In this talk, we discuss functional inequalities and gradient bounds for the heat kernel on the Vicsek set. The Vicsek set has both fractal and tree structure, whereas neither analogue of curvature nor obvious differential structure exists. We introduce Sobolev spaces in that setting and prove several characterizations based on a metric, a discretization or a weak gradient approach. We also obtain $L^{p}$ Poincaré inequalities and pointwise gradient bounds for the heat kernel. These properties have important applications in harmonic analysis like Sobolev inequalities and the Riesz transform. Moreover, several of our techniques and results apply to more general fractals and trees.[-]

About fifteen years ago, Patrick Gérard and I introduced the cubic Szegö equation$$\begin{aligned}i \partial_{t} u & =\Pi\left(|u|^{2} u\right), \quad u=u(x, t), \quad x \in \mathbb{T}, t \in \mathbb{R} \\u(x, 0) & =u_{0}(x) .\end{aligned}$$Here $\Pi$ denotes the Szegö projector which maps $L^{2}(\mathbb{T})$-functions into the Hardy space of $L^{2}(\mathbb{T})$-traces of holomorphic functions in the unit disc. It turned out that the dynamics of this equation were unexpected. This motivated us to try to understand whether the cubic Szegö equation is an isolated phenomenon or not. This talk is part of this project.
We consider a family of perturbations of the cubic Szegö equation and look for their traveling waves. Let us recall that traveling waves are particular solutions of the form$$u(x, t)=\mathrm{e}^{-i \omega t} u_{0}\left(\mathrm{e}^{-i c t} x\right), \quad \omega, c \in \mathbb{R}$$We will explain how the spectral analysis of some operators allows to characterize them.
From joint works with Patrick Gérard.[-]

We investigate the Dirichlet problem for a non-divergence form elliptic operator $L=a^{i j}(x) D_{i j}+b^{i}(x) D_{i}-c(x)$ in a bounded domain of $\mathbb{R}^{d}$. Under certain conditions on the coefficients of $L$, we first establish the existence of a unique Green's function in a ball and derive two-sided pointwise estimates for it. Utilizing these results, we demonstrate the equivalence of regular points for $L$ and those for the Laplace operator, characterized via the Wiener test. This equivalence facilitates the unique solvability of the Dirichlet problem with continuous boundary data in regular domains. Furthermore, we construct the Green's function for $L$ in regular domains and establish pointwise bounds for it.[-]

We are interested in parabolic differential equations $(\partial_t-a\partial_x^2)u=f$ with a very irregular forcing $f$ and only mildly regular coefficients $a$. This is motivated by stochastic differential equations, where $f$ is random, and quasilinear equations, where $a$ is a (nonlinear) function of $u$.
Below a certain threshold for the regularity of $f$ and $a$ (on the Hölder scale), giving even a sense to this equation requires a renormalization. In the framework of the above setting, we present recent ideas from the area of stochastic differential equations (Lyons' rough path, Gubinelli's controlled rough paths, Hairer's regularity structures) that allow to build a solution theory. We make a connection with Safonov's approach to Schauder theory.
This is based on joint work with H. Weber, J. Sauer, and S. Smith.[-]

This talk introduces, in a simplified setting, a novel commutator method to obtain averaging lemma estimates. Averaging lemmas are a type regularizing effect on averages in velocity of solutions to kinetic equations. We introduce a new bilinear approach that naturally leads to velocity averages in $L^{2}\left ( \left [ 0,T \right ],H_{x}^{s} \right )$. The new method outperforms classical averaging lemma results when the right-hand side of the kinetic equation has enough integrability. It also allows a perturbative approach to averaging lemmas which provides, for the first time, explicit regularity results for non-homogeneous velocity fluxes.[-]

This talk introduces, in a simplified setting, a novel commutator method to obtain averaging lemma estimates. Averaging lemmas are a type regularizing effect on averages in velocity of solutions to kinetic equations. We introduce a new bilinear approach that naturally leads to velocity averages in $L^{2}\left ( \left [ 0,T \right ],H_{x}^{s} \right )$. The new method outperforms classical averaging lemma results when the right-hand side of the kinetic equation has enough integrability. It also allows a perturbative approach to averaging lemmas which provides, for the first time, explicit regularity results for non-homogeneous velocity fluxes.[-]

An early motivation of smooth ergodic theory was to provide a mathematical account for the unpredictable, chaotic behavior of real-world fluids. While many interesting questions remain, in the last 25 years significant progress has been achieved in understanding models of fluid mechanics, e.g., the Navier-Stokes equations, in the presence of stochastic driving. Noise is natural for modeling purposes, and certain kinds of noise have a regularizing effect on asymptotic statistics. These kinds of noise provide an effective technical tool for rendering tractable otherwise inaccessible results on chaotic regimes, e.g., positivity of Lyapunov exponents and the presence of a strange attractor supporting a physical (SRB) measure. In this talk I will describe some of my work in this vein, including a recent result with Jacob Bedrossian and Sam Punshon-Smith providing positive Lyapunov exponents for f inite-dimensional (a.k.a. Galerkin) truncations of the Navier-Stokes equations.[-]

This talk introduces, in a simplified setting, a novel commutator method to obtain averaging lemma estimates. Averaging lemmas are a type regularizing effect on averages in velocity of solutions to kinetic equations. We introduce a new bilinear approach that naturally leads to velocity averages in $L^{2}\left ( \left [ 0,T \right ],H_{x}^{s} \right )$. The new method outperforms classical averaging lemma results when the right-hand side of the kinetic equation has enough integrability. It also allows a perturbative approach to averaging lemmas which provides, for the first time, explicit regularity results for non-homogeneous velocity fluxes.[-]