# Seminars & Events for Analysis of Fluids and Related Topics

##### Some results on singular transport equations arising in fluid mechanics

We will discuss a few recent results in the study of fluid equations which stem from studying the dynamics of transport equations with non-local forcing. These are equations of the form: $f_t +u\cdot\nabla f =R(f)$ where $R$ is a singular integral operator and $u$ is a divergence-free vector field possibly depending upon $f$. These types of equations arise in a variety of physical scenarios. Depending upon the symbol of $R$, these equations can exhibit a number of different properties. We will briefly examine four different cases as they show up in different physical situations. The types of results we will discuss include: (1) Ill-posedness in critical spaces (2) Cascading solutions (3) Dispersion (4) "Inviscid damping" We will also mention several interesting open problems.

##### Onsager's Conjecture

In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy; conversely, he conjectured the existence of solutions belonging to any Hölder space with exponent less than 1/3 which dissipate energy. The first part of this conjecture has since been confirmed (cf. Eyink 1994, Constantin, E and Titi 1994). During this talk we will discuss recent work by Camillo De Lellis, László Székelyhidi Jr., Phil Isett and myself related to resolving the second component of Onsager's conjecture. In particular, we will discuss the construction of weak non-conservative solutions to the Euler equations whose Hölder $1/3-\epsilon$ norm is Lebesgue integrable in time.

##### Thin knotted vortex tubes in stationary solutions to the Euler equation

In this talk we will discuss the proof of the existence of thin vortex tubes for stationary solutions to the incompressible Euler equation in R^3. More precisely, given a finite collection of (possibly linked and knotted) disjoint thin tubes in R^3, we will show that they can be transformed using a small diffeomorphism into a set of vortex tubes of a Beltrami field that tends to zero at infinity.

##### Infinite volume limit for the Nonlinear Schrodinger Equation and Weak turbulence

The theory of weak turbulence has been put forward by applied mathematicians to describe the asymptotic behavior of NLS set on a compact domain - as well as many other infinite dimensional Hamiltonian systems. It is believed to be valid in a statistical sense, in the weakly nonlinear, infinite volume limit. I will present how these limits can be taken rigorously, and give rise to new equations.

##### Propagation enhancement of reaction-diffusion fronts by a line of fast diffusion

We discuss here a new model to describe biological invasions in the plane when a strong diffusion takes place on a line. By 'strong diffusion', we mean a large multiple of the Laplacian, or the fractional laplacian. The question is the asymptotic (as time goes to infinity) speed of spreading in the direction of the line and in the plane. In the case of a standard diffusion on the line, and for low diffusion, the line has no effect. Conversely, past a threshold, the line enhances the overall propagation in the plane. When the diffusion on the line is given by the fractional laplacian, this is even more dramatic: the propagation is exponential in time.

##### Interface Singularities for the Euler Equations

The fluid interface ``splash'' singularity was introduced by Castro, C\'{o}rdoba, Fefferman, Gancedo, \& G\'{o}mez-Serrano. A splash singularity occurs when a fluid interface remains locally smooth but self-intersects in finite time. In this talk, I will very briefly discuss how we construct splash singularities for the one-phase 3-D Euler and Navier-Stokes equations. I will then discuss the problem of two-phase Euler flow. Recently, Fefferman, Ionescu, and Lie have shown that a locally smooth vortex sheet cannot self-intersect in finite time. I will explain our proof of this result, which is based on elementary arguments and some precise blow-up rates for the gradient of the velocity of the fluid through which the interfaces tries to self-intersect. This is joint work with D. Coutand.

##### TBA - Saint-Raymond

**This is a joint Analysis of Fluids and Related Topics - Analysis seminar.**

##### On the stability of Prandtl boundary layer expansions of Navier-Stokes in the inviscid limit

I will present two recent results concerning the stability of boundary layer asymptotic expansions of solutions of Navier-Stokes with small viscosity. First, we show that the linearization around an arbitrary stationary shear flow (other than the Couette flow) admits an unstable eigenfunction with small wave number, when viscosity is sufficiently small. In boundary-layer variables, this yields an exponentially growing sublayer near the boundary and hence instability of the asymptotic expansions, within an arbitrarily small time, in the inviscid limit. The proof introduces a new functional-analytic approach to construct exact solutions of Orr-Sommerfeld in presence of critical layers (primitive Airy solutions). On the other hand, we show that the Prandtl asymptotic expansions hold for steady flows.

##### Anomalous diffusion in fast cellular flows

In '53, GI Taylor estimated the effective dispersion rate of a solute diffusing in the presence of a laminar flow in a pipe. It turns out that the length scales involved in typical pipes are too short for Taylor's result to apply. The goal of my talk will be to establish a preliminary estimate for the effective dispersion rate in a model problem at time scales much shorter than those required in Taylor's result. Precisely, I will study a diffusive tracer in the presence of a fast cellular flow. The main result (joint with A. Novikov) shows that the variance at intermediate time scales is of order $\sqrt{t}$. This was conjectured by W. Young, and is consistent with an anomalous diffusive behaviour.

##### Radiative transport or homogenization?

Radiative transport equations and other macroscopic kinetic models are widely used to describe multiple scattering of wave energy in random media. They account for the incoherent wave energy, and their validity is associated with the randomness of the wave field. The opposite regime is homogenization - here, the wave field has a deterministic macroscopic limit. I will try to describe the transition from one regime to the other, and understand which one is more generic. This is a joint work with G. Bal, T. Chen and T. Komorowski.

