Publications
42 results found
Coti Zelati M, Dolce M, Lo CC, 2024, Diffusion Enhancement and Taylor Dispersion for Rotationally Symmetric Flows in Discs and Pipes, Journal of Mathematical Fluid Mechanics, Vol: 26, ISSN: 1422-6928
In this note, we study the long-time dynamics of passive scalars driven by rotationally symmetric flows. We focus on identifying precise conditions on the velocity field in order to prove enhanced dissipation and Taylor dispersion in three-dimensional infinite pipes. As a byproduct of our analysis, we obtain an enhanced decay for circular flows on a disc of arbitrary radius.
Bedrossian J, Bianchini R, Zelati MC, et al., 2023, Nonlinear inviscid damping and shear-buoyancy instability in the two-dimensional Boussinesq equations, COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, Vol: 76, Pages: 3685-3768, ISSN: 0010-3640
Coti Zelati M, Gallay T, 2023, Enhanced dissipation and Taylor dispersion in higher-dimensional parallel shear flows, JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, Vol: 108, Pages: 1358-1392, ISSN: 0024-6107
Blumenthal A, Coti Zelati M, Gvalani RS, 2023, Exponential mixing for random dynamical systems and an example of Pierrehumbert, The Annals of Probability, Vol: 51, Pages: 1559-1601, ISSN: 0091-1798
We consider the question of exponential mixing for random dynamical systems on arbitrary compact manifolds without boundary. We put forward a robust, dynamics-based framework that allows us to construct space-time smooth, uniformly bounded in time, universal exponential mixers. The framework is then applied to the problem of proving exponential mixing in a classical example proposed by Pierrehumbert in 1994, consisting of alternating periodic shear flows with randomized phases. This settles a longstanding open problem on proving the existence of a space-time smooth (universal) exponentially mixing incompressible velocity field on a two-dimensional periodic domain while also providing a toolbox for constructing such smooth universal mixers in all dimensions.
Coti Zelati M, Elgindi TM, Widmayer K, 2023, Stationary Structures Near the Kolmogorov and Poiseuille Flows in the 2d Euler Equations, Archive for Rational Mechanics and Analysis, Vol: 247, ISSN: 0003-9527
<jats:title>Abstract</jats:title><jats:p>We study the behavior of solutions to the incompressible 2<jats:italic>d</jats:italic> 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 that are arbitrarily close to the Kolmogorov flow on the square torus <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathbb {T}^2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> in analytic regularity. This situation contrasts strongly with the setting of some monotone shear flows, such as the Couette flow: there the linearized problem exhibits an “inviscid damping” mechanism that leads to relaxation of perturbations of the base flows back to nearby shear flows. Our results show that such a simple description of the long-time behavior is not possible for solutions near the Kolmogorov flow on <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathbb {T}^2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>. Our construction of the new stationary states builds on a degenerac
Bianchini R, Coti Zelati M, Dolce M, 2022, Linear Inviscid Damping for Shear Flows Near Couette in the 2D Stably Stratified Regime
We investigate the linear stability of shears near the Couette flow for a class of 2D incompressible stably stratified fluids. Our main result consists of nearly optimal decay rates for perturbations of stationary states whose velocities are monotone shear flows (U(y), 0) and have an exponential density profile. In the case of the Couette flow U(y) = y, we recover the rates predicted by Hartman in 1975, by adopting an explicit point-wise approach in frequency space. As a byproduct, this implies optimal decay rates as well as Lyapunov instability in L2 for the vorticity. For the previously unexplored case of more general shear flows close to Couette, the inviscid damping results follow by a weighted energy estimate. Each outcome concerning the stably stratified regime applies to the Boussinesq equations as well. Remarkably, our results hold under the celebrated Miles-Howard criterion for stratified fluids.
