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

Let ΩΩ be an open domain of class C2C2 contained in R3R3, let L2(Ω)3L2(Ω)3 be the Hilbert space of square integrable functions on ΩΩ and let H[Ω]≔HH[Ω]≔H be the completion of the set, {u∈(C0∞[Ω])3∣∇⋅u=0}, with respect to the inner product of L2(Ω)3L2(Ω)3. A well-known unsolved problem is that of the construction of a sufficient class of functions in HH 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 HH, which provides strong bounds on the nonlinear term. This allows us to prove that, under appropriate conditions, there exists a number u+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 DD contained in the closed ball B(Ω)≕BB(Ω)≕B of radius 12u+ in HH, the Navier–Stokes equations have unique, strong, solutions in C1((0,∞),H)C1((0,∞),H).