# Seminars & Events for Analysis of Fluids and Related Topics

##### Fluid models from plasma physics

we will give an overview of some (charged) fluid models from plasma physics, mainly introducing the models and stating some important results and open questions.

##### Of logarithms and exponents: convection at infinite Prandtl numbers with mixed thermal boundary conditions

For decades, experiments (and more recently numerical simulations) have attempted to determine how the effective transport of heat (measured by the non-dimensional Nusselt number Nu) scales with the driving force (as measured by the Rayleigh number Ra) --when said driving force is asymptotically strong--in Rayleigh-Benard convection where an incompressible, Boussinesq fluid is driven by an imposed temperature gradient. To date the results are inconclusive for experiments, and finite limitations on computational resources restrict the potential usefulness of direct numerical simulation. In contrast to these approaches, rigorous upper bounds on the heat transport have been derived for this system using a variational technique commonly referred to as the background method. After providing a brief survey of some of the more recent resul

##### Ill-posedness results for some of the equations of hydrodynamics

We will begin by presenting some new instability and ill-posedness results for the incompressible Euler equations. These ill-posedness results are based upon some special exact solutions to the 3D Euler equations which bring out both the non-locality and the non-linear nature of the vortex stretching term. We will then discuss a general ill-posedness result for a large class of equations arising in hydrodynamics.

##### An L log L bound on the vorticity in the 3D NSE

The goal of this lecture is to present a spatially localized L log L bound on the vorticity in the 3D Navier-Stokes equations, assuming a very mild, _purely geometric_ condition. Besides being of an independent interest, the bound transforms a physically, numerically, and mathematical-analysis motivated large-data criticality scenario (based on vortex stretching and anisotropic diffusion) into a no blow-up scenario.

##### Universality in interface growth models

Over the past few years, there has been growing evidence, at the heuristic, the mathematically rigorous, and even the experimental level, that models of one-dimensional interface growth exhibit a "universal" behaviour at large scales. More precisely, it is conjectured that there exists a self-similar space-time process called the "KPZ fixed point" which attracts a very large class of microscopic models under suitable rescaling. It has also emerged that a certain ill-posed nonlinear stochastic PDE, the KPZ equation, has a "weak universality" property in the sense that large classes of models with a tuneable parameter converge to its solutions at intermediate scalings in the limit where the tuneable parameter is small.

##### Ill-posedness / Well-posedness Results for a Class of Active Scalar Equations.

**Please note special day (Tuesday). **We discuss a class of active scalar equations where the transport velocities are more singular than the active scalar. There is a significant difference in the well-posedness properties of the problem depending on whether the Fourier multiplier symbol for the velocity is even or odd. The "even" symbol non-diffusive or weakly diffusive equations are ill-posed in Sobolev spaces. However the critically diffusive equations are globally well posed in both the odd and even cases. Examples of "even" equations are the magnetogeostrophic equation that is a model for the geodynamo and the modified porous media equation. This is joint work with Francisco Gancedo, Weiran Sun and Vlad Vicol.

##### Potentially Singular Solutions of the 3D Incompressible Euler Equations

Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view, by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### TBA - Kiselev

**PLEASE NOTE SPECIAL DAY (WEDNESDAY) AND LOCATION.**

##### Scale invariant solutions to Navier Stokes equation and implications to Leray-Hopf weak solutions

In this talk, I will first discuss the existence of scale invariant solutions to Navier Stokes equation with arbitrary $-1$ homogeneous initial data. Since these solutions may not be small, linearized analysis seem to suggest nontrivial bifurcations. Under a quite plausible spectral assumption, we show rigorously that such bifurcations do occur and they imply non-uniqueness of scale-invariant solutions. By appropriately localizing such solutions, we then obtain non-uniqueness of Leray-Hopf weak solutions with initial data which are compactly supported, smooth away from origin, and having a singularity at the origin of the type $O(\frac{1}{`|x|`

})$, which will be sharp. The verification of the spectral assumption involves only smooth and decaying functions, and seems to be doable numerically.

##### On the well-posedness of an interface damped free boundary fluid-structure model

We address a fluid-structure system which consists of the incompressible Navier-Stokes equations and a damped linear wave equation defined on two dynamic domains. The equations are coupled through transmission boundary conditions and additional boundary stabilization effects imposed on the free moving interface separating the two domains. We will discuss local existence and uniqueness of solutions and establish global existence for small initial data. This is a joint work with M. Ignatova, I. Lasiecka, and A. Tuffaha.

