On large periodic traveling wave solutions to the free boundary Stokes and Navier-Stokes equations

We study the free boundary problem for a finite-depth layer of viscous incompressible fluid in arbitrary dimension, modeled by the Stokes or Navier-Stokes equations. In addition to the gravitational field acting in the bulk, the free boundary is acted upon by surface tension and an external stress tensor posited to be in traveling wave form. We prove that for any isotropic stress tensor with periodic profile, there exists a locally unique periodic traveling wave solution, which can have large amplitude. Moreover, we prove that the constructed traveling wave solutions are asymptotically stable for the dynamic free boundary Stokes equations. Our proofs rest on the analysis of the nonlocal normal-stress to normal-Dirichlet operators for the Stokes and Navier-Stokes equations in domains of Sobolev regularity.

💡 Research Summary

The paper investigates a free‑boundary problem for a viscous incompressible fluid of finite depth in arbitrary dimension, governed either by the Stokes (α=0) or Navier‑Stokes (α=1) equations. The fluid is acted upon by gravity, surface tension, and an external stress tensor that is prescribed in a traveling‑wave form T(x,y,t)=T₀(x−γe₁t). The authors focus on isotropic stresses T₀=φ I with φ a periodic scalar function, which may be arbitrarily large.

The first main result (Theorem 1.1) shows that for any φ∈H^{s−½}(T^d)∩C¹(T^d) with s>(d+1)/2 and satisfying a natural positivity condition, there exists a large static free‑surface profile η* = (σH+gI)^{-1}(−φ). By perturbing η* and allowing a small wave speed |γ|<γ_δ, they prove the existence of a unique periodic traveling‑wave solution (η, v, p) of the full free‑boundary system, with η close to η* in the Sobolev space H^{s+3/2}(T^d). The proof hinges on recasting the problem as a single nonlinear equation G(γ,η)=0, where

 G(γ,η)=Ψ₀