# Seminars & Events for Analysis of Fluids and Related Topics

##### Bayesian Inversion for Functions and Geometry

**This is a joint seminar with the PACM Colloquium. Please note special location. **Many problems in the physical sciences require the determination of an unknown function from a finite set of indirect measurements. Examples include oceanography, medical imaging, oil recovery, water resource management and weather forecasting. Furthermore there are numerous inverse problems where geometric characteristics, such as interfaces, are key unknown features of the overall inversion. Applications include the determination of layers and faults within subsurface formations, and the detection of unhealthy tissue in medical imaging. We describe a theoretical and computational Bayesian framework relevant to the solution of inverse problems for functions, and for geometric features.

##### A probabilistic approach to the Navier-Stokes equations

In this preliminary talk, we consider the regularity issues of the incompressible Navier-Stokes equations on the basis of probabilistic methods. In the stream function (vector potential) formulation, the Navier-Stokes equations are recast in path-integral forms by "the Feynman-Kac formula". We discuss the regularity of solutions on this basis.

##### The Taylor model in magnetohydrodynamics

**Please note double seminar on this date. **We shall discuss a model introduced by J.B. Taylor in 1963, that comes from a formal asymptotic limit of MHD (magnetohydrodynamics) equations with rotation. This asymptotic model, relevant to the Earth's dynamo problem, should in principle allow for easier numerics. However, its simulation has been unsuccessful so far, due to unclear stability properties. The aim of the talk is to present recent mathematical results on this stability issue (joint work with I. Gallagher, L. Saint-Raymond).

##### The Hele-Shaw and Muskat problem

**Please note double seminar on this date. **In this talk we are going to review some recent results concerning the evolution of a free boundary under Darcy's flow. This is known as the Muskat problem or the Hele-Shaw cell problem with gravity. In particular, we will present a new method based on the study of the bulk rather than the appropriate integral equation.

##### Space-time resonances and high-frequency instabilities in the two-fluid Euler-Maxwell system

We show that space-time resonances induce high-frequency instabilities in the two-fluid Euler-Maxwell system. This implies in particular that the Zakharov approximation to Euler-Maxwell is stable if and only if the group velocity vanishes in the Schrödinger equation satisfied by the envelope of the WKB electrical field. Our analysis further shows that time resonances may fail to induce fast instabilities, even in the case of incompatible nonlinearities, in the presence of fast transverse variations of the WKB profile. This is joint work with Eric Dumas (Grenoble) and Lu Yong (Prague).

##### Convection enhanced mixing and spectral properties of the advection-diffusion equation in the semi-classical limit for vanishing diffusivity

We consider the two-dimensional advection-diffusion equation on a bounded domain subject to Dirichlet or von Neumann boundary conditions involving a Liouville integrable Hamiltonian. Transformation to action-angle coordinates permits averaging in time and angle, resulting in an equation that allows for separation of variables. The Fourier transform in the angle coordinate transforms the equation into an effective diffusive equation and a countable family of non-self-adjoint Schrödinger equations. For the corresponding Liouville-Sturm problems, complex-plane WKB methods were applied to study the spectrum in the semi-classical limit of vanishing diffusivity. The spectral limit graph is found to consist of analytic curves (branches) related to Stokes graphs forming a tree-structure.

##### Probabilistic global well-posedness of the energy-critical defocusing nonlinear wave equation bellow the energy space

We consider the energy-critical defocusing nonlinear wave equation (NLW) on $R^d$, $d = 3, 4, 5$. In the deterministic setting, Christ, Colliander, and Tao showed that this equation is ill-posed below the energy space $H^1× L^2$. In this talk, we take a probabilistic approach. More precisely, we prove almost sure global existence and uniqueness for NLW with rough random initial data below the energy space. The randomization that we use is naturally associated with the Wiener decomposition and with modulation spaces. The proof is based on a probabilistic perturbation theory and on probabilistic energy bounds. Secondly, we prove analogous results in the periodic setting, for the energy-critical NLW on $T^d$, $d = 3, 4, 5$.

##### The vanishing viscosity limit in porous media

We consider the flow of a viscous, incompressible, Newtonian fluid in a perforated domain in the plane. We study the simultaneous limit of vanishing pore size and inter-pore distance, and vanishing viscosity. Under suitable conditions on the pore size, distance between pores, and viscosity, we prove that solutions of the Navier-Stokes system in the perforated domain converges to solution of the Euler system in the full plane. That is, the flow is not disturbed by the porous medium and becomes inviscid in the limit. This is joint work with Christophe Lacave.

##### Enhanced dissipation and hypoellipticity in shear flows

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 case of space-periodic and the case of a bounded channel with no-flux boundary conditions. In the infinite Péclet number limit, 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.

##### On the V-states for some transport models

We shall discuss in this talk some aspects of the vortex motion for different nonlinear transport models arising in fluid dynamics such as Euler equations and the inviscid generalized surface quasi-geostrophic equations. The main concern is to establish the existence of rotating vortex patches (also called V-states) for different topological structures: simply connected and doubly connected patches. The proofs are based on the bifurcation theory combined with special functions.The existence of the V-states in a disc and their interaction with the boundary will also be analyzed. The content of this lecture is based on joint papers with de-La Hoz, Hmidi and Mateu.

