Analysis of moving incompressible fluid interfaces

Project Information

FLUID-INTERFACE

Grant agreement ID: 639227

DOI

10.3030/639227

Project closed

EC signature date 6 February 2015

Start date 1 September 2015

End date 31 August 2022

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 106 936,00

EU contribution

€ 1 106 936,00

Coordinated by

UNIVERSIDAD DE SEVILLA
Spain

Periodic Reporting for period 5 - FLUID-INTERFACE (Analysis of moving incompressible fluid interfaces)

Reporting period: 2021-09-01 to 2022-08-31

Summary of the context and overall objectives of the project

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

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

- Viscous waves (Navier-Stokes free boundary models):
1"Splash singularities for the free boundary Navier-Stokes equations'', Annals of PDE, 5: 12, 2019
2 "Global regularity for 2D Boussinesq temperature patches with no diffusion'', Annals of PDE, 3: 14, 2017
3 "Global regularity of 2D density patches for inhomogeneous Navier-Stokes'', Arch. Ration. Mech. Anal., 229, no. 1, 339–360, 2018
4 "Regularity results for viscous 3D Boussinesq temperature fronts", Comm. Math. Phys., 376, 1705-1736, 2020
5 "Global regularity of 2D Navier-Stokes free boundary with small viscosity contrast", Ann. Inst. H. Poincaré Anal. Non Linéaire, Accepted, 2022.

- Muskat and Interface Hele-Shaw problems:
6 "On the Muskat problem with viscosity jump: global in time results", Adv. Math., 345, 552-597, 2019
7 "Global Regularity for Gravity Unstable Muskat Bubbles", Mem. Amer. Math. Soc., Accepted, 2021
8 "Global well-posedness for the 3D Muskat problem in the critical Sobolev space", Arch. Ration. Mech. Anal., Accepted 2022
9 "Global well-posedness for the one-phase Muskat problem", Comm. Pure Appl. Math., Accepted 2021

- SQG Sharp Fronts:
10 "Uniqueness for SQG patch solutions", Trans. Amer. Math. Soc. Ser. B, 5, 1-31, 2018
11 “On the local existence and blow-up for generalized SQG patches” Ann. PDE, 7:4, 2021
12 “Well-posedness for SQG sharp fronts with unbounded curvature”, Math. Models Methods Appl. Sci., Accepted 2022.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The project has been very successful and we have completed many of the goals in most of the blocks of the proposal:

-Viscous waves: In the paper "Splash singularities for the free boundary Navier-Stokes equations", Annals of PDE, 5: 12, 2019 we shows that some 2D domains of incompressible fluid moving by Navier-Stokes develop finite-time splash singularities. The initial velocity can be chosen for the system, the compatibility conditions make challenging to find the appropriate geometry for the contour to be close to a splat. Then a stability theorem has to be proven to the system given by applying the conformal mapping. This is the first finite-time blow-up result for the incompressible Navier-Stokes system.

- Muskat and Interface Hele-Shaw problems: In the paper "Global well-posedness for the one-phase Muskat problem", Comm. Pure Appl. Math., Accepted, 2021 the 2D one-fluid Muskat problem is considered. It is proved that if the initial free boundary is the graph of a periodic Lipschitz function, then there exists a global-in-time Lipschitz solution in the strong $L^\infty$-time $L^2$-space sense. The construction provides viscosity solutions in the sense of Crandall-Lions, property used to show uniqueness in this class. Therefore, considering graph as initial data, global existence is obtained instead of blow-up. This is the first construction of unique global strong solutions for the Muskat problem with initial data of arbitrary size in a critical space. Considering higher regularity it is easy to get maximum principles but difficult to propagate regularity. On the other hand, with critical regularity it is difficult to get the maximum principles, to satisfies the contour equation and to get uniqueness. Even though, this paper provides tools to avoid these last three difficulties with new ideas, cancellations and using the nonlocal structure of the system. In particular it requires new critical quantitative estimates for layer potentials and pointwise elliptic regularity in Lipschitz domains.

- SQG Sharp Fronts: In this section we got the following result: On the local existence and blow-up for generalized SQG patches, Annals of PDE, 7: 4, 2021. Recent results have shown finite-time blow-up for the $\alpha$-model, $0<\alpha<1/12$, in the half space with slip boundary condition. Two fronts of opposite temperature on the fixed boundary approach to each other until colliding in finite time. In this work, we have extended the singularity scenarios to the cases $0<\alpha<1/3$. Meanwhile the cancellations known give local existence for $H^3$ regularity, using a new point of view it is possible to get the result with $H^2$ regularity. It allows to extend the range of $\alpha$ in previous results as, the higher regularity needed, the singular part of the equation due to the fixed boundary has higher singular effect. This result increases the very short list of finite time singularity scenarios for incompressible fluids.

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