* Solitary waves.

In this series of lectures we will discuss the problem of the growth in time of the Lipschitz norm of the vorticity for 2D incompressible ideal flow. We start by showing that the growth can be at most double exponential in time. We present next several examples of solutions that exhibit: unbounded growth (Yudovich 1974), linear growth (Nadirashvili 1991), optimal double exponential growth (Kiselev and Sverak 2013) and exponential growth in the absence of boundaries (Zlatos 2013).

We will discuss what is known, and not known, about existence and uniqueness of solutions to the Euler equations for various boundary conditions. We will focus mostly on strong solutions in two or three dimensions. Time allowing, we will discuss boundary conditions in relation to the Navier-Stokes equations as well as the vanishing viscosity limit.

In this series of lectures we discuss the analysis of free and moving boundary value problems for fluids of Navier-Stokes type. Moreover, we analyze various types of complex fluid models such as liquid crystals or multicomponent fluids of Maxwell-Stefan type.

In this series of lectures we discuss the analysis of free and moving boundary value problems for fluids of Navier-Stokes type. Moreover, we analyze various types of complex fluid models such as liquid crystals or multicomponent fluids of Maxwell-Stefan type.

Two real-valued functions are rearrangements if they are equimeasurable, that is, if inverse images of the same (Borel) set have equal measure. Classical rearrangement inequalities compare a function with a more symmetric rearrangement of itself, for example a symmetric decreasing rearrangement. Certain optimisation problems posed on the class of all rearrangements of a function give rise to solutions that represent the vorticity in a steady planar ideal fluid flow. In some instances the optimisation problem illuminates the question of stability of the flow.

In these 3 lectures  I will present a brief mathematical history of the  non-Newtonian alpha-fluids. After an introductory part on this class of fluids I will focus on some mathematical results for the following models: the second grade fluid, the alpha-Euler and alpha Navier-Stokes models, the third grade fluid (if there is time). There will be old results, new results and open problems, in various situations.

These lectures aim to serve as an introduction to the study of unique ergodicity for the basic nonlinear equations of fluids dynamics subject to a spatially degenerate, white in time gaussian forcing. In order to provide an accessible introduction to some recent developments in the subject we will focus our attention on a finite dimensional model, indicating how the methods generalize to an infinite dimensional setting.  Some relationships to turbulence will be discussed.

Shortwave instabilities are instabilities with respect to highly localized and rapidly oscillating perturbations. They were first discovered in the early 80s in the context of homogeneous incompressible fluids as a mechanism of fast breaking of coherent structures into a fully developed turbulent flow. When mathematical theory came along in the 90s it was found that these instabilities are associated with the continuous spectrum of the linearized equations and can be described by a finite dimensional dynamical system. A general approach further developed in 2000s revealed a broad range of models shearing a certain common structure where these kind of instabilities occur. Those include general non-homogeneous fluids, porous media and Darcy's law, SQG, visco-elastic fluids, alpha-models, etc. In the first part of these lectures we will overview the history and discuss the geometric optics approach to shortwave instabilities. Emphasis will be made on developing a range of example to which this approach applies and finding easy instability criteria for various flows. In the second part we introduce the mathematical language in terms of which we give a full description of the continuous spectra of the corresponding equations and draw further details about shortwave instabilities. Some familiarity of harmonic analysis and PDEs will be assumed.

Since their inception in the mid 18th century, the Euler equations remain subject of both intense study and debate. In three dimensions, the question of whether the Cauchy problem is globally well-posed for smooth initial data remains famously unresolved. However, when one relaxes one's notion of solution and considers weak solutions, then the solutions are known to exhibit bad and in some cases paradoxical behavour (cf. Scheffer 1993, Shnirelman 1997, Camillo De Lellis, László Székelyhidi Jr. 2009). Despite this, weak solutions remain the subject of study due to their perceived connection with the theory of turbulence.

A fundamental feature of turbulent flow is that of dissipation of kinetic energy. In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the Euler equations belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve kinetic energy; conversely, he conjectured the existence of solutions belonging to any Hölder space with exponent less than 1/3 which dissipate kinetic energy.  

The first part of this conjecture has since been confirmed (cf. Eyink 1994, Constantin, E and Titi 1994).  During this lecture series we will discuss recent work by Camillo De Lellis, László Székelyhidi Jr., Phil Isett and myself related to resolving the second component of Onsager's conjecture. In particular, we will outline the construction of weak non-conservative solutions to the Euler equations whose Hölder 1/3-ϵ norm is Lebesgue integrable in time.