##### On viscous incompressible flows around a rotating obstacle in two dimensions

In this talk we consider the Navier-Stokes equations for viscous incompressible flows in a two-dimensional exterior domain subject to the no-slip boundary condition. Due to the boundary condition the motion of the obstacle (complement of the exterior domain) naturally create a nontrivial flow. The most typical case is that the obstacle is translating with a constant velocity, known as the Oseen problem, and the stationary Navier-Stokes flows in this case were obtained by Finn and Smith in 1960's. Another typical motion of the obstacle is the rotation with a constant angular velocity, however, the existence of the corresponding stationary Navier-Stokes flows (in the reference frame) has been open in the two-dimensional case.

##### Dynamics of self-gravitating fluids

We discuss the dynamics of polytropic gaseous stars described by the compressible Euler-Poisson system. For the static equilibria and for nontrivially accelerated fluids, the so-called physical vacuum naturally arises and we are led to study the vacuum free boundary value problems. We give a brief overview of the existence and stability theory, and we present a recent joint work with Mahir Hadzic on the nonlinear stability of self-similar expanding solutions in the mass-critical regime.

##### An introduction to asymptotic coupling to prove unique ergodicity

I will give an overview of the idea of asymptotic coupling and how it can be used to prove unique ergodicity in a number of settings. In particular, I will consider some of Stochastically forced PDEs. Examples will include: the Navier stokes equations, a fractionally dissipative Euler equation, Stochastically Forced Euler-Voigt with damping, and a Damped Nonlinear Wave Equation. I will also discuss a Stochastic delay equations and the scaling limits of MCMC. These ideas date back to work with Ya Sinai and Weinan E, yet seem to have been under appreciated. Though in many cases, it is not much harder to prove exponential convergence to equilibrium, I will emphasis the simplest framework with an eye to proving only uniqueness of the invariant measure.

##### Rigorous validations of periodic orbits in the Kuramoto-Sivashinsky PDE

We will give a-posteriori results suitable for proving the existence of periodic orbits for some PDEs. Among these we could have parabolic PDEs or ill-posed equations like Boussinesq equation. We will show how to implement these a-posteriori results as Computer-Assisted Proofs in the 1D Kuramoto-Sivashinsky PDE: \[ \partial_t u = \partial_{xxxx} u+\alpha \partial_{xx} u+\partial_x (u^2), \] where $\alpha > 0$ and $u: \mathbb R\times [0, 1] \rightarrow \mathbb R$ is odd-periodic ($u(t,x)=u(t, x+1)$, $u(t,0) =0$. The validations are based on proving that certain smooth functionals have an isolated zero. For doing so, a combination of functional analysis and rigorous estimates using computers are needed. We will finish the talk showing some examples.

##### Quasineutral limit for the Vlasov Poisson system

We shall study the quasi-neutral limit of the Vlasov-Poisson system. This is a very singular limit due to the presence of the two streams instability in plasmas. In particular the convergence towards the expected limit system cannot be always true in Sobolev spaces. We will justify this limit and prove the well-posedness of the limit system for initial data in Sobolev spaces for which the profile in the velocity variable satisfies some linear stability condition. (joint work with D. Han-Kwan)

##### Finite depth gravity water waves in holomorphic coordinates

We consider irrotational gravity water waves with finite bottom in 2d. We discuss the local well-posedness of this problem in holomorphic coordinates and establish cubic lifespan bounds for small initial data. This is joint work with Mihaela Ifrim and Daniel Tataru.

##### Ill-posedness of the Euler equations in the critical Sobolev space

The question of well-posedness of the Euler equations in various critical spaces was only recently solved. Ill-posedness in the critical Sobolev spaces was obtained by Bourgain-Li, and ill-posedness in the integer C^m spaces was settled again by Bourgain-Li and independently by Elgindi-Masmoudi. In this talk, we give a simple proof of the ill-posedness in the critical Sobolev space for the 2D Euler equations. The proof is based on a hyperbolic flow scenario, utilized in the recent works of Denissov, Kiselev-Sverak, and Zlatos. This is joint work with Tarek Elgindi.

##### Fluid-composite structure interaction

Fluid-structure interaction (FSI) problems arise in many applications. The widely known examples are aeroelasticity and biofluids. In biofluidic applications, including the interaction between blood flow and cardiovascular tissue, the coupling between the fluid and structure is highly nonlinear because the density of the structure (tissue) and the density of the fluid (blood) are roughly the same. In such problems, geometric nonlinearities of the fluid-structure interface and significant exchange in the energy between the moving fluid and structure play important roles in the physical and mathematical description of the underlying biological problem. The problems are further exacerbated by the fact that the walls of major arteries are composed of several layers, each with different mechanical characteristics.

##### Solitary water waves in finite and infinite depth

We will discuss the qualitative properties of spatially localized traveling water waves. First we will review some results for waves in a 2D finite-depth fluid with vorticity and possibly density stratification but no surface tension. Next we will consider waves in a 2D or 3D infinite-depth fluid with or without surface tension but with an irrotational velocity field. In this second case we prove asymptotic formulas for the velocity potential and free surface, and relate the constants in these formulas to the kinetic energy. As a consequence, we find that nontrivial waves must have infinite angular momentum. The first part includes joint work with Robin Ming Chen and Samuel Walsh, and also Walter Strauss.