# Seminars & Events for Analysis Seminar

##### Regularity and blow up in models of fluid mechanics

I will discuss a family of modified SQG equations that varies between 2D Euler and SQG with patch-like initial data defined on half-plane. The family is modulated by a parameter that sets the degree of the kernel in the Biot-Savart law. The main result I would like to describe is the phase transition in the behavior of solutions that happens right beyond the 2D Euler case. Namely, for the 2D Euler equation the patch solution stays globally regular, while for a range of nearby models there exist regular initial data that lead to finite time blow up. The geometry of the blow up example involves hyperbolic fixed point of the flow at the boundary. If time permits, I will discuss some other recent works designed to better understand vorticity growth in a similar setting for solutions of the 3D Euler equation.

##### Dynamical relativistic liquid bodies

In this talk, I will discuss a new approach to establishing the well-posedness of the relativistic Euler equations for liquid bodies in vacuum. The approach is based on a wave formulation of the relativistic Euler equations that consists of a system of non-linear wave equations in divergence form together with a combination of acoustic and Dirichlet boundary conditions. The equations and boundary conditions of the wave formulation differs from the standard one by terms proportional to certain constraints, and one of the main technical problems to overcome is to show that these constraints propagate, which is necessary to ensure that solutions of the wave formulation determine solutions to the Euler equations with vacuum boundary conditions.

##### Nonuniqueness of weak solutions to the Navier-Stokes equation

For initial datum of finite kinetic energy Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this talk, I will discuss very recent joint work with Vlad Vicol in which we prove that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy

##### A proof of the instability of AdS spacetime for the Einstein–null dust system

The AdS instability conjecture, suggested by Dafermos and Holzegel in 2006, states that generic, arbitrarily small perturbations to the initial data of the AdS spacetime, under evolution by the vacuum Einstein equations with reflecting boundary conditions on conformal infinity, lead to the formation of black holes. Following the work of Bizon and Rostworowski in 2011, a vast amount of numerical and heuristic works have been dedicated to the study of this conjecture, focusing

mainly on the simpler setting of the spherically symmetric Einstein--scalar field system.

##### Maximal globally hyperbolic developments of subluminal and superluminal quasilinear wave equations

**Please note the different time.**

##### How to play with the three spheres theorem.

**Please note the different time.**

Let u be a solution of second order elliptic differential equation Lu = 0. Fix three concentric balls B_1 \subset B_2 \subset B_3. The three spheres theorem claims that if |u| < 1 on $B_3$ and |u| < \varepsilon on B_1, then |u| < C \varepsilon^\alpha on B_2, where \alpha \in (0,1) and C>0 depend on L and the balls, but not on u.

We are going to demonstrate how to apply this theorem.

##### High frequency back reaction for the Einstein equations

It has been observed by physicists (Isaacson, Burnett, Green-Wald) that metric perturbations of a background solution, which are small amplitude but with high frequency, yield at the limit to a non trivial contribution which corresponds to the presence of a stress-energy tensor in the equation for the background metric. This non trivial contribution is due to the nonlinearities in Einstein equations, which involve products of derivatives of the metric. It has been conjectured by Burnett that the only tensors which can be obtained this way are massless Vlasov, and it has been proved by Green and Wald that the limit tensor must be traceless and satisfy the dominant energy condition.

##### Nonlinear stability of Minkowski spacetime for self-gravitating massive fields

I will discuss the global evolution problem for self-gravitating massive matter in the context of Einstein's theory and, more generally, of the f(R)-theory of gravity. In collaboration with Yue Ma (Xian), by analyzing the Einstein equations in wave gauge coupled to Klein-Gordon equations, I have established that Minkowski spacetime is globally nonlinearly stable in presence of massive fields. This extends fundamental works by Christodoulou and Klainerman and by Lindblad and Rodnianski, who were concerned with vacuum spacetimes and massless fields.

##### Almost global existence of solutions for space periodic capillarity-gravity water waves equations

We prove that any solution of the Cauchy problem for the capillarity-gravity water waves equations, in one space dimension, with periodic, even in space, initial data of small size \ep , is almost globally defined in time on Sobolev spaces, i.e. it exists on a time interval of length of magnitude \ep^{-N} for any N, as soon as the initial data are smooth enough, and the gravity-capillarity parameters are taken outside an exceptional subset of zero measure.

##### Recent developments in dimensional free estimates in harmonic analysis

**Please note the different time.**

We will discuss some recent developments in dimensional-free bounds for the Hardy--Littlewood averaging operators defined over convex symmetric bodies in $\mathbb R^d$. Specifically we will

be interested in $r$-variational bounds. nWe also prove the dimension-free bounds on $\ell^p(\mathbb Z^d)$ with $p>3/2$ for the discrete maximal function associated with cubes in $\mathbb Z^d$. Using similar methods we also give a new simplified proof for the dimension-free bounds on $L^p(\mathbb R^d)$ with $p>3/2$ for maximal functions corresponding to symmetric convex bodies in $\mathbb R^d.$ If the time permits we will discuss problems for discrete Euclidean balls in $\mathbb Z^d$ as well.

This is joint project with J. Bourgain, E.M. Stein and B. Wr\'obel.

##### Regularity and structure of scalar conservation laws

Scalar conservation law equations develop jump discontinuities even when the initial data is smooth. Ideally, we would expect these discontinuities to be confined to a collection of codimension-one surfaces, and the solution to be relatively smoother away from these jumps. The picture is less clear for rough initial data which is merely bounded. While a linear transport equation may have arbitrarily rough solutions, genuinly nonlinear conservation laws have a subtle regularization effect. We prove that the entropy solution will become immediately continuous outside of a codimension-one rectifiable set, that all entropy dissipation is concentrated on the closure of this set, and that the L^\infty norm of the solution decays at a certain rate as t goes to infinity.

##### Stochastic homogenization: renormalization, regularity, and quantitative estimates

There has been a lot of work in recent years on the problem of understanding the behavior of solutions of PDEs with random coefficients, with most of the work focused on linear elliptic equations in divergence form. I will give an overview of recent joint works with Smart, Kuusi and Mourrat in which we introduce a ``renormalization group" method, which leads to a very precise, quantitative description of the solutions.