# Seminars & Events for Analysis of Fluids and Related Topics

##### Transporting microstructure and dissipative Euler flows

##### Nonlinear maximum principles and applications to active scalar equations

We discuss regularity questions for fluid models motivated by the Navier-Stokes and Euler equations. We begin by analyzing the toy-example of the fractal Burgers equation, and then focus on the surface quasi-geostrophic (SQG) equation. The main tool presented here is a new nonlinear maximum principle for equations with drift and dissipation. As an application we give the proof of global regularity for the critical SQG equation.

##### Nonlinear maximum principles and applications to active scalar equations (Part II)

We present a new nonlinear maximum principle for equations with drift and dissipation. As an application we prove global regularity for the critical SQG equation.

##### Global existence for the gravity water waves system in 2d

We start by introducing the free boundary problem for the Euler equations and discuss some aspects related to its local well-posedness, in both Eulerian and Lagrangian coordinates, together with some known results. We will then describe some general tools that have been used to construct global solutions for the irrotational problem with gravity in 3 space dimensions. In the second part we will focus on the gravity water waves system in 2 dimensions and sketch the proof of global existence of small solutions. This is based on energy estimates performed in some special Lagrangian coordinates due to S. Wu, and on decay estimates which are proven using the Eulerian formulation of the system.

##### Global existence for the gravity water waves system in 2d

We will focus on the gravity water waves system in 2 dimensions and sketch the proof of global existence of small solutions. This is based on energy estimates performed in some special Lagrangian coordinates due to S. Wu, and on decay estimates which are proven using the Eulerian formulation of the system.

##### Ergodicity of 2D Navier-Stokes equations

We will discuss the paper by Martin Hairer and Jonathan Mattingly "Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing". We will present some background material, eventually leading to an outline of the proof of the main theorem.

##### Nonlinear inviscid damping in 2D Euler

**Please note different day (Friday).** We prove the global asymptotic stability of shear flows close to planar Couette flow in the 2D incompressible Euler equations. Specifically, given an initial perturbation of the Couette flow which is small in a suitable regularity class we show that the velocity converges strongly in L2 to another shear flow which is not far from Couette. This strong convergence is usually referred to as "inviscid damping" and is roughly analogous to Landau damping in the Vlasov equations. Joint work in progress with Nader Masmoudi.

##### Ergodicity of 2D Navier-Stokes equations (Part II)

We will discuss the paper by Martin Hairer and Jonathan Mattingly "Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing". We will present some background material, eventually leading to an outline of the proof of the main theorem.