Existence Theorems for Regular Spatially Periodic Solutions to the Navier-