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1. Traveling Wave Solutions to the Free Boundary Incompressible Navier‐Stokes Equations

   https://doi.org/10.1002/cpa.22084  

   Leoni, Giovanni; Tice, Ian (October 2023, Communications on Pure and Applied Mathematics)

   Abstract In this paper we study a finite‐depth layer of viscous incompressible fluid in dimension , modeled by the Navier‐Stokes equations. The fluid is assumed to be bounded below by a flat rigid surface and above by a free, moving interface. A uniform gravitational field acts perpendicularly to the flat surface, and we consider the cases with and without surface tension acting on the free interface. In addition to these gravity‐capillary effects, we allow for a second force field in the bulk and an external stress tensor on the free interface, both of which are posited to be in traveling wave form, i.e., time‐independent when viewed in a coordinate system moving at a constant velocity parallel to the rigid lower boundary. We prove that, with surface tension in dimension and without surface tension in dimension , for every nontrivial traveling velocity there exists a nonempty open set of force and stress data that give rise to traveling wave solutions. While the existence of inviscid traveling waves is well‐known, to the best of our knowledge this is the first construction of viscous traveling wave solutions. Our proof involves a number of novel analytic ingredients, including: the study of an overdetermined Stokes problem and its underdetermined adjoint problem, a delicate asymptotic development of the symbol for a normal‐stress to normal‐Dirichlet map defined via the Stokes operator, a new scale of specialized anisotropic Sobolev spaces, and the study of a pseudodifferential operator that synthesizes the various operators acting on the free surface functions. © 2022 The Authors.Communications on Pure and Applied Mathematicspublished by Wiley Periodicals LLC. 

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2. Traveling wave solutions to the inclined or periodic free boundary incompressible Navier-Stokes equations

   https://doi.org/10.1016/j.jfa.2023.110057  

   Koganemaru, Junichi; Tice, Ian (October 2023, Journal of Functional Analysis)

   This paper concerns the construction of traveling wave solutions to the free boundary incompressible Navier-Stokes system. We study a single layer of viscous fluid in a strip-like domain that is bounded below by a flat rigid surface and above by a moving surface. The fluid is acted upon by a bulk force and a surface stress that are stationary in a coordinate system moving parallel to the fluid bottom. We also assume that the fluid is subject to a uniform gravitational force that can be resolved into a sum of a vertical component and a component lying in the direction of the traveling wave velocity. This configuration arises, for instance, in the modeling of fluid flow down an inclined plane. We also study the effect of periodicity by allowing the fluid cross section to be periodic in various directions. The horizontal component of the gravitational field gives rise to stationary solutions that are pure shear flows, and we construct our solutions as perturbations of these by means of an implicit function argument. An essential component of our analysis is the development of some new functional analytic properties of a scale of anisotropic Sobolev spaces, including that these spaces are an algebra in the supercritical regime, which may be of independent interest. 

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3. Traveling capillary waves on the boundary of a fluid disc

   https://doi.org/10.1111/sapm.12435  

   Dyachenko, Sergey A. (August 2021, Studies in Applied Mathematics)

   Abstract The equation for a traveling wave on the boundary of a two‐dimensional droplet of an ideal fluid is derived by using the conformal variables technique for free surface waves. The free surface is subject only to the force of surface tension and the fluid flow is assumed to be potential. We use the canonical Hamiltonian variables discovered and map the lower complex plane to the interior of a fluid droplet conformally. The equations in this form have been originally discovered for infinitely deep water and later adapted to a bounded fluid domain.The new class of solutions satisfies a pseudodifferential equation similar to the Babenko equation for the Stokes wave. We illustrate with numerical solutions of the time‐dependent equations and observe the linear limit of traveling waves in this geometry. 

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4. Islands in stable fluid equilibria

   https://doi.org/10.1090/proc/16951  

   Drivas, Theodore; Ginsberg, Daniel (November 2024, Proceedings of the American Mathematical Society)

   We prove that stable fluid equilibria with trivial homology on curved, reflection-symmetric periodic channels must posses “islands”, or cat’s eye vortices. In this way, arbitrarily small disturbances of a flat boundary cause a change of streamline topology of stable steady states. 

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5. 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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