The purpose of these lectures is to present general methods to construct boundary layers both in linear and nonlinear contexts. We will explain how the boundary layer sizes and profiles can be predicted in linear cases, together with some decay estimates. We will illustrate this method with several explicit examples: Ekman layers, reflection of internal waves in a stratified fluid. . . We will also tackle semilinear problems, adding for instance a convection term to the previous examples. Eventually, we will introduce some tools for the study of the Prandtl equation.[-]

In this course I will be covering three main topics. The first part will be concerning the NavierStokes and Euler equations - a quick survey. The second part will discuss the question of global regularity of certain geophysical flows. The third part about coupling the atmospheric models with the microphysics dynamics of moisture in warm clouds formation.
The basic problem faced in geophysical fluid dynamics is that a mathematical description based only on fundamental physical principles, which are called the 'Primitive Equations', is often prohibitively expensive computationally, and hard to study analytically. In these introductory lectures, aimed toward graduate students and postdocs, I will survey the mathematical theory of the 2D and 3D Navier-Stokes and Euler equations, and stress the main obstacles in proving the global regularity for the 3D case, and the computational challenge in their direct numerical simulations. In addition, I will emphasize the issues facing the turbulence community in their turbulence closure models. However, taking advantage of certain geophysical balances and situations, such as geostrophic balance and the shallowness of the ocean and atmosphere, I will show how geophysicists derive more simplified models which are easier to study analytically. In particular, I will prove the global regularity for 3D planetary geophysical models and the Primitive equations of large scale oceanic and atmospheric dynamics with various kinds of anisotropic viscosity and diffusion. Moreover, I will also show that for certain class of initial data the solutions of the inviscid 2D and 3D Primitive Equations blowup (develop a singularity).[-]

The purpose of these lectures is to present general methods to construct boundary layers both in linear and nonlinear contexts. We will explain how the boundary layer sizes and profiles can be predicted in linear cases, together with some decay estimates. We will illustrate this method with several explicit examples: Ekman layers, reflection of internal waves in a stratified fluid. . . We will also tackle semilinear problems, adding for instance a convection term to the previous examples. Eventually, we will introduce some tools for the study of the Prandtl equation.[-]

The purpose of these lectures is to present general methods to construct boundary layers both in linear and nonlinear contexts. We will explain how the boundary layer sizes and profiles can be predicted in linear cases, together with some decay estimates. We will illustrate this method with several explicit examples: Ekman layers, reflection of internal waves in a stratified fluid. . . We will also tackle semilinear problems, adding for instance a convection term to the previous examples. Eventually, we will introduce some tools for the study of the Prandtl equation.[-]

Many hydrodynamic instabilities take place near a solid boundary at high Reynolds number. This reflects into the mathematical theory of the classical Prandtl model for the boundary layer: it exhibits high frequency instabilities, limiting its well-posedness to infinite regularity (Gevrey) spaces. After reviewing shortly this fact, we will turn to the Triple Deck model, a refinement of the Prandtl system that is commonly accepted to be more stable. We will show that this is actually wrong, and that the recent result of analytic well-posedness obtained by Iyer and Vicol is more or less optimal. This is based on joint work with Helge Dietert.[-]

Weak solutions of the incompressible Euler equations which are weak limits of vanishing viscosity Navier-Stokes solutions inherit, in two dimensions, conservation properties which are not available for general weak solutions. Research has focused on the behavior of energy, enstrophy and, more generally, the distribution function of vorticity, always in fluid domains with no boundary, with and without forcing. In this talk I will report on recent work in this direction.[-]

I will consider the motion of an incompressible viscous fluid on compact surfaces without boundary. Local in time well-posedness is established in the framework of $L_{p}$-$L_{q}$ maximal regularity for initial values in critical spaces. It will be shown that the set of equilibria consists exactly of the Killing vector fields. Each equilibrium is stable and any solution starting close to an equilibrium converges at an exponential rate to a (possibly different) equilibrium. In case the surface is two-dimensional, it will be shown that any solution with divergence free initial value in $L_{2}$ exists globally and converges to an equilibrium.[-]

We study the global existence of the parabolic-parabolic Keller–Segel system in $\mathbb{R}^{d}$. We prove that initial data of arbitrary size give rise to global solutions provided the diffusion parameter $\tau$ is large enough in the equation for the chemoattractant. This fact was observed before in the two-dimensional case by Biler, Guerra and Karch (2015) and Corrias, Escobedo and Matos (2014). Our analysis improves earlier results and extends them to any dimension d ≥ 3. Our size conditions on the initial data for the global existence of solutions seem to be optimal, up to a logarithmic factor in $\tau$ , when $\tau\gg 1$: we illustrate this fact by introducing two toy models, both consisting of systems of two parabolic equations, obtained after a slight modification of the nonlinearity of the usual Keller–Segel system. For these toy models, we establish in a companion paper finite time blowup for a class of large solutions.[-]