## Periodic Reporting for period 3 - FLUID-INTERFACE (Analysis of moving incompressible fluid interfaces)

Reporting period: 2018-09-01 to 2020-02-29

The research of this proposal is focused on solving problems that involve the evolution of fluid interfaces. The project will investigate the dynamics of free boundaries arising between incompressible fluids of different nature. The main concern is well-posed scenarios which include the possible formation of singularities in finite time or existence of solutions for all time. These contour dynamics issues are governed by fundamental fluid mechanics equations such as the Euler, Navier-Stokes, Darcy and quasi-geostrophic systems. They model important problems such as water waves, viscous waves, Muskat, interface Hele-Shaw and SQG sharp front evolution. All these contour dynamics frameworks will be studied with emphasis on singularity formation and global existence results, not only for their importance in mathematical physics, but also for their mathematical interest. This presents huge challenges which will in particular require the use of different tools and methods from several areas of mathematics. A new technique, introduced to the field by the Principal Investigator, has already enabled the analysis of several singularity formations for the water waves and Muskat problems, as well as to obtain global existence results for Muskat. The main goal of this proposal is to develop upon this work, going far beyond the state of the art in these contour dynamics problems for incompressible uids.

Successful analysis of singularities in incompressible flows would solve a major problem of mathematics and would establish a new method for addressing blow-up formation in non-linear PDE. A fluid dynamics understanding of these singularities could lead to important insights on the structure of turbulence, one of the major open scientific problems of classical physics. This in turn could lead to important new methods for understanding and simulating turbulent flows, with potential for great impact throughout science and technology.

The overall objectives are to understand the dynamics of the interaction among different immiscible fluids in the context of finite time blow-up or the global existence of the solutions for all time. The motion takes place on the interface between fluids that evolves with the flow. These contour dynamics problems are given by fundamental fluid mechanics laws, such as Euler's and Navier-Stokes equations, Darcy’s law and Quasi-geostrophic systems. These give rise to problems such as vortex sheets, water waves, Muskat, two-phase Hele-Shaw, evolution of sharp fronts or free boundary Navier-Stokes.

Successful analysis of singularities in incompressible flows would solve a major problem of mathematics and would establish a new method for addressing blow-up formation in non-linear PDE. A fluid dynamics understanding of these singularities could lead to important insights on the structure of turbulence, one of the major open scientific problems of classical physics. This in turn could lead to important new methods for understanding and simulating turbulent flows, with potential for great impact throughout science and technology.

The overall objectives are to understand the dynamics of the interaction among different immiscible fluids in the context of finite time blow-up or the global existence of the solutions for all time. The motion takes place on the interface between fluids that evolves with the flow. These contour dynamics problems are given by fundamental fluid mechanics laws, such as Euler's and Navier-Stokes equations, Darcy’s law and Quasi-geostrophic systems. These give rise to problems such as vortex sheets, water waves, Muskat, two-phase Hele-Shaw, evolution of sharp fronts or free boundary Navier-Stokes.

"The work performed from the beginning of the project has yielded the following results:

- Viscous waves (Navier-Stokes free boundary models): Together with Ángel Castro, Diego Córdoba, Charles Fefferman and Javier Gómez-Serrano, we proved formation of splash singularities for Navier-Stokes free boundary for the case of vacuum-fluid interaction. Preprint: ""Splash singularities for the free boundary Navier-Stokes equations'', ArXiv:1504.02775. A new version will be posted soon. The paper is still under referee. This is the first finite-time blow-up result for a well-posed incompressible Navier-Stokes fluid model. Furthermore, during Eduardo García-Juárez's PhD studies, we have proved together global regularity for Boussinesq temperature patches with no diffusion. ""Global regularity for 2D Boussinesq temperature patches with no diffusion'', Annals of PDE, 3: 14, 2017. Using before approach together with new ideas we also have proved global existence for 1996 P.L. Lions' conjecture for density patches evolving by inhomogeneous Navier-Stokes equations. ""Global regularity of 2D density patches for inhomogeneous Navier-Stokes'', Arch. Ration. Mech. Anal., 229, no. 1, 339–360, 2018.

- Muskat and Interface Hele-Shaw problems: In this topic I worked with Peter Constantin, Vlad Vicol and Roman Shvydkoy. We were able to prove the first blow-up criteria in terms of a critical norm for the slope of the two-dimensional Muskat problem with density jump. It also shows the first local existence result for any supercritical norm in $L^p$ spaces, $p>1$, for the curvature and global existence results of smooth solutions with small slope. The results can be found in ""Global regularity for 2D Muskat equations with finite slope"", Ann. Inst. H. Poincaré Anal. Non Linéaire"", 34, no. 4, 1041-1074, 2017. Together with Eduardo García-Juárez, Neel Patel and Robert M. Strain we proved the first global existence results in critical spaces for the model with jump of viscosities and densities starting from medium size data. This case is very important from the physical point of view as in nature usually the interaction is among immiscible fluids with different densities and viscosities. Preprint: ""On The Muskat Problem With Viscosity Jump: Global In Time Results"", ArXiv: 1710.11604v2.

