More Like this

1. Sharp Hadamard Local Well-Posedness, Enhanced Uniqueness and Pointwise Continuation Criterion for the Incompressible Free Boundary Euler Equations

   https://doi.org/10.1007/s40818-025-00204-4  

   Ifrim, Mihaela; Pineau, Ben; Tataru, Daniel; Taylor, Mitchell A (June 2025, Annals of PDE)

   Abstract We provide a complete local well-posedness theory inH^sbased Sobolev spaces for the free boundary incompressible Euler equations with zero surface tension on a connected fluid domain. Our well-posedness theory includes: (i) Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data, all in low regularity Sobolev spaces; (ii) Enhanced uniqueness: Our uniqueness result holds at the level of the Lipschitz norm of the velocity and the$$C^{1,\frac{1}{2}}$$regularity of the free surface; (iii) Stability bounds: We construct a nonlinear functional which measures, in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) Energy estimates: We prove refined, essentially scale invariant energy estimates for solutions, relying on a newly constructed family of elliptic estimates; (v) Continuation criterion: We give the first proof of a sharp continuation criterion in the physically relevant pointwise norms, at the level of scaling. In essence, we show that solutions can be continued as long as the velocity is in$$L_T^1W^{1,\infty}$$and the free surface is in$$L_T^1C^{1,\frac{1}{2}}$$, which is at the same level as the Beale-Kato-Majda criterion for the boundaryless case; (vi) A novel proof of the construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in more general fluid domains. 

   more » « less
2. The relativistic Euler equations with a physical vacuum boundary: Hadamard local well-posedness, rough solutions, and continuation criterion

   https://doi.org/10.1007/s00205-022-01783-3  

   Disconzi, Marcelo M.; Ifrim, Mihaela; Tataru, Daniel (May 2022, Archive for Rational Mechanics and Analysis)

   Abstract In this paper we provide a complete local well-posedness theory for the free boundary relativistic Euler equations with a physical vacuum boundary on a Minkowski background. Specifically, we establish the following results: (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and continuous dependence on the data; (ii) low regularity solutions: our uniqueness result holds at the level of Lipschitz velocity and density, while our rough solutions, obtained as unique limits of smooth solutions, have regularity only a half derivative above scaling; (iii) stability: our uniqueness in fact follows from a more general result, namely, we show that a certain nonlinear functional that tracks the distance between two solutions (in part by measuring the distance between their respective boundaries) is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions; (v) a sharp continuation criterion, at the level of scaling, showing that solutions can be continued as long as the velocity is in$$L^1_t Lip$$ $L_t^{1}Lip$and a suitable weighted version of the density is at the same regularity level. Our entire approach is in Eulerian coordinates and relies on the functional framework developed in the companion work of the second and third authors on corresponding non relativistic problem. All our results are valid for a general equation of state$$p(\varrho )= \varrho ^\gamma $$ $p\left(\varrho \right)={\varrho }^{\gamma }$,$$\gamma > 1$$ $\gamma >1$. 

   more » « less
3. Regularity and nondegeneracy for nonlocal Bernoulli problems with variable kernels

   https://doi.org/10.1090/tran/9343  

   Snelson, Stanley; Teixeira, Eduardo (June 2025, Transactions of the American Mathematical Society)

   We consider a generalization of the Bernoulli free boundary problem where the underlying differential operator is a nonlocal, non-translation-invariant elliptic operator of order $2s\in <\#comment/>(0,2)$. Because of the lack of translation invariance, the Caffarelli-Silvestre extension is unavailable, and we must work with the nonlocal problem directly instead of transforming to a thin free boundary problem. We prove global Hölder continuity of minimizers for both the one- and two-phase problems. Next, for the one-phase problem, we show Hölder continuity at the free boundary with the optimal exponent $s$. We also prove matching nondegeneracy estimates. A key novelty of our work is that all our findings hold without requiring any regularity assumptions on the kernel of the nonlocal operator. This characteristic makes them crucial in the development of a universal regularity theory for nonlocal free boundary problems. 

   more » « less
4. Global well‐posedness for the one‐phase Muskat problem

   https://doi.org/10.1002/cpa.22124  

   Dong, Hongjie; Gancedo, Francisco; Nguyen, Huy Q. (December 2023, Communications on Pure and Applied Mathematics)

   Abstract The free boundary problem for a two‐dimensional fluid permeating a porous medium is studied. This is known as the one‐phase Muskat problem and is mathematically equivalent to the vertical Hele‐Shaw problem driven by gravity force. We prove 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 sense and it is the unique viscosity solution. The proof requires quantitative estimates for layer potentials and pointwise elliptic regularity in Lipschitz domains. This is the first construction of unique global strong solutions for the Muskat problem with initial data of arbitrary size. 

   more » « less
5. Optimal Regularity and Structure of the Free Boundary for Minimizers in Cohesive Zone Models

   https://doi.org/10.1007/s00205-020-01509-3  

   Caffarelli, L.; Cagnetti, F.; Figalli, A. (July 2020, Archive for Rational Mechanics and Analysis)

   null (Ed.)

   Abstract We study optimal regularity and free boundary for minimizers of an energy functional arising in cohesive zone models for fracture mechanics. Under smoothness assumptions on the boundary conditions and on the fracture energy density, we show that minimizers are $$C^{1, 1/2}$$ C 1 , 1 / 2 , and that near non-degenerate points the fracture set is $$C^{1, \alpha }$$ C 1 , α , for some $$\alpha \in (0, 1)$$ α ∈ ( 0 , 1 ) . 

   more » « less