The purpose of these lectures will be to review mathematical results on fluid boundary layers, presenting both classical methods and recent developments. We will mostly focus on the Prandtl boundary layer model, which describes the behavior of an incompressible fluid with small viscosity in the vicinity of a solid wall. In the stationary case, it is known since the work of Oleinik in the 60's that the 2d equation is well-posed when there is no reversed flow (or recirculation bubble) close to the boundary. However, in the vicinity of the separation point, and in the recirculating zone, singularities generically appear, which heuristically invalidate the model. We will also spend some time reviewing open problems: which model could be used as a replacement for the Prandtl system close to the separation point? How could the system be modified in the recirculation zone to avoid singularities? In the time dependent case, the system is well-posed in Sobolev spaces when the tangential velocity is monotone in the normal variable. This assumption is essentially optimal since instabilities develop in the vicinity of non monotone shear flows, which prevent the system from being well posed in Sobolev spaces. We will also present related results on variants of the Prandtl system: interactive boundary layer system, triple deck system.[-]

These lectures review the physical and mathematical foundations of the phenomenon of spontaneous stochasticity. The essence is that stochastic classical equations and/or with randomness in their initial data, when considered along a sequence where the randomness vanishes but also the dynamics becomes singular, can have limits described by a probability measure over the non-unique weak solutions of the limiting deterministic dynamics with deterministic initial data. Furthermore, the limiting probability measure is often universal, independent of the precise sequence considered, so that the stochastic limit is then the well-posed solution of the Cauchy problem for the limiting deterministic dynamics. In the firstlecture, we discuss Lagrangian spontaneous stochasticity, which has its origin in the 1926 paper of Lewis Fry Richardson on turbulent 2-particle dispersion. As first realized by Krzysztof Gawędzki and collaborators in 1997, Lagrangian spontaneous stochasticity is necessary for anomalous dissipation of a scalar advected by a turbulent fluid flow. In the second lecture, we discuss Eulerian spontaneous stochasticity, which was anticipated in the 1969 work of Edward Lorenz on predictability of turbulent flows. After the convex integration studies of De Lellis, Székelyhidi, and others showed that Euler equations with suitable initial data may admit infinitely many, non-unique admissible weak solutions, it became clear that Lorenz' pioneering work could be understood in the framework of spontaneous stochasticity. Finally, in the third lecture we discuss outstanding problems and more recent work on spontaneous stochasticity, both Lagrangian and Eulerian. We focus in particular on statistical-mechanical analogies, on the chaotic dynamical properties necessary to achieve universality,on the use of renormalization group methods to calculate spontaneous statistics in dynamics with scale symmetries, and finally on the challenge of observing spontaneous stochasticity in laboratory experiments.[-]

The Ocean hosts a combination of slowly evolving balanced mean flows and rapidly evolving inertia-gravity waves (IGWs). While reduced models describing the slow balanced flow are standard textbook material [1], the description of the wave field is an active area of current research. We will first consider near-inertial waves (NIWs) induced by wind blowing over the Ocean, with the goal of determining their spatial reorganization by a background mean flow [2]. After deriving the standard asymptotic model governing NIW mean-flow interaction [3], we will highlight an analogy with the (quantum) dynamics of charged particles in a background electromagnetic field. The analogy offers a shortcut for predicting the spatial organization of the wave field using elementary methods from quantum physics and statistical mechanics. Time-permitting, we will then move on to fully 3D IGWs interacting with a slow background, focusing on the resulting cascade of wave action to small scales [4]. This is an elementary example of a turbulent cascade in a linear wave system, paving the way for subsequent lectures by G. Krstulovic on turbulent cascades in nonlinear wave systems.[-]

Internal and inertial waves propagate in the bulk of rotating stratified fluids and play an important role in oceans and atmospheres. They interact nonlinearly, triggering instabilities and energy transfers along scales in a cascade process. In the first lecture, I will briefly introduce the general wave turbulence theory, including somehints on the derivation of the wave kinetic equation (WKE) in the general case. Then, I will discuss how we use the WKE to describe wave turbulent cascades and when this description is valid mathematically and then expected to be realisable in experiments and nature. In the second lecture, I will apply the concept of wave turbulence to the case of rotating andstratified fluids, explain the main theoretical difficulties and give an overview of some of the current experiments and simulations.[-]

The purpose of these lectures will be to review mathematical results on fluid boundary layers, presenting both classical methods and recent developments. We will mostly focus on the Prandtl boundary layer model, which describes the behavior of an incompressible fluid with small viscosity in the vicinity of a solid wall. In the stationary case, it is known since the work of Oleinik in the 60's that the 2d equation is well-posed when there is no reversed flow (or recirculation bubble) close to the boundary. However, in the vicinity of the separation point, and in the recirculating zone, singularities generically appear, which heuristically invalidate the model. We will also spend some time reviewing open problems: which model could be used as a replacement for the Prandtl system close to the separation point? How could the system be modified in the recirculation zone to avoid singularities? In the time dependent case, the system is well-posed in Sobolev spaces when the tangential velocity is monotone in the normal variable. This assumption is essentially optimal since instabilities develop in the vicinity of non monotone shear flows, which prevent the system from being well posed in Sobolev spaces. We will also present related results on variants of the Prandtl system: interactive boundary layer system, triple deck system.[-]

