## Periodic Reporting for period 4 - BLOC (Mathematical study of Boundary Layers in Oceanic Motions)

Reporting period: 2020-03-01 to 2021-02-28

Boundary layer theory is a large component of fluid dynamics. It is ubiquitous in Oceanography, where boundary layer currents, such as the Gulf Stream, play an important role in the global circulation. Comprehending the underlying mechanisms in the formation of boundary layers is therefore crucial for applications. However, the treatment of boundary layers in ocean dynamics remains poorly understood at a theoretical level, due to the variety and complexity of the forces at stake.

The goal of this project is to develop several tools to bridge the gap between the mathematical state of the art and the physical reality of oceanic motion. We will address separately Ekman layers, which are horizontal boundary layers that take place at the top and at the bottom of the ocean, and western boundary layers, which are vertical boundary layers that are localized in the vicinity of the western coast of an oceanic basin. There are four points on which we mainly focus:

1. degeneracy issues: roughly speaking, degeneracy occurs when the expected size of the boundary layer changes drastically at some point of the boundary. In oceanographic models, this might happen in two different contexts. Western boundary layers degenerate at the North and South end of an oceanic basin (this is called the "geostrophic degeneracy"). And Ekman layers degenerate near the equator (Stewartson layers).

2. rough boundaries, meaning either non-smooth boundaries, or boundaries with small amplitude and high frequency variations;

3. the inclusion of the advection term in the construction of stationary boundary layers;

4. the linear and nonlinear stability of the boundary layers.

This project will allow us to have a better understanding of small scale phenomena in fluid mechanics, and in particular of the inviscid limit of incompressible fluid

The goal of this project is to develop several tools to bridge the gap between the mathematical state of the art and the physical reality of oceanic motion. We will address separately Ekman layers, which are horizontal boundary layers that take place at the top and at the bottom of the ocean, and western boundary layers, which are vertical boundary layers that are localized in the vicinity of the western coast of an oceanic basin. There are four points on which we mainly focus:

1. degeneracy issues: roughly speaking, degeneracy occurs when the expected size of the boundary layer changes drastically at some point of the boundary. In oceanographic models, this might happen in two different contexts. Western boundary layers degenerate at the North and South end of an oceanic basin (this is called the "geostrophic degeneracy"). And Ekman layers degenerate near the equator (Stewartson layers).

2. rough boundaries, meaning either non-smooth boundaries, or boundaries with small amplitude and high frequency variations;

3. the inclusion of the advection term in the construction of stationary boundary layers;

4. the linear and nonlinear stability of the boundary layers.

This project will allow us to have a better understanding of small scale phenomena in fluid mechanics, and in particular of the inviscid limit of incompressible fluid

Important progress has been made on the construction of nonlinear Ekman layers. In collaboration with D. Gérard-Varet and Y. Maekawa, the PI has proved the existence and uniqueness of solutions of nonlinear boundary layer equations for rotating fluids, in general environments.

The PI has also obtained a breakthrough result with N. Masmoudi on boundary layer separation. They obtained the first mathematical proof of separation for the stationary Prandtl equation, and showed the relevance of the “Goldstein singularity”. The corresponding paper has been published in Publications mathématiques de l'IHES.

The techniques developed by the PI and N. Masmoudi could likely be extended to western boundary layers in oceanography, and therefore give a mathematical description of separation of the Gulf Stream.

On the other hand, the PI and M. Paddick studied the stabilizing effect of rotation on western boundary layers and gave a quantitative condition on the profile of the coastline that prevents recirculation.

Concerning boundary layer denegeracy, the PI and L. Saint-Raymond have solved the problem of geostrophic degeneracy. J. Rax studied several degenerate boundary layer problems: the equatorial Ekman layer, boundary layers in MHD, and recirculating solutions of the stationary Burgers equation with transverse viscosity.

A team composed of the PI, H. Dietert, D. Gérard-Varet and F. Marbach has also studied alternate boundary layer models, which were claimed to have a better behavior than the Prandtl system. However, the afore mentioned team proved that the behaviour of the time-dependent versions of these models is actually worse: there are less stable profiles, and the instabilities present in the models are stronger than the ones for Prandtl.

G. Lopez-Ruiz proved the well-posedness of the boundary layer equations for western boundary currents in the vicinity of rough shores.

Other achievements include:

- Criteria for the convergence of weak solutions of the 2D incompressible Navier-Stokes equations with Navier slip boundary conditions (M. Paddick, Y. Maekawa)

- Quantitative description of critical reflection phenomena for internal waves (R. Bianchini, L. Saint-Raymond, PI)

- Stability properties of traveling fronts within a soft congestion model (C. Perrin, PI)

- 1d compressible Euler equations with a singular pressure law (R. Bianchini, C. Perrin)

- Linear stability of shears near the Couette flow for a class of 2D incompressible stably stratified fluids (R. Bianchini, M. Coti-Zelati, M. Dolce)

- Global stability of a class of semilinear waves in 2+1 dimension under a null condition and global stability of a wave and Klein-Gordon system with mixed coupling (S. Dong)

- Correction to Einstein's formula for the effective viscosity in a dilute suspension of particles (A. Mecherbet, D. Gérard-Varet)

- A nalysis of a transport-Stokes equation describing sedimentation of a droplet in viscous fluid (A. Mecherbet)

The PI has also obtained a breakthrough result with N. Masmoudi on boundary layer separation. They obtained the first mathematical proof of separation for the stationary Prandtl equation, and showed the relevance of the “Goldstein singularity”. The corresponding paper has been published in Publications mathématiques de l'IHES.

The techniques developed by the PI and N. Masmoudi could likely be extended to western boundary layers in oceanography, and therefore give a mathematical description of separation of the Gulf Stream.

On the other hand, the PI and M. Paddick studied the stabilizing effect of rotation on western boundary layers and gave a quantitative condition on the profile of the coastline that prevents recirculation.

Concerning boundary layer denegeracy, the PI and L. Saint-Raymond have solved the problem of geostrophic degeneracy. J. Rax studied several degenerate boundary layer problems: the equatorial Ekman layer, boundary layers in MHD, and recirculating solutions of the stationary Burgers equation with transverse viscosity.

A team composed of the PI, H. Dietert, D. Gérard-Varet and F. Marbach has also studied alternate boundary layer models, which were claimed to have a better behavior than the Prandtl system. However, the afore mentioned team proved that the behaviour of the time-dependent versions of these models is actually worse: there are less stable profiles, and the instabilities present in the models are stronger than the ones for Prandtl.

G. Lopez-Ruiz proved the well-posedness of the boundary layer equations for western boundary currents in the vicinity of rough shores.

Other achievements include:

- Criteria for the convergence of weak solutions of the 2D incompressible Navier-Stokes equations with Navier slip boundary conditions (M. Paddick, Y. Maekawa)

- Quantitative description of critical reflection phenomena for internal waves (R. Bianchini, L. Saint-Raymond, PI)

- Stability properties of traveling fronts within a soft congestion model (C. Perrin, PI)

- 1d compressible Euler equations with a singular pressure law (R. Bianchini, C. Perrin)

- Linear stability of shears near the Couette flow for a class of 2D incompressible stably stratified fluids (R. Bianchini, M. Coti-Zelati, M. Dolce)

- Global stability of a class of semilinear waves in 2+1 dimension under a null condition and global stability of a wave and Klein-Gordon system with mixed coupling (S. Dong)

- Correction to Einstein's formula for the effective viscosity in a dilute suspension of particles (A. Mecherbet, D. Gérard-Varet)

- A nalysis of a transport-Stokes equation describing sedimentation of a droplet in viscous fluid (A. Mecherbet)

The project has already delivered results going far beyond the state of the art in several aspects. Let us highlight here three achievements, and point out new directions and open problems raised by these results:

1. Boundary layer separation: the PI and N. Masmoudi have proved the validity of the Goldstein singularity in the Prandtl system, using modulation of variables techniques combined with maximum principle arguments and with new energy estimates. This result has already attracted a lot of attention from the mathematical community, and the PI has given a large number of talks on this subject. It is likely that the tools developed for the analysis (e.g. the new energy estimates) will be useful for the study of the Prandtl equation in other contexts (for instance, in order to prove the relevance of the Prandtl Ansatz). Furthermore, the same strategy can be used to analyze variants of the Prandtl equation, such as the IBL system or the equation describing western boundary layers in a nonlinear setting, and determine whether the Goldstein singularity is an artefact of the Prandtl system, or whether it is present in these variants as well.

2. Instabilities in boundary layer models: in order to describe the retroaction of the boundary layer flow onto the interior of the fluid, physicists and applied mathematicians use boundary layer models that are variants of the Prandtl system (e.g. IBL system, triple deck system). The PI, H. Dietert, D. Gérard-Varet and F. Marbach have studied the IBL system and the Prandtl system with prescribed displacement thickness, and proved that both these models develop strong linear instabilities. These instabilities grow at a rate that is much stronger than the one of the typical instabilities in the Prandtl system. Such results were highly unexpected, since it was believed that these models were better behaved than the Prandtl system. They somehow question the relevance of a boundary layer Ansatz for the 2d incompressible Navier-Stokes system with small viscosity.

3. Degenerate boundary layers: the PI and L. Saint-Raymond have developed new tools to study boundary layer degeneracy and to address problems in which several boundary layers coexist. These tools can be applied in several other contexts, such as the study of Stewartson layers close to the equator, for instance.

Expected results until the end of the project:

- Construction of 2d flows with recirculation;

- Existence and uniqueness of western boundary layers near rough coasts in a non-periodic environment.

- Construction of stationary solutions of alternate boundary layer models (models with prescribed displacement thickness, IBL model).

1. Boundary layer separation: the PI and N. Masmoudi have proved the validity of the Goldstein singularity in the Prandtl system, using modulation of variables techniques combined with maximum principle arguments and with new energy estimates. This result has already attracted a lot of attention from the mathematical community, and the PI has given a large number of talks on this subject. It is likely that the tools developed for the analysis (e.g. the new energy estimates) will be useful for the study of the Prandtl equation in other contexts (for instance, in order to prove the relevance of the Prandtl Ansatz). Furthermore, the same strategy can be used to analyze variants of the Prandtl equation, such as the IBL system or the equation describing western boundary layers in a nonlinear setting, and determine whether the Goldstein singularity is an artefact of the Prandtl system, or whether it is present in these variants as well.

2. Instabilities in boundary layer models: in order to describe the retroaction of the boundary layer flow onto the interior of the fluid, physicists and applied mathematicians use boundary layer models that are variants of the Prandtl system (e.g. IBL system, triple deck system). The PI, H. Dietert, D. Gérard-Varet and F. Marbach have studied the IBL system and the Prandtl system with prescribed displacement thickness, and proved that both these models develop strong linear instabilities. These instabilities grow at a rate that is much stronger than the one of the typical instabilities in the Prandtl system. Such results were highly unexpected, since it was believed that these models were better behaved than the Prandtl system. They somehow question the relevance of a boundary layer Ansatz for the 2d incompressible Navier-Stokes system with small viscosity.

3. Degenerate boundary layers: the PI and L. Saint-Raymond have developed new tools to study boundary layer degeneracy and to address problems in which several boundary layers coexist. These tools can be applied in several other contexts, such as the study of Stewartson layers close to the equator, for instance.

Expected results until the end of the project:

- Construction of 2d flows with recirculation;

- Existence and uniqueness of western boundary layers near rough coasts in a non-periodic environment.

- Construction of stationary solutions of alternate boundary layer models (models with prescribed displacement thickness, IBL model).