Learning seminars on free surface flows
Goal
The goal of the learning seminar held at Brown is to study the well-posedness problems of the evolution of the free surface separating air from an incompressible perfect fluid. Necessary mathematics tools will be introduced as a crash course.
Time/Venue
Every Wednesday 9:00 – 10:15 (US Eastern Time) / Location : Zoom
Contents and tentative schedule
Mainly I will review the work of Benoit and Hongjie. I will mainly get into the one-phase Muskat problem case but I will give brief explanation for two-phase case as well.
- Reformulation of the Muskat problem via the Dirichlet-to-Neumann operator
- Construction of Dirichlet-to-Neumann operator
- A crash course on paradifferential calculus
- A good unknown of Alinhac and continuity estimates for Dirichlet-to-Neumann map
- Local well-posedness of Muskat problem for subcritical Sobolev initial data I-III
- A crash course on layer potential theory
- Global well-posedness of one-phase Muskat problem for periodic Lipschitz data I-III
Courselog
9/11 - We introduced the Muskat problem and reformulated the original problem via the Dirichlet-to-Neumann operator
9/18 - We proved the continuity of the Dirichlet-to-Neumann operator from $H^{1/2}$ to $H^{-1/2}$. Moreover, we explained a motivation to study paradifferential calculus.
9/25 - We introduced paradifferential operators and some estimates on paraproduct.
10/2 - We continue the discussion on the paradifferential calculus. We proved the paracomposition.
10/9 - Application of paradifferential calculus to Dirichlet-to-Neumann operator
10/16 - A priori estimates for Muskat problems
10/23 - Finish the proof of theorem of Nguyen-Pausader and begin the discussion on Dong-Nguyen-Gancedo.
10/30 - Crash course on layer potential theory and reformulation of the Dirichlet-to-Neumann operator via layer potentials
11/06 - A precise bound for the inverse of the layer potential operators
11/13 - Global existence of the regularized Muskat problem
11/20 - Comparision principle for the Dirichlet-to-Neumann operator and the motivation of Viscosity solution
11/27 - No seminar
12/04 - Comparison principle for viscosity solution of the Muskat problem (end)
Notes
- upon request