- Abstract
- Open Access Link
- Cite
- Citations: 21
Coti Zelati M, Dolce M, Feng Y, et al., 2021, Global existence for the two-dimensional Kuramoto–Sivashinsky equation with a shear flow, Publisher: Springer Science and Business Media LLC
<jats:title>Abstract</jats:title><jats:p>We consider the Kuramoto–Sivashinsky equation (KSE) on the two-dimensional torus in the presence of advection by a given background shear flow. Under the assumption that the shear has a finite number of critical points and there are linearly growing modes only in the direction of the shear, we prove global existence of solutions with data in <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>, using a bootstrap argument. The initial data can be taken arbitrarily large.</jats:p>
Zelati MC, Gallay T, 2021, Enhanced dissipation and Taylor dispersion in higher-dimensional parallel shear flows
We consider the evolution of a passive scalar advected by a parallel shearflow in an infinite cylinder with bounded cross section, in arbitrary spacedimension. The essential parameters of the problem are the moleculardiffusivity $\nu$, which is assumed to be small, and the wave number $k$ in thestreamwise direction, which can take arbitrary values. Under genericassumptions on the shear velocity $v$, we obtain optimal decay estimates forlarge times, both in the enhanced dissipation regime $\nu \ll |k|$ and in theTaylor dispersion regime $|k| \ll \nu$. Our results can be deduced fromresolvent estimates using a quantitative version of the Gearhart-Pr\"usstheorem, or can be established more directly via the hypocoercivity method.Both approaches are explored in the present example, and their relativeefficiency is compared.
Coti Zelati M, D Drivas T, 2021, A stochastic approach to enhanced diffusion, ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, Pages: 811-834
Coti Zelati M, Glatt-Holtz N, Trivisa K, 2021, Invariant Measures for the Stochastic One-Dimensional Compressible Navier–Stokes Equations, Applied Mathematics & Optimization, Vol: 83, Pages: 1487-1522, ISSN: 0095-4616
<jats:title>Abstract</jats:title><jats:p>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.</jats:p>
Coti Zelati M, Hairer M, 2021, A Noise-Induced Transition in the Lorenz System, Communications in Mathematical Physics, Vol: 383, Pages: 2243-2274, ISSN: 0010-3616
<jats:title>Abstract</jats:title><jats:p>We consider a stochastic perturbation of the classical Lorenz system in the range of parameters for which the origin is the global attractor. We show that adding noise in the last component causes a transition from a unique to exactly two ergodic invariant measures. The bifurcation threshold depends on the strength of the noise: if the noise is weak, the only invariant measure is Gaussian, while strong enough noise causes the appearance of a second ergodic invariant measure.</jats:p>
Bedrossian J, Bianchini R, Zelati MC, et al., 2021, Nonlinear inviscid damping and shear-buoyancy instability in the two-dimensional Boussinesq equations
We investigate the long-time properties of the two-dimensional inviscidBoussinesq equations near a stably stratified Couette flow, for an initialGevrey perturbation of size $\varepsilon$. Under the classical Miles-Howardstability condition on the Richardson number, we prove that the systemexperiences a shear-buoyancy instability: the density variation and velocityundergo an $O(t^{-1/2})$ inviscid damping while the vorticity and densitygradient grow as $O(t^{1/2})$. The result holds at least until the natural,nonlinear timescale $t \approx \varepsilon^{-2}$. Notice that the densitybehaves very differently from a passive scalar, as can be seen from theinviscid damping and slower gradient growth. The proof relies on severalingredients: (A) a suitable symmetrization that makes the linear terms amenableto energy methods and takes into account the classical Miles-Howard spectralstability condition; (B) a variation of the Fourier time-dependent energymethod introduced for the inviscid, homogeneous Couette flow problem developedon a toy model adapted to the Boussinesq equations, i.e. tracking the potentialnonlinear echo chains in the symmetrized variables despite the vorticitygrowth.
