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

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 Mathematics published by Wiley Periodicals LLC.

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

“…As is the case for the traveling wave problems studied in [55,58,62,91,92], the stationary boundary value problem (1.13) lies in an unbounded domain of infinite measure and possesses a non-compact free boundary. The equations are quasilinear and do not enjoy a variational formulation.…”

“…The second line of inquiry, which dates back essentially to the beginning of mathematical fluid mechanics, concerns the search for traveling wave solutions moving with speed γ = 0. In this context, a huge literature exists for the corresponding inviscid problem, but progress on the viscous problem was initiated much more recently in the work of Leoni and Tice [58], and further developed by Stevenson and Tice [91,92], Koganemaru and Tice [55], and Nguyen and Tice [62]. The analysis in [55,58,62,91,92] crucially relies on the condition γ = 0 to provide an estimate for the free surface function in a scale of anisotropic Sobolev spaces.…”

“…In this context, a huge literature exists for the corresponding inviscid problem, but progress on the viscous problem was initiated much more recently in the work of Leoni and Tice [58], and further developed by Stevenson and Tice [91,92], Koganemaru and Tice [55], and Nguyen and Tice [62]. The analysis in [55,58,62,91,92] crucially relies on the condition γ = 0 to provide an estimate for the free surface function in a scale of anisotropic Sobolev spaces. When γ = 0, this estimate degenerates, and [58] fails to construct solutions within their functional framework.…”

“…The analysis in [55,58,62,91,92] crucially relies on the condition γ = 0 to provide an estimate for the free surface function in a scale of anisotropic Sobolev spaces. When γ = 0, this estimate degenerates, and [58] fails to construct solutions within their functional framework. Thus, a natural question is whether there exists an alternate functional framework in which solutions can be constructed for all γ in a neighborhood of 0.…”

“…In contrast, progress on the viscous traveling wave problem has only recently commenced. Leoni and Tice [58] developed a well-posedness theory for small forcing and stress data, provided γ = 0. This was generalized to multi-layer, inclined, and periodic configurations by Stevenson and Tice [91] and Koganemaru and Tice [55].…”