The purpose of these lectures will be to review mathematical results on fluid boundary layers, presenting both classical methods and recent developments. We will mostly focus on the Prandtl boundary layer model, which describes the behavior of an incompressible fluid with small viscosity in the vicinity of a solid wall. In the stationary case, it is known since the work of Oleinik in the 60's that the 2d equation is well-posed when there is no reversed flow (or recirculation bubble) close to the boundary. However, in the vicinity of the separation point, and in the recirculating zone, singularities generically appear, which heuristically invalidate the model. We will also spend some time reviewing open problems: which model could be used as a replacement for the Prandtl system close to the separation point? How could the system be modified in the recirculation zone to avoid singularities? In the time dependent case, the system is well-posed in Sobolev spaces when the tangential velocity is monotone in the normal variable. This assumption is essentially optimal since instabilities develop in the vicinity of non monotone shear flows, which prevent the system from being well posed in Sobolev spaces. We will also present related results on variants of the Prandtl system: interactive boundary layer system, triple deck system.[-]

These lectures review the physical and mathematical foundations of the phenomenon of spontaneous stochasticity. The essence is that stochastic classical equations and/or with randomness in their initial data, when considered along a sequence where the randomness vanishes but also the dynamics becomes singular, can have limits described by a probability measure over the non-unique weak solutions of the limiting deterministic dynamics with deterministic initial data. Furthermore, the limiting probability measure is often universal, independent of the precise sequence considered, so that the stochastic limit is then the well-posed solution of the Cauchy problem for the limiting deterministic dynamics. In the firstlecture, we discuss Lagrangian spontaneous stochasticity, which has its origin in the 1926 paper of Lewis Fry Richardson on turbulent 2-particle dispersion. As first realized by Krzysztof Gawędzki and collaborators in 1997, Lagrangian spontaneous stochasticity is necessary for anomalous dissipation of a scalar advected by a turbulent fluid flow. In the second lecture, we discuss Eulerian spontaneous stochasticity, which was anticipated in the 1969 work of Edward Lorenz on predictability of turbulent flows. After the convex integration studies of De Lellis, Székelyhidi, and others showed that Euler equations with suitable initial data may admit infinitely many, non-unique admissible weak solutions, it became clear that Lorenz' pioneering work could be understood in the framework of spontaneous stochasticity. Finally, in the third lecture we discuss outstanding problems and more recent work on spontaneous stochasticity, both Lagrangian and Eulerian. We focus in particular on statistical-mechanical analogies, on the chaotic dynamical properties necessary to achieve universality,on the use of renormalization group methods to calculate spontaneous statistics in dynamics with scale symmetries, and finally on the challenge of observing spontaneous stochasticity in laboratory experiments.[-]

These lectures review the physical and mathematical foundations of the phenomenon of spontaneous stochasticity. The essence is that stochastic classical equations and/or with randomness in their initial data, when considered along a sequence where the randomness vanishes but also the dynamics becomes singular, can have limits described by a probabilitymeasure over the non-unique weak solutions of the limiting deterministic dynamics with deterministic initial data. Furthermore, the limiting probability measure is often universal, independent of the precise sequence considered, so that the stochastic limit is then the well-posed solution of the Cauchy problem for the limiting deterministic dynamics. In the firstlecture, we discuss Lagrangian spontaneous stochasticity, which has its origin in the 1926 paper of Lewis Fry Richardson on turbulent 2-particle dispersion. As first realized by Krzysztof Gawędzki and collaborators in 1997, Lagrangian spontaneous stochasticity is necessary for anomalous dissipation of a scalar advected by a turbulent fluid flow. In the second lecture, we discuss Eulerian spontaneous stochasticity, which was anticipated in the 1969 work of Edward Lorenz on predictability of turbulent flows. After the convex integration studies of De Lellis, Székelyhidi, and others showed that Euler equations with suitable initial data may admit infinitely many, non-unique admissible weak solutions, it became clear thatLorenz' pioneering work could be understood in the framework of spontaneous stochasticity. Finally, in the third lecture we discuss outstanding problems and more recent work on spontaneous stochasticity, both Lagrangian and Eulerian. We focus in particular on statistical-mechanical analogies, on the chaotic dynamical properties necessary to achieve universality,on the use of renormalization group methods to calculate spontaneous statistics in dynamics with scale symmetries, and finally on the challenge of observing spontaneous stochasticity in laboratory experiments.[-]

The Ocean hosts a combination of slowly evolving balanced mean flows and rapidly evolving inertia-gravity waves (IGWs). While reduced models describing the slow balanced flow are standard textbook material [1], the description of the wave field is an active area of current research. We will first consider near-inertial waves (NIWs) induced by wind blowing over the Ocean, with the goal of determining their spatial reorganization by a background mean flow [2]. After deriving the standard asymptotic model governing NIW mean-flow interaction [3], we will highlight an analogy with the (quantum) dynamics of charged particles in a background electromagnetic field. The analogy offers a shortcut for predicting the spatial organization of the wave field using elementary methods from quantum physics and statistical mechanics. Time-permitting, we will then move on to fully 3D IGWs interacting with a slow background, focusing on the resulting cascade of wave action to small scales [4]. This is an elementary example of a turbulent cascade in a linear wave system, paving the way for subsequent lectures by G. Krstulovic on turbulent cascades in nonlinear wave systems.[-]

Internal and inertial waves propagate in the bulk of rotating stratified fluids and play an important role in oceans and atmospheres. They interact nonlinearly, triggering instabilities and energy transfers along scales in a cascade process. In the first lecture, I will briefly introduce the general wave turbulence theory, including somehints on the derivation of the wave kinetic equation (WKE) in the general case. Then, I will discuss how we use the WKE to describe wave turbulent cascades and when this description is valid mathematically and then expected to be realisable in experiments and nature. In the second lecture, I will apply the concept of wave turbulence to the case of rotating andstratified fluids, explain the main theoretical difficulties and give an overview of some of the current experiments and simulations.[-]