{H}^2}^{1-γ/2}\leq R$, where {\em $R$ is arbitrarily large}, there exists $γ_1 = γ_1(R) \in (0,1)$ such that for $γ\geq γ_1$, the solution of the supercritical SQG equation with dissipation $Λ^γ$ does not blow up in finite time. The main ingredient in the proof is a new concise proof of eventual regularity for the supercritical SQG equation, that relies solely on nonlinear lower bounds for the fractional Laplacian and the maximum principle.

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arXiv2014

Minimality properties of set-valued processes and their pullback attractors

We discuss the existence of pullback attractors for multivalued dynamical systems on metric spaces. Such attractors are shown to exist without any assumptions in terms of continuity of the solution maps, based only on minimality properties with respect to the notion of pullback attraction. When invariance is required, a very weak closed graph condition on the solving operators is assumed. The presentation is complemented with examples and counterexamples to test the sharpness of the hypotheses involved, including a reaction-diffusion equation, a discontinuous ordinary differential equation and an irregular form of the heat equation.

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arXiv2014

The primitive equations of the atmosphere in presence of vapor saturation

A modification of the classical primitive equations of the atmosphere is considered in order to take into account important phase transition phenomena due to air saturation and condensation. We provide a mathematical formulation of the problem that appears to be new in this setting, by making use of differential inclusions and variational inequalities, and which allows to develop a rather complete theory for the solutions to what turns out to be a nonlinearly coupled system of non-smooth partial differential equations. Specifically we prove global existence of quasi-strong and strong solutions, along with uniqueness results and maximum principles of physical interest.

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arXiv2012

Multivalued Attractors and their Approximation: Applications to the Navier-Stokes equations

This article is devoted to the study of multivalued semigroups and their asymptotic behavior, with particular attention to iterations of set-valued mappings. After developing a general abstract framework, we present an application to a time discretization of the two-dimensional Navier-Stokes equations. More precisely, we prove that the fully implicit Euler scheme generates a family of discrete multivalued dynamical systems, whose global attractors converge to the global attractor of the continuous system as the time-step parameter approaches zero.

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