International Summer School on Evolution Equations, Prague, Czech Republic, 11.–15. 7. 2016

The aim of the course is the study of the *pullback equation*
\[
\varphi^{\ast}\left( g\right) =f.
\]
More precisely, given two \(k-\)forms \(f\) and \(g,\) we want to find a
diffeomorphism \(\varphi:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) which solves
the equation. In terms of components this reads as
\[
\sum_{1\leq i_{1} < \cdots < i_{k}\leq n}g_{i_{1}\cdots i_{k}}\left(
\varphi\left( x\right) \right) d\varphi^{i_{1}}\wedge\cdots\wedge
d\varphi^{i_{k}}=\sum_{1\leq i_{1} < \cdots < i_{k}\leq n}f_{i_{1}\cdots i_{k}%
}\left( x\right) dx^{i_{1}}\wedge\cdots\wedge dx^{i_{k}}\,.
\]
The pullback equation is thus a system of \(\binom{n}{k}\) first order partial
differential equations. If, for example, \(k=n\) the equation is just
\[
g\left( \varphi\left( x\right) \right) \det\nabla\left( \varphi\left(
x\right) \right) =f\left( x\right) .
\]
Our course will be divided into three parts.

I) We will start with a detailed introduction and historical background. Special emphasis will be on the symplectic case \(k=2\) and the case of volume forms \(k=n\). We will also briefly recall some preliminary results (Hodge decomposition, Poincaré lemma, the flow method of Moser...).

II) We will next turn to the case \(k=n\) of volume forms. We will also discuss the question of ellipticity and of uniqueness.III) We will then consider the symplectic case when \(k=2\) and \(n\) is even. We will discuss local as well as global results; emphasizing the ellipticity, the uniqueness and the regularity of the problem. We will also briefly speak of the so-called symplectic decomposition.

Some of the results as well as the introductory topics can be found in the
book by Csato G., Dacorogna B. and Kneuss O., *The pullback equation for
differential forms*, Birkhaüser, PNLDE Series, New York, **83** (2012).

These lectures are dedicated to the study of the stationary Prandtl equation, which arises in the low viscosity limit of the Navier-Stokes system. More precisely, the Prandtl equation describes the behavior of an incompressible fluid with low viscosity in the vicinity of an obstacle. We will first recall some previous results due to O. Oleinik on the local well-posedness of the equation. The key is that the stationary Prandtl equation can in fact be seen as a parabolic equation, with the spatial tangential variable along the obstacle playing the role of time. We will also give sufficient conditions for global well-posedness. We will then turn to recent results obtained in collaboration with Nader Masmoudi on singularity formation for the stationary Prandtl equation. Indeed, it can be proved that the solutions constructed by Oleinik may "blow-up" in finite time, in the sense that they cannot be extended after a certain value of the tangential variable. This phenomenon describes in mathematical terms what is known as "boundary layer separation". One will not need prior knowledge related to the stationary Prandtl equation to attend the lectures.

Data assimilation describes a methodology for tracking the trajectory of a dynamical system, with the aid of partial observations of the trajectory. The idea is to "blend" data from the observations into the dynamical system itself, with the intention of steering the evolution towards observations in a "probabilistically consistent" manner.

In this mini-course, we will introduce several data assimilation methods that are widespread throughout applied sciences and are specifically targeted to infinite dimensional dynamical systems. We will also discuss how one can prove important properties of the modified dynamical systems, such as whether the trajectory estimate asymptotically synchronizes with the true underlying trajectory (accuracy) and whether estimates are stable to perturbations in initialization (ergodicity).

In these lectures, we will discuss classical and recent results on the Navier-Stokes equations, the Euler equations, which are the classical models for an incompressible fluid, and systems involving these equations.

In the first lecture, we will summarize classical and more recent results on the regularity and partial regularity of solutions of the Navier-Stokes equations and known regularity results for the Euler equations. In the second lecture, we will consider the free-boundary Euler equations with special emphasis on local well-posedness theory. These will involve both cases of zero and nonzero surface tension. In the last lecture we will talk about results and challenges in the fluid-structure systems, which are the models for an elastic body immersed in an incompressible fluid.

We consider the problem of the time asymptotics for large solution for non-prepared initial data. The question is whether the evolution asymptotically decouples for large time into a sum of modulated solitons and a free radiation term (the soliton resolution conjecture) for global solutions of energy critical wave equations.

We will introduce a strategy to consider this problem. Then we solve the conjecture in the radial case and give the state of art in the nonradial case. One will not need prior knowledge related to this question to attend the lecture.

In the first part of these lectures I will present certain, by now classic, results on local and global well-posedness for the nonlinear wave (NLW) and the nonlinear Schrodinger (NLS) equations via Strichartz estimates.

In the second part I will show how one can use randomization of the initial data to prove well-posedness almost surely even when the problem lacks enough regularity for a more deterministic approach. In this context I will also introduce certain Gibbs type measures and how in certain cases their invariance can be used to extend local solutions to global ones.