- SQG Sharp Fronts: In this subject I worked with Diego Córdoba and Antonio Córdoba. We were able to prove uniqueness for SQG Sharp Fronts, improving recent results of A. Kiselev, Y. Yao and A. Zlatos in the paper “Local regularity for the modified SQG patch equation” Commun. Pure and Applied Math., vol. 70, 1253–1315, 2017. Our result can be found in ""Uniqueness for SQG patch solutions"", Trans. Amer. Math. Soc. Ser. B, 5, 1-31, 2018."

- Viscous waves (Navier-Stokes free boundary models): Together with Ángel Castro, Diego Córdoba, Charles Fefferman and Javier Gómez-Serrano, we proved formation of splash singularities for Navier-Stokes free boundary for the case of vacuum-fluid interaction. Preprint: ""Splash singularities for the free boundary Navier-Stokes equations'', ArXiv:1504.02775. A new version will be posted soon. The paper is still under referee. This is the first finite-time blow-up result for a well-posed incompressible Navier-Stokes fluid model. Furthermore, during Eduardo García-Juárez's PhD studies, we have proved together global regularity for Boussinesq temperature patches with no diffusion. ""Global regularity for 2D Boussinesq temperature patches with no diffusion'', Annals of PDE, 3: 14, 2017. Using before approach together with new ideas we also have proved global existence for 1996 P.L. Lions' conjecture for density patches evolving by inhomogeneous Navier-Stokes equations. ""Global regularity of 2D density patches for inhomogeneous Navier-Stokes'', Arch. Ration. Mech. Anal., 229, no. 1, 339–360, 2018.

- Muskat and Interface Hele-Shaw problems: In this topic I worked with Peter Constantin, Vlad Vicol and Roman Shvydkoy. We were able to prove the first blow-up criteria in terms of a critical norm for the slope of the two-dimensional Muskat problem with density jump. It also shows the first local existence result for any supercritical norm in $L^p$ spaces, $p>1$, for the curvature and global existence results of smooth solutions with small slope. The results can be found in ""Global regularity for 2D Muskat equations with finite slope"", Ann. Inst. H. Poincaré Anal. Non Linéaire"", 34, no. 4, 1041-1074, 2017. Together with Eduardo García-Juárez, Neel Patel and Robert M. Strain we proved the first global existence results in critical spaces for the model with jump of viscosities and densities starting from medium size data. This case is very important from the physical point of view as in nature usually the interaction is among immiscible fluids with different densities and viscosities. Preprint: ""On The Muskat Problem With Viscosity Jump: Global In Time Results"", ArXiv: 1710.11604v2.

- SQG Sharp Fronts: In this subject I worked with Diego Córdoba and Antonio Córdoba. We were able to prove uniqueness for SQG Sharp Fronts, improving recent results of A. Kiselev, Y. Yao and A. Zlatos in the paper “Local regularity for the modified SQG patch equation” Commun. Pure and Applied Math., vol. 70, 1253–1315, 2017. Our result can be found in ""Uniqueness for SQG patch solutions"", Trans. Amer. Math. Soc. Ser. B, 5, 1-31, 2018."

"In the paper ""Splash singularities for the free boundary Navier-Stokes equations'', ArXiv:1504.02775 we can not use a backward in time argument to move from a splash singularity to a regular scenario. This argument was used in ``Finite time singularities for the free boundary incompressible Euler equations'', A. Castro, D. Córdoba, C. Fefferman, F. Gancedo and J. Gómez-Serrano, Annals of Math., 178, no. 3, 1061-1134, 2013. The Navier-Stokes model, due to the parabolic character of the equations, does not allows to go backward in time. This is the reason why we consider a perturbative argument, in such a way that we can find a family of solutions which is close in an appropriate norm to a curve with a splash singularity. The family evolves close enough to the splash curve during enough time to conclude that it blows-up in finite time. The geometry of the splash curve is of a great variety.

In the work ""Global regularity for 2D Boussinesq temperature patches with no diffusion'' we find a new cancellation in the heat kernel for non-homogeneous linear equations which allows to propagate regularity for 2D temperature patches driven by Boussinesq equations. We find that due to regularity of weak solutions it is possible to propagate $C^{1+\gamma}$ regularity for the patch, $0<\gamma<1$, by using maximal regularity of the heat equation. The new cancellation found, together with a bootstrap argument allow to propagate bounded curvature regularity. This regularity allows us to bootstrap again, so that we are able to propagate $C^{2+\gamma}$ regularity.

