A sufficiency class for global (in time) solutions to the 3D Navier–Stokes equations

Abstract

Let Ω be an open domain of class C 2 contained in R 3 , let L 2 ( Ω ) 3 be the Hilbert space of square integrable functions on Ω and let H [ Ω ] ≔ H be the completion of the set, { u ∈ ( C 0 ∞ [ Ω ] ) 3 ∣ ∇ ⋅ u = 0 } , with respect to the inner product of L 2 ( Ω ) 3 . A well-known unsolved problem is that of the construction of a sufficient class of functions in H which will allow global, in time, strong solutions to the three-dimensional Navier–Stokes equations. These equations describe the time evolution of the fluid velocity and pressure of an incompressible viscous homogeneous Newtonian fluid in terms of a given initial velocity and given external body forces. In this paper, we use the analytic nature of the Stokes semigroup to construct an equivalent norm for H , which provides strong bounds on the nonlinear term. This allows us to prove that, under appropriate conditions, there exists a number u + , depending only on the domain, the viscosity, the body forces and the eigenvalues of the Stokes operator, such that, for all functions in a dense set D contained in the closed ball B ( Ω ) ≕ B of radius 1 2 u + in H , the Navier–Stokes equations have unique, strong, solutions in C 1 ( ( 0 , ∞ ) , H ) .