ABSTRACT. 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 T 2 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 H 1 in the limit ν → 0, we show that viscous invariant measures still converge to a unique inviscid measure.

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) t −1 and u θ (t) t −2 in the appropriate radially weighted L 2 spaces. We moreover prove that the vorticity weakly converges back to radial symmetry as t → ∞, 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 → ∞, 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.Remark 1.2. By density we can extend the results to cover any ω in k ∈ L 2 (satisfying (1.7)) for which the norms appearing on the right-hand sides above are finite.Remark 1.3. The L 2 norms we are using in Theorem 1.1, namely (1.10), are natural in light of (1.8) and are wellsuited for studying vorticity depletion. However, these norms are quite strong at the origin (and infinity). Note that≈ k −1 (as opposed to k −2 as one might expect), which explains why some of the powers of k in Theorem 1.1 are slightly higher than might be at first expected. Similarly, note that F k contains information about the second term in the expansion (1.8).Remark 1.4. The correct analogue of propagation of regularity for mixing problems is the regularity of: e iktu(r) ω k (t, r), the object which measures the difference between the passive scalar and full linearized (or nonlinear) dynamics. Regularity of this object is often studied in dispersive equations and it is sometimes called the 'profile'; see e.g.[14] for more discussions (note that regularity of this type was called 'gliding regularity' in [52]). Understanding regularity of the profile plays a major role in all of the works involving nonlinear inviscid/Landau damping [5-8, 10, 14, 15, 19, 52] including those which obtain results in Sobolev spaces [9,31]. Theorem 1.1 deduces higher regularity of the vorticity profile than is necessary to prove the (1.11), at least for k ≥ 3. However, as regularity plays a crucial role in the nonlinear theory, it seems appropriate to study it as carefully as possible in the linear problem. This goal has motivated many of the primary aspects of our approach.Remark 1.5. Because in this work we were only concerned with obtaining finite Sobolev regularity of the vorticity profile, we have not carefully quantified how the constants in (1.13) depend on n. This is...

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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.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.

Abstract. 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.

ABSTRACT. We consider the global attractor of the critical SQG semigroup S(t) on the scale-invariant space

We prove that statistically stationary martingale solutions of the 3D Navier-Stokes equations on 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 → 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.2000 Mathematics Subject Classification. 35Q30, 60H30, 76F05.

We consider the initial value problem for the fractionally dissipative quasi-geostrophic equationon T 2 = [0, 1] 2 , with γ ∈ (0, 1). The coefficient in front of the dissipative term Λ γ = (−∆) γ/2 is normalized to 1. We show that given a smooth initial datum with θ0where R is arbitrarily large, there exists γ1 = γ1(R) ∈ (0, 1) such that for γ ≥ γ1, 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.August 3, 20182000 Mathematics Subject Classification. 35Q35, 76D03.