In the manuscript ""Global regularity of 2D density patches for inhomogeneous Navier-Stokes'', we were able to prove propagation of $C^{1+\gamma}$, $W^{2,\infty}$ and $C^{2+\gamma}$ without any smallness condition on the initial data and without any restriction on the density jump by using new ideas. In this work it is shown that for the $C^{1+\gamma}$ low regularity result one cannot expect to get the needed regularity for the velocity through the chain of Sobolev spaces. On the other hand, it is possible to take advantage of the fact that $\rho$ remains as a patch with Lipschitz boundary. The quasilinear character of the coupling between density and velocity makes it harder to propagate the regularity of the velocity and hence that of the patch. This extra difficulty, compared to the same problem for Boussinesq system, is overcome by noticing that the characteristic function of a Lipschitz patch belongs to a multiplier space in negative Hölder spaces. This fact allows to bootstrap the regularity of the velocity to obtain the proof. This low regularity theorem combined with new ideas allow to show that the curvature of patches with initial $W^{2,\infty}$ regularity remains bounded for all time, together with $C^{2+\gamma}$. It is done by finding extra cancellation dealing directly with singular integral operators.

The next novel idea was used in the paper ""Global regularity for 2D Muskat equations with finite slope"". We found that any modulus of continuity of the slope of the interface in the case of density jump provides global regularity for classical solutions. The sharp energy estimates found to delve in the regularity criteria allow to get the first local existence result on any supercritical space $L^p$, $p>1$, in terms of the curvature. They also provide global regularity for small slopes. The main tool is a new approach to check the evolution of the curvature which together with nonlocal lower bounds provide that as long as the slope of the interface remains uniformly bounded, the curvature remains controlled.

In the work ""On The Muskat Problem With Viscosity Jump: Global In Time Results"", ArXiv: 1710.11604v2 we provide new ideas for the Muskat problem to deal with the case with viscosity jump. We used new weight spaces which allow us to invert the implicit operator relating the amplitude of the vorticity with the free boundary. Those critical spaces also provide we"

In the work ""Global regularity for 2D Boussinesq temperature patches with no diffusion'' we find a new cancellation in the heat kernel for non-homogeneous linear equations which allows to propagate regularity for 2D temperature patches driven by Boussinesq equations. We find that due to regularity of weak solutions it is possible to propagate $C^{1+\gamma}$ regularity for the patch, $0<\gamma<1$, by using maximal regularity of the heat equation. The new cancellation found, together with a bootstrap argument allow to propagate bounded curvature regularity. This regularity allows us to bootstrap again, so that we are able to propagate $C^{2+\gamma}$ regularity.

In the manuscript ""Global regularity of 2D density patches for inhomogeneous Navier-Stokes'', we were able to prove propagation of $C^{1+\gamma}$, $W^{2,\infty}$ and $C^{2+\gamma}$ without any smallness condition on the initial data and without any restriction on the density jump by using new ideas. In this work it is shown that for the $C^{1+\gamma}$ low regularity result one cannot expect to get the needed regularity for the velocity through the chain of Sobolev spaces. On the other hand, it is possible to take advantage of the fact that $\rho$ remains as a patch with Lipschitz boundary. The quasilinear character of the coupling between density and velocity makes it harder to propagate the regularity of the velocity and hence that of the patch. This extra difficulty, compared to the same problem for Boussinesq system, is overcome by noticing that the characteristic function of a Lipschitz patch belongs to a multiplier space in negative Hölder spaces. This fact allows to bootstrap the regularity of the velocity to obtain the proof. This low regularity theorem combined with new ideas allow to show that the curvature of patches with initial $W^{2,\infty}$ regularity remains bounded for all time, together with $C^{2+\gamma}$. It is done by finding extra cancellation dealing directly with singular integral operators.

The next novel idea was used in the paper ""Global regularity for 2D Muskat equations with finite slope"". We found that any modulus of continuity of the slope of the interface in the case of density jump provides global regularity for classical solutions. The sharp energy estimates found to delve in the regularity criteria allow to get the first local existence result on any supercritical space $L^p$, $p>1$, in terms of the curvature. They also provide global regularity for small slopes. The main tool is a new approach to check the evolution of the curvature which together with nonlocal lower bounds provide that as long as the slope of the interface remains uniformly bounded, the curvature remains controlled.

In the work ""On The Muskat Problem With Viscosity Jump: Global In Time Results"", ArXiv: 1710.11604v2 we provide new ideas for the Muskat problem to deal with the case with viscosity jump. We used new weight spaces which allow us to invert the implicit operator relating the amplitude of the vorticity with the free boundary. Those critical spaces also provide we"