Coti Zelati M, Pavliotis GA, 2020, Homogenization and hypocoercivity for Fokker–Planck equations driven by weakly compressible shear flows, IMA Journal of Applied Mathematics, Vol: 85, Pages: 951-979, ISSN: 0272-4960
<jats:title>Abstract</jats:title> <jats:p>We study the long-time dynamics of 2D linear Fokker–Planck equations driven by a drift that can be decomposed in the sum of a large shear component and the gradient of a regular potential depending on one spatial variable. The problem can be interpreted as that of a passive scalar advected by a slightly compressible shear flow, and undergoing small diffusion. For the corresponding stochastic differential equation, we give explicit homogenization rates in terms of a family of time-scales depending on the parameter measuring the strength of the incompressible perturbation. This is achieved by exploiting an auxiliary Poisson problem, and by computing the related effective diffusion coefficients. Regarding the long-time behavior of the solution of the Fokker–Planck equation, we provide explicit decay rates to the unique invariant measure by employing a quantitative version of the classical hypocoercivity scheme. From a fluid mechanics perspective, this turns out to be equivalent to quantifying the phenomenon of enhanced diffusion for slightly compressible shear flows.</jats:p>
Coti Zelati M, Dolce M, 2020, Separation of time-scales in drift-diffusion equations on R<sup>2</sup>, Journal des Mathematiques Pures et Appliquees, Vol: 142, Pages: 58-75, ISSN: 0021-7824
We deal with the problem of separation of time-scales and filamentation in a linear drift-diffusion problem posed on the whole space R2. The passive scalar considered is stirred by an incompressible flow with radial symmetry. We identify a time-scale, much faster than the diffusive one, at which mixing happens along the streamlines, as a result of the interaction between transport and diffusion. This effect is also known as enhanced dissipation. The proofs are based on an adaptation of a hypocoercivity scheme and yield a linear semigroup estimate in a suitable weighted L2-based space.
- Abstract
- Open Access Link
- Cite
- Citations: 13
Colombo M, Zelati MC, Widmayer K, 2020, Mixing and diffusion for rough shear flows
This article addresses mixing and diffusion properties of passive scalarsadvected by rough ($C^\alpha$) shear flows. We show that in general, one cannotexpect a rough shear flow to increase the rate of inviscid mixing to more thanthat of a smooth shear without critical points. On the other hand, diffusionmay be enhanced at a much faster rate. This shows that in the setting of lowregularity, the interplay between inviscid mixing properties and enhanceddissipation is more intricate, and in fact contradicts some of the naturalheuristics that are valid in the smooth setting.
Coti Zelati M, Elgindi TM, Widmayer K, 2020, Enhanced Dissipation in the Navier–Stokes Equations Near the Poiseuille Flow, Communications in Mathematical Physics, Vol: 378, Pages: 987-1010, ISSN: 0010-3616
Coti Zelati M, 2020, Stable mixing estimates in the infinite Péclet number limit, Journal of Functional Analysis, Vol: 279, Pages: 108562-108562, ISSN: 0022-1236
Bedrossian J, Coti Zelati M, Punshon-Smith S, et al., 2020, Sufficient Conditions for Dual Cascade Flux Laws in the Stochastic 2d Navier–Stokes Equations, Archive for Rational Mechanics and Analysis, Vol: 237, Pages: 103-145, ISSN: 0003-9527
Zelati MC, Delgadino MG, Elgindi TM, 2020, On the Relation between Enhanced Dissipation Timescales and Mixing Rates, Communications on Pure and Applied Mathematics, Vol: 73, Pages: 1205-1244, ISSN: 0010-3640
<jats:title>Abstract</jats:title><jats:p>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 that 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 timescale 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 timescales. 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 an optimization step in the frequency cutoff.</jats:p><jats:p>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 timescales. To the best of our knowledge, this is the first example of a flow that induces an enhanced dissipation timescale faster than polynomial. Other applications include passive scalar evolution in both planar and radial settings and fractional diffusion. © 2019 Wiley Periodicals, Inc.</jats:p>
Bedrossian J, Coti Zelati M, Vicol V, 2019, Vortex Axisymmetrization, Inviscid Damping, and Vorticity Depletion in the Linearized 2D Euler Equations, Annals of PDE, Vol: 5, ISSN: 2524-5317
Bedrossian J, Coti Zelati M, Punshon-Smith S, et al., 2019, A Sufficient Condition for the Kolmogorov 4/5 Law for Stationary Martingale Solutions to the 3D Navier–Stokes Equations, Communications in Mathematical Physics, Vol: 367, Pages: 1045-1075, ISSN: 0010-3616
We prove that statistically stationary martingale solutions of the 3D Navier–Stokes equations on T3 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→ 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→ 0 imply that the 4/5 law (or any similar scaling law) cannot hold.
