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

Abstract

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.

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

“…A few remarks are in order. First, we note again that in the physically relevant case n = 3, we cannot solve (1.29) in the stated spaces if Σ = L 1 T × R. Second, Theorem 1.2 is analogous to Theorem 1.1 in [18] for the problem without incline and periodicity, and Theorem 1 in [25] for the multilayer problem. However, our choice of the function space X s is slightly different from the one used there because we have formulated the problem (1.29) in a slightly different manner, with the η term shifted from the dynamic boundary condition into the bulk.…”

“…The existence of traveling wave solutions to the equations of fluid mechanics has been a subject of intense study for nearly two centuries (see Section 1.4 for a brief summary). Until recently, most of the mathematical results in this area focused on inviscid fluids, but work in the last few years constructed traveling wave solutions to the free boundary Navier-Stokes equations with a single horizontally infinite but finite depth layer of incompressible fluid [18], and with multiple layers [25] in a uniform, downward-pointing gravitational field. The purpose of the present paper is to extend these constructions into more general physical configurations by considering two effects: inclination of the fluid domain, which results in a component of the gravitational field parallel to the fluid layer; and, periodicity of the fluid layer in certain directions.…”

“…The purpose of the present paper is to extend these constructions into more general physical configurations by considering two effects: inclination of the fluid domain, which results in a component of the gravitational field parallel to the fluid layer; and, periodicity of the fluid layer in certain directions. The key to the constructions in [18,25] was the identification and utilization of a new scale of anisotropic Sobolev spaces, which serve as the container space for the function describing the free surface of the fluid. The results in this paper rely crucially on some new functional analytic properties of these spaces, which we prove here: the development of this scale of spaces on domains with periodicity; and, the fact that these spaces form an algebra in the supercritical regime.…”

“…The existence of traveling wave solutions without background shear flows to (1.3) first appeared in the recent work of Leoni-Tice [18], and the results therein were extended by Stevenson-Tice [25] to a multilayer configuration. To the best of our knowledge, there are no known results on the existence of traveling wave solutions around shear flows in the case when κ = 0 or when the cross section Σ is periodic.…”

“…However, our choice of the function space X s is slightly different from the one used there because we have formulated the problem (1.29) in a slightly different manner, with the η term shifted from the dynamic boundary condition into the bulk. This results in the pressure belonging to a standard Sobolev space rather than a specialized anisotropic one, as in [18,25]. Third, we can say something about the set of (γ, κ) parameters for which we can produce solutions,…”