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1. 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. 

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2. Concavity of solutions to semilinear equations in dimension two

   https://doi.org/10.1112/blms.12750  

   Chau, Albert; Weinkove, Ben (November 2022, Bulletin of the London Mathematical Society)

   Abstract We consider the Dirichlet problem for a class of semilinear equations on two‐ dimensional convex domains. We give a sufficient condition for the solution to be concave. Our condition uses comparison with ellipses, and is motivated by an idea of Kosmodem'yanskii. We also prove a result on propagation of concavity of solutions from the boundary, which holds in all dimensions. 

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3. Traveling Wave Solutions to the One-Phase Muskat Problem: Existence and Stability

   https://doi.org/10.1007/s00205-023-01951-z  

   Nguyen, Huy Q; Tice, Ian (February 2024, Archive for Rational Mechanics and Analysis)

   We study the Muskat problem for one fluid in an arbitrary dimension, bounded below by a flat bed and above by a free boundary given as a graph. In addition to a fixed uniform gravitational field, the fluid is acted upon by a generic force field in the bulk and an external pressure on the free boundary, both of which are posited to be in traveling wave form. We prove that, for sufficiently small force and pressure data in Sobolev spaces, there exists a locally unique traveling wave solution in Sobolev-type spaces. The free boundary of the traveling wave solutions is either periodic or asymptotically flat at spatial infinity. Moreover, we prove that small periodic traveling wave solutions induced by external pressure only are asymptotically stable. These results provide the first class of nontrivial stable solutions for the problem. 

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4. Well-posedness of free boundary hard phase fluids in Minkowski background and their Newtonian limit

   https://doi.org/10.4310/CJM.2021.v9.n2.a1  

   Miao, Shuang; Shahshahani, Sohrab; Wu, Sijue (October 2021, Cambridge journal of mathematics)

   The hard phase model describes a relativistic barotropic irrotational fluid with sound speed equal to the speed of light. In this paper, we prove the local well-posedness for this model in the Minkowski background with free boundary. Moreover, we show that as the speed of light tends to infinity, the solution of this model converges to the solution of the corresponding Newtonian free boundary problem for incompressible fluids. In the appendix we explain how to extend our proof to the general barotropic fluid free boundary problem. 

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5. Segregation and domain formation in non-local multi-species aggregation equations

   https://doi.org/10.1016/j.physd.2023.133936  

   Glasner, Karl (December 2023, Physica D: Nonlinear Phenomena)

   A system of aggregation equations describing nonlocal interaction of two species is studied. When interspecies repulsive forces dominate intra-species repulsion, phase segregation may occur. This leads to the formation of distinct phase domains, separated by moving interfaces. The one dimensional interface problem is formulated variationally, and conditions for existence and nonexistence are established. The singular limit of large and short-ranged repulsion in two dimensions is then considered, leading to a two-phase free boundary problem describing the evolution of phase interfaces. Long term dynamics are investigated computationally, demonstrating coarsening phenomenon quantitatively different from classical models of phase separation. Finally, the interplay between long-range interspecies attraction and interfacial energy is illustrated, leading to pattern formation. 

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