- Abstract
- Open Access Link
- Cite
- Citations: 11
Coti Zelati M, Zillinger C, 2019, On degenerate circular and shear flows: the point vortex and power law circular flows, Communications in Partial Differential Equations, Vol: 44, Pages: 110-155, ISSN: 0360-5302
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. By using a Fourier multiplier approach, we construct energy functionals to deduce stability in a perturbative setting. For sufficiently high spherical harmonics, we can handle any circular flows with power law singularities or zeros as r↓0 or r↑∞, while for low frequencies we can treat circular flows close to the Taylor–Couette flow. Similar results apply in the planar shear flow case close to Couette.
- Abstract
- Open Access Link
- Cite
- Citations: 18
Coti Zelati M, 2018, Long-Time Behavior and Critical Limit of Subcritical SQG Equations in Scale-Invariant Sobolev Spaces, Journal of Nonlinear Science, Vol: 28, Pages: 305-335, ISSN: 0938-8974
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∞ bounds for this problem. As a by-product, 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.
Bedrossian J, Coti Zelati M, 2017, Enhanced Dissipation, Hypoellipticity, and Anomalous Small Noise Inviscid Limits in Shear Flows, Archive for Rational Mechanics and Analysis, Vol: 224, Pages: 1161-1204, ISSN: 0003-9527
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 T2 setting and the case of a bounded channel T×[ 0 , 1 ] with no-flux boundary conditions. In the infinite Péclet number limit (diffusivity ν→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 H1 in the limit ν→ 0 , we show that viscous invariant measures still converge to a unique inviscid measure.
Coti Zelati M, Kalita P, 2017, Smooth attractors for weak solutions of the SQG equation with critical dissipation, Discrete & Continuous Dynamical Systems - B, Vol: 22, Pages: 1857-1873, ISSN: 1553-524X
Bedrossian J, Coti Zelati M, Glatt-Holtz N, 2016, Invariant Measures for Passive Scalars in the Small Noise Inviscid Limit, Communications in Mathematical Physics, Vol: 348, Pages: 101-127, ISSN: 0010-3616
Coti Zelati M, Vicol V, 2016, On the global regularity for the supercritical SQG equation, Indiana University Mathematics Journal, Vol: 65, Pages: 535-552, ISSN: 0022-2518
Constantin P, Coti Zelati M, Vicol V, 2016, Uniformly attracting limit sets for the critically dissipative SQG equation, Nonlinearity, Vol: 29, Pages: 298-318, ISSN: 0951-7715
Coti Zelati M, Gal CG, 2015, Singular Limits of Voigt Models in Fluid Dynamics, Journal of Mathematical Fluid Mechanics, Vol: 17, Pages: 233-259, ISSN: 1422-6928
Coti Zelati M, Kalita P, 2015, Minimality Properties of Set-Valued Processes and their Pullback Attractors, SIAM Journal on Mathematical Analysis, Vol: 47, Pages: 1530-1561, ISSN: 0036-1410
This data is extracted from the Web of Science and reproduced under a licence from Thomson Reuters. You may not copy or re-distribute this data in whole or in part without the written consent of the Science business of Thomson Reuters.