Electroconvection is the motion of charged fluids driven by electrical forces. The fractional Laplacian in bounded domains appears naturally in modelling electroconvection. I will discuss nonlinear bounds for the fractional Laplacian, in the spirit of the nonlinear maximum principle, and applications to regularity issues for electroconvection.

Interface dynamics for incompressible flows in 2D: SQG, Muskat and Water Waves, Diego Cordoba (ICMAT, Madrid)We will study contour dynamics that are driven by basic fluid mechanics systems; Euler's equation, Darcy's law and the Quasi-geostrophic equation. These give rise to problems such as water wave, Muskat, and the evolution of sharp fronts of temperature. We will present the main ideas and arguments for well-posedness, global-existence and finite time singularities for these models.

The ergodic theory of the stochastically forced Navier Stokes and related equations, Jonathan Mattingly (Duke University)I will describe sufficient conditions on the forcing of a class of stochastic PDEs to ensure that the system is uniquely ergodic. I will take my examples from fluid and related equations. My goal is to show how different the probability theory related to the equations is depending on the structure of the forcing and the implications for the qualitative structure of the equations from a probabilistic point of view. I will roughly follow the following program.

- Lecture 1: Over view and the truly elliptic setting
- Lecture 2: The essentially elliptic setting
- Lecture 3: More on essentially elliptic setting and overview of the Hypoelliptic setting
- Lecture 4: Malliavin Calculus and the hypoelliptic setting

- 1. Shocks in the 1D compressible Euler equations
- - Glimm's theory of the Cauchy problem --> shocks
- - Formal derivation of the Euler equations from the Boltzmann equation
- - Some partial convergence results (Caflisch-Nicolaenko, Liu, Zumbrun-Métivier)
- - Are entropic solutions the physical ones? (Slemrod)

- 2. Vortices in the 2D incompressible Euler equations
- - Delort's theory of the Cauchy problem --> vortices
- - Formal derivation of the incompressible Euler equations from the Boltzmann equation
- - Convergence towards Lipschitz and dissipative solutions (Golse, SR)
- - What could be the enstrophy at the microscopic level?

- 3. Energy defects in the incompressible Navier-Stokes equations
- - Weak solutions à la Leray --> possible loss of energy
- - From renormalized solutions to the Boltzmann equation to weak solutions of the Navier-Stokes equations (Bardos-Golse-Levermore, Lions-Masmoudi, Golse-SR)
- - Acoustic oscillations, initial and boundary layers
- - Energy vs entropy?

- 4. Concentrations in the Boltzmann equation
- - Renormalized solutions --> possible concentrations
- - Formal derivation from system of particles
- - Short time convergence in the low density limit
- - What is the physical meaning of renormalization (and its microscopic counterpart)?

While solutions to the 2D Euler equation have been known to be globally regular since the 1930s, this question has not yet been resolved for the surface quasi-geostrophic (SQG) equation. The latter state of affairs is also true for the modified SQG equations, a natural family of PDE which interpolate between these two models. I will present two results about the patch dynamics version of these equations in the half-plane. The first is global-in-time regularity for the Euler patch model, even if the patches initially touch the boundary of the half-plane. The second is local regularity for those modified SQG patch models which are only slightly more singular than 2D Euler, but also existence of their solutions which blow up in finite time.