On Some Electroconvection Models

We consider a model of electroconvection motivated by studies of the motion of a two-dimensional annular suspended smectic film under the influence of an electric potential maintained at the boundary by two electrodes. We prove that this electroconvection model has global in time unique smooth solutions.     

Viscous flow down a membrane trough

An inclined, gravity driven, open membrane trough is used as a low-cost fluid transport conduit. The membrane shape and the fluid velocity are determined numerically. The optimum opening width for maximum flow is found to be 0.651 of the membrane perimeter.     

Long Time Stability for Solutions of a β‐Plane Equation

We prove stability for arbitrarily long times of the zero solution for the so‐called β‐plane equation, which describes the motion of a two‐dimensional inviscid, ideal fluid under the influence of the Coriolis effect. The Coriolis force introduces a linear dispersive operator into the two‐dimensional incompressible Euler equations, thus making this problem amenable to an analysis from the point of view of nonlinear dispersive equations. The dispersive operator, , exhibits good decay, but has numerous unfavorable properties, chief among which are its anisotropy and its behavior at small frequencies.© 2016 Wiley Periodicals, Inc.     

Global Regularity for Some Oldroyd‐B Type Models

We investigate some critical models for visco‐elastic flows of Oldroyd‐B type in dimension 2. We use a transformation that exploits the Oldroyd‐B structure to prove an L^∞ bound on the vorticity which allows us to prove global regularity for our systems. © 2015 Wiley Periodicals, Inc.     

Homotopy method for the eigenvalues of symmetric tridiagonal matrices

We will present the homotopy method for finding eigenvalues of symmetric, tridiagonal matrices. This method finds eigenvalues separately, which can be a large advantage on systems with parallel processors. We will introduce the method and establish some bounds that justify the use of Newton’s method in constructing the homotopy curves.     

Osgood’s Lemma and Some Results for the Slightly Supercritical 2D Euler Equations for Incompressible Flow

We investigate the (slightly) super-critical two-dimensional Euler equations. The paper consists of two parts. In the first part we prove well-posedness in C ^s spaces for all s > 0. We also give growth estimates for the C ^s norms of the vorticity for ${0 < s \leqq 1}$ . In the second part we prove global regularity for the vortex patch problem in the super-critical regime. This paper extends the results of Chae et al. where they prove well-posedness for the so-called LogLog-Euler equation. We also extend the classical results of Chemin and Bertozzi–Constantin on the vortex patch problem to the slightly supercritical case. Both problems we study in the setting of the whole space.     

On the Relation between Enhanced Dissipation Timescales 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 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. © 2019 Wiley Periodicals, Inc.     

Inviscid Limit of Vorticity Distributions in the Yudovich Class

We prove that given initial data , forcing and any T > 0, the solutions u^ν of Navier-Stokes converge strongly in for any p ∈ [1, ∞) 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 by-product 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. © 2020 Wiley Periodicals LLC.     

Symmetries and Critical Phenomena in Fluids

We study two-dimensional active scalar systems arising in fluid dynamics in critical spaces in the whole plane. We prove an optimal well-posedness result that allows for the data and solutions to be scale-invariant. These scale-invariant solutions are new and their study seems to have far-reaching consequences. More specifically, we first show that the class of bounded vorticities satisfying a discrete rotational symmetry is a global existence and uniqueness class for the two-dimensional Euler squation. That is, in the well-known L¹∩L^∞ theory of Yudovich, the L¹-assumption can be dropped upon having an appropriate symmetry condition. We also show via explicit examples the necessity of discrete symmetry for the uniqueness. This already answers problems raised by Lions in 1996 and Bendetto, Marchioro, and Pulvirenti in 1993. Next, we note that merely bounded vorticity allows for one to look at solutions that are invariant under scaling—the class of vorticities that are 0-homo-geneous in space. Such vorticity is shown to satisfy a new one-dimensional evolution equation on 𝕊¹. Solutions are also shown to exhibit a number of interesting properties. In particular, using this framework, we construct time quasi-periodic solutions to the two-dimensional Euler equation exhibiting pendulum-like behavior. Finally, using the analysis of the one-dimensional equation, we exhibit strong solutions to the two-dimensional Euler equation with compact support for which angular derivatives grow at least (almost) quadratically in time (in particular, superlinear) 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 that 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 one-dimensional model that bears great resemblance to the so-called De Gregorio model. We then show that finite-time singularity formation for the one-dimensional model implies finite-time singularity formation in the class of Lipschitz solutions to the SQG equation that are compactly support. While the study of special infinite energy (i.e., nondecaying) solutions to fluid models is classical, this appears to be the first case where these special solutions 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 direct bearing on the global regularity problem for finite-energy solutions. © 2019 Wiley Periodicals, Inc.