##### The 2D Magnetohydrodynamic Equations with Partial Dissipation

The magnetohydrodynamic (MHD) equations model electrically conducting fluids such as plasmas and are important in understanding many natural phenomena such as solar flares and the formation of Northern Lights. Mathematically the MHD equations can be difficult to analyze due to the nonlinear coupling between the induction equation and the Navier-Stokes equations with the Lorentz force. One fundamental problem on the MHD equations is whether or not their solutions exist for all time. This problem has attracted considerable interest recently for the 2D MHD equations with partial dissipation. This talk presents some very recent global regularity results for various partial dissipation cases.

##### Mathematical study of a degenerate boundary layer

The goal of this talk is to analyze asymptotically an equation stemming from oceanographic models describing the motion of large scale currents. This equation is known to give rise to boundary layers on the east and west coasts of the domain. One of the major issues of our study lies in the fact that the size of these lateral boundary layers becomes very large as one approaches the north and south end points of the domain. In a neighbourhood of these zones, the classical construction of boundary layers must therefore be completely changed. We prove that the north and south boundary layers are the solutions of some evolutionary equation, and that their profile is thus non-intrinsic. We also exhibit discontinuity boundary layers, which penetrate the interior of the domain when the latter has islands, for instance.

##### Gradient structures and dissipation distances for reaction-diffusion systems

We discuss reaction-diffusion systems with reactions satisfying mass-action kinetics and the detailed-balance condition. They allow for a gradient structure where the driving functional is the relative entropy and the dissipation potential is the sum of a Wasserstein part for diffusion and a reaction part. This formulation highlights the additive splitting of the dissipative processes for diffusion in terms of optimal transport and for reaction in terms of transformation in chemical space. We emphasize the geometric structure induced by the interaction of these two dissipative processes. For the simple case of a scalar reaction-diffusion equation the induced dissipation distance can be characterized explicitly, giving the so-called Hellinger-Kantorovich distance on the set of all measure-valued concentration profiles.

##### The mathematical theory of the Prandtl equation.

At high Reynolds number, viscous flows near a solid wall exhibit a ***boundary layer*** behavior, that is strong velocity gradients near the wall. To describe this boundary layer, a formal asymptotic model was written down by Prandtl in 1904. Although very classical, the Prandtl model still raises mathematical difficulties. We shall discuss these difficulties and recent progress in the talk (based on a joint work with Nader Masmoudi).

##### Analysis, computing, and numerical analysis for 3D interfacial flows with surface tension

In this talk, we will discuss the initial value problem for 3D interfacial fluid flows with surface tension. We will emphasize the case in which the fluid velocities are given by Darcy's Law; this can serve as a

model for intefacial flow in a porous medium. We will discuss a well-posedness proof for the problem, with the initial data in Sobolev spaces. We will also discuss a non-stiff numerical method; this

is similar in spirit to the method of Hou, Lowengrub, and Shelley for the corresponding 2D problem. Finally, we will give some of the details of a convergence proof for a variant of this numerical method;

this convergence proof requires estimates which are similar in spirit to the estimates required to demonstrate well-posedness. The results discussed may include joint work with Nader Masmoudi,

##### An introduction to Computer-Assisted Proofs with applications to incompressible fluids

The first half of the talk will be devoted to explain the fundamentals of this technique, stressing the difference between rigorous and non-rigorous methods. In the second half we will discuss applications to problems in incompressible fluids such as the confined Muskat, the $\alpha$-patches or the water waves problem. Some proofs will be presented in real time. Joint and ongoing work with Angel Castro, Diego Cordoba, Charles Fefferman, Francisco Gancedo, Rafael Granero-Belinchon, and Alberto Martin Zamora.

##### Unique Ergodicity and Mixing for the Degenerately forced Boussinesq Equations and related systems

We establish the existence, uniqueness and attraction properties of an ergodic invariant measure for the Boussinesq Equations in the presence of a degenerate stochastic forcing acting only in the temperature equation and only at the largest spatial scales. The central challenge is to establish time asymptotic smoothing properties of the Markovian dynamics corresponding to this system. Towards this aim we encounter a Lie bracket structure in the associated vector fields with a complicated dependence on solutions. This leads us to develop a novel Hormander-type condition for infinite-dimensional systems. Demonstrating the sufficiency of this condition requires new techniques for the spectral analysis of the Malliavin covariance matrix. This is joint work with Juraj Foldes, Geordie Richards and Enrique Thomann.

##### On active scalar equations with nonlocal velocity.

The problem of finite-time singularity versus global regularity for active scalar equations with nonlocal velocities has attracted much attention in recent years. In this talk, I will discuss some recent results in this direction.

##### Long time behavior of the Navier-Stokes and related equations

I will review some resent results on the long time behavior of equations arising in fluid dynamics, such as the 3D Navier-Stokes and critical surface quasi-geostrophic equations.