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.[-]

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 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.[-]

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.[-]