##### From particles to fluids: a derivation of the heat and the Stokes equations from Newton's laws

**This is a joint seminar with Analysis. Please note special day, time and location. **We aim at obtaining equations of fluid dynamics from Newton's laws governing the motion of the particles of fluid, in the limit when the number of particles goes to infinity and their diameter simultaneously goes to zero under the Boltzmann-Grad scaling. As suggested by Hilbert in his sixth problem, it is useful to use Boltzman's equation as an intermediate level of description. In this talk we shall report on some progress on this question in the linear case, when one derives the heat and the Stokes equations. This is a joint work with Thierry Bodineau and Laure Saint-Raymond.

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##### Global weak solutions to the inviscid 3d quasi-geostrophic equation

We will show the existence of global weak solutions to the inviscid three-dimensional quasi-geostrophic system of equations. This system of equations models the evolution of the temperature in the atmosphere. It is widely used in geophysics and meteorology. It involves a coupling between a transport equation in the whole domain, and an other one on the boundary of the domain, at the surface of the earth. This is a joint work with Marjolaine Puel.

##### Diffusions with Rough Drifts and Stochastic Symplectic Maps

**This is a joint seminar with the Probability Seminar. Please note special day and time. **According to DiPerna-Lions theory, velocity fields with weak derivatives in $L^p$ spaces possess weakly regular flows. When a velocity field is perturbed by a white noise, the corresponding (stochastic) flow is far more regular in spatial variables; a $d$-dimensional diffusion with a drift in $L^{r,q}$ space ($r$ for the spatial variable and $q$ for the temporal variable) possesses weak derivatives with stretched exponential bounds, provided that $d/r+2/q<1$. As an application one show that a Hamiltonian system that is perturbed by a white noise produces a symplectic flow provided that the corresponding Hamiltonian function $H$ satisfies $\nabla H\in L^{r,q}$ with $d/r+2/q<1$.

##### New regularity estimates for compressible Navier-Stokes

We present a new approach offering explicit regularity estimates for solutions to transport equations, and in particular for the density of the solutions to the compressible Navier-Stokes equations. This new method removes several constraints of the classical Lions-Feireisl theory. It thus allows us to treat a larger class of models with ***non*** ***monotone*** ***pressure*** terms, or ***anisotropic*** ***viscosity*** for instance, leading to many applications from geophysical flows (eddy viscosity) to solar events (virial pressure law) and some biological situations.

##### Long wave limit for water waves with vorticity.

**Please note special time. **In this talk we will present a Hamiltonian Formulation of the water waves problem with vorticity and we will study the long wave limit. In the shallow-water regime (long waves) the characteristic lenght of the horizontal motion is assumed much larger than the depth. Because this difference in the scales, the water waves equation can be simplified. In the first part of the presentation we will justify the existence of this limit in presence of vorticity and in the second one we will dicuss some properties of the models obtained.

##### Stochastic Three-Dimensional Rotating Navier-Stokes Equations: Averaging, Convergence and Regularity

**Please note special day, time and location. **We consider stochastic three-dimensional rotating Navier-Stokes equations and prove averaging theorems for stochastic problems in the case of strong rotation. Regularity results are established by bootstrapping from global regularity of the limit stochastic equations and convergence theorems. The energy injected in the system by the noise is large, the initial condition has large energy, and the regularization time horizon is long. Regularization is the consequence of a precise mechanism of relevant three-dimensional nonlinear dynamics. We establish multiscale averaging and convergence theorems for the stochastic dynamics. References [1] Flandoli F. , Mahalov A.

##### The compressible viscous surface-internal wave problem

This talk concerns the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another. The lower fluid is bounded below by a rigid bottom, and the upper fluid is bounded above by a trivial fluid of constant pressure. This is a free boundary problem: the interfaces between the fluids and above the upper fluid are free to move. The fluids are acted on by gravity in the bulk, and at the free interfaces we consider both the case of surface tension and the case of no surface forces. We establish a sharp nonlinear global-in-time stability criterion and give the explicit decay rates to the equilibrium. When the upper fluid is heavier than the lower fluid along the equilibrium interface, we characterize the set of surface tension values in which the equilibrium is nonlinearly stable.

##### Finite time singularity of a vortex patch model in the half plane

The question of global regularity vs. finite time blow-up remains open for many fluid equations. In this talk, I will discuss an active scalar equation which is an interpolation between the 2D Euler equation and the surface quasi-geostrophic equation. We study the patch dynamics for this equation in the half-plane, and prove that the solutions can develop a finite-time singularity. This is a joint work with A. Kiselev, L. Ryzhik and A. Zlatos.

##### Invariant Gibbs measures for Hamiltonian PDEs

In this talk, I will first discuss the construction of invariant Gibbs measures for Hamiltonian PDEs on the circle by Bourgain '94. Then, I will discuss the situation on the two dimensional torus and in the infinite volume case.

##### Schauder estimates for a class of non-local parabolic equations and applications

Regularity theory for non-local elliptic and parabolic equations has undergone rapid development in recent years and reached applications in several models of fluid dynamics. In this talk we will present new Schauder estimates for parabolic equations with general kernels (lacking any traditional symmetry of evenness assumptions) and show their use in studying well-posedness of an Euler-like scalar model.