Well-posedness of the stationary and slowly traveling wave problems for the free boundary incompressible Navier-Stokes equations

We establish that solitary stationary waves in three dimensional viscous incompressible fluids are a generic phenomenon and that every such solution is a vanishing wave-speed limit along a one parameter family of traveling waves. The setting of our result is a horizontally-infinite fluid of finite depth with a flat, rigid bottom and a free boundary top. A constant gravitational field acts normal to bottom, and the free boundary experiences surface tension. In addition to these gravity-capillary effects, we allow for applied stress tensors to act on the free surface region and applied forces to act in the bulk. These are posited to be in either stationary or traveling form. In the absence of any applied stress or force, the system reverts to a quiescent equilibrium; in contrast, when such sources of stress or force are present, stationary or traveling waves are generated. We develop a small data well-posedness theory for this problem by proving that there exists a neighborhood of the origin in stress, force, and wave speed data-space in which we obtain the existence and uniqueness of stationary and traveling wave solutions that depend continuously on the stress-force data, wave speed, and other physical parameters. To the best of our knowledge, this is the first proof of well-posedness of the solitary stationary wave problem and the first continuous embedding of the stationary wave problem into the traveling wave problem. Our techniques are based on vector-valued harmonic analysis, a novel method of indirect symbol calculus, and the implicit function theorem.

Well-posedness of the traveling wave problem for the free boundary compressible Navier-Stokes equations

We prove that traveling waves in viscous compressible liquids are a generic phenomenon. The setting for our result is a horizontally infinite, finite depth layer of compressible, barotropic, viscous fluid, modeled by the free boundary compressible Navier-Stokes equations in dimension n≥2. The bottom boundary of the fluid is flat and rigid, while the top is a moving free boundary. A constant gravitational field acts normal to the flat bottom. We allow external forces to act in the fluid's bulk and external stresses to act on its free surface. These are posited to be in traveling wave form, i.e. time-independent when viewed in a coordinate system moving at a constant, nontrivial velocity parallel to the lower rigid boundary. In the absence of such external sources of stress and force, the fluid system reverts to equilibrium, which corresponds to a flat, quiescent fluid layer with vertically stratified density. In contrast, when such sources of stress or force are present, the system admits traveling wave solutions. We establish a small data well-posedness theory for this problem by proving that for every nontrivial traveling wave speed there exists a nonempty open set of stress and forcing data that give rise to unique traveling wave solutions, and that these solutions depend continuously on the data and the wave speed. When n≥3 we prove this with surface tension accounted for at the free boundary, while in the case n=2 we prove this with or without surface tension. To the best of our knowledge, this result constitutes the first general construction of traveling wave solutions to any free boundary compressible fluid equations.

Traveling Wave Solutions to the Multilayer Free Boundary Incompressible Navier-Stokes Equations

For a natural number m≥2, we study m layers of finite depth, horizontally infinite, viscous, and incompressible fluid bounded below by a flat rigid bottom. Adjacent layers meet at free interface regions, and the top layer is bounded above by a free boundary as well. A uniform gravitational field, normal to the rigid bottom, acts on the fluid. We assume that the fluid mass densities are strictly decreasing from bottom to top and consider the cases with and without surface tension acting on the free surfaces. In addition to these gravity-capillary effects, we allow a force to act on the bulk and external stress tensors to act on the free interface regions. Both of these additional forces are posited to be in traveling wave form: time-independent when viewed in a coordinate system moving at a constant, nontrivial velocity parallel to the lower rigid boundary. Without surface tension in the case of two dimensional fluids and with all positive surface tensions in the higher dimensional cases, we prove that for each sufficiently small force and stress tuple there exists a traveling wave solution. The existence of traveling wave solutions to the one layer configuration (m=1) was recently established and, to the best of our knowledge, this paper is the first construction of traveling wave solutions to the incompressible Navier-Stokes equations in the m-layer arrangement.

A truncated real interpolation method and characterizations of screened Sobolev spaces

In this paper we prove structural and topological characterizations of the screened Sobolev spaces with screening functions bounded below and above by positive constants. We generalize a method of interpolation to the case of seminormed spaces. This method, which we call the truncated method, generates the screened Sobolev subfamily and a more general screened Besov scale. We then prove that the screened Besov spaces are equivalent to the sum of a Lebesgue space and a homogeneous Sobolev space and provide a Littlewood-Paley frequency space characterization.

Analysis of micropolar fluids: existence of potential microflow solutions, nearby global well-posedness, and asymptotic stability

In this paper we concern ourselves with an incompressible, viscous, isotropic, and periodic micropolar fluid. We find that in the absence of forcing and microtorquing there exists an infinite family of well-behaved solutions, which we call potential microflows, in which the fluid velocity vanishes identically, but the angular velocity of the microstructure is conservative and obeys a linear parabolic system. We then prove that nearby each potential microflow, the nonlinear equations of motion are well-posed globally-in-time, and solutions are stable. Finally, we prove that in the absence of force and microtorque, solutions decay exponentially, and in the presence of force and microtorque obeying certain conditions, solutions have quantifiable decay rates.