## Final Report Summary - TRANSIC (Global existence vs Blow-up in some nonlinear PDEs arising in fluid mechanics.)

PUBLISHABLE SUMMARY

A summary description of the project objectives;

The proposal TRANSIC was divided into 3 problems. Part 1 and Part 2 deal with the study of transport equations with nonlocal velocity, whereas Part 3 deals with the analytical study of the Muskat problem.

The main objective of this project was to construct solutions to some Partial Differential Equations that arise in fluid mechanic problems. One of the central equation that has played an important role during these two years was the so-called Muskat Problem. Specifically, this problem deals with the evolution of the interface between two immiscible fluids (for instance oil and water in a sand). Since the two fluids do not mix, one may consider their commun interface leading to the Muskat equation. Mathematically speaking, it is important to study the geometry of this interface, to do so, one has to use some specific functional space that allows for a nice geometric interpretation (i.e. one could even draw the interface using its mathematical properties). The easiest quantity one may consider is the slope of the curve or equivalently of the evolving interface. It is well-known that the fluid stay for all time very flat if intinially the interface is also flat. We say that the flateness is preserved by the evolution (or equivalently as time evolves). At the contrary, it was just conjectured (thanks to some numerical evidences performed by numerical analysists) that the slope may be arbitray large initially with no "turning phenomena" (in the sense that the fluid interface never goes back as the time evolves), we say in that case that the solution exists globally in time in the class of arbitrary big slopes.

During the fellowship, D. Córdoba (the scientist in charge) and the fellow have solved this major open problem. They proved that high slope solutions are, as the flat solutions, preserved along the evolution. The proof releases on techniques coming from Functional Analysis. This has allowed the fellow and the supervisor to solve the problem mentionned in Part 3 of the proposal.

The two other parts deal with the study of 1D nonlocal transport equations which modelise other important fluid mechanics problems such as the 3D Euler equation or the 2D Surface-Quasi-Geostrophic (SQG) equation. Both equations are physically relevant and major open questions remain unsolved. The idea of this part is to show in what extent those 1D models can help to go beyond the known theory with respect to these important fluid mechanic equations. The main objectives consisted in proving new global existence results by using the so-called modulus of continuity method. This method turns out to be powerful when dealing with regularity issues. In particular, the fellow used this method to improve the well-known theory for the dissipative and dispersive SQG equation. This was done jointly with L. Xue. Also, one of the objectif of this proposal was to construct finite and infinite energy solutions to some one dimensional transport equations with nonlocal velocity. This have been done in collaboration with R. Granero from the University of Davis in California (USA) and Hantaek Bae from the Ulsan University in Seoul (Korea).

Overall, the project was well managed and the impact (in particular with the breakthrough results for the Muskat problem) will be really important for the expert in the field. The result on the Muskat problem will undoubtely published in a top class journal since many research teams have tried to solve this conjectured and no improvements have been obtained since then. We expect also the two other works to be published in good mathematical journals.

A summary description of the project objectives;

The proposal TRANSIC was divided into 3 problems. Part 1 and Part 2 deal with the study of transport equations with nonlocal velocity, whereas Part 3 deals with the analytical study of the Muskat problem.

The main objective of this project was to construct solutions to some Partial Differential Equations that arise in fluid mechanic problems. One of the central equation that has played an important role during these two years was the so-called Muskat Problem. Specifically, this problem deals with the evolution of the interface between two immiscible fluids (for instance oil and water in a sand). Since the two fluids do not mix, one may consider their commun interface leading to the Muskat equation. Mathematically speaking, it is important to study the geometry of this interface, to do so, one has to use some specific functional space that allows for a nice geometric interpretation (i.e. one could even draw the interface using its mathematical properties). The easiest quantity one may consider is the slope of the curve or equivalently of the evolving interface. It is well-known that the fluid stay for all time very flat if intinially the interface is also flat. We say that the flateness is preserved by the evolution (or equivalently as time evolves). At the contrary, it was just conjectured (thanks to some numerical evidences performed by numerical analysists) that the slope may be arbitray large initially with no "turning phenomena" (in the sense that the fluid interface never goes back as the time evolves), we say in that case that the solution exists globally in time in the class of arbitrary big slopes.

During the fellowship, D. Córdoba (the scientist in charge) and the fellow have solved this major open problem. They proved that high slope solutions are, as the flat solutions, preserved along the evolution. The proof releases on techniques coming from Functional Analysis. This has allowed the fellow and the supervisor to solve the problem mentionned in Part 3 of the proposal.

The two other parts deal with the study of 1D nonlocal transport equations which modelise other important fluid mechanics problems such as the 3D Euler equation or the 2D Surface-Quasi-Geostrophic (SQG) equation. Both equations are physically relevant and major open questions remain unsolved. The idea of this part is to show in what extent those 1D models can help to go beyond the known theory with respect to these important fluid mechanic equations. The main objectives consisted in proving new global existence results by using the so-called modulus of continuity method. This method turns out to be powerful when dealing with regularity issues. In particular, the fellow used this method to improve the well-known theory for the dissipative and dispersive SQG equation. This was done jointly with L. Xue. Also, one of the objectif of this proposal was to construct finite and infinite energy solutions to some one dimensional transport equations with nonlocal velocity. This have been done in collaboration with R. Granero from the University of Davis in California (USA) and Hantaek Bae from the Ulsan University in Seoul (Korea).

Overall, the project was well managed and the impact (in particular with the breakthrough results for the Muskat problem) will be really important for the expert in the field. The result on the Muskat problem will undoubtely published in a top class journal since many research teams have tried to solve this conjectured and no improvements have been obtained since then. We expect also the two other works to be published in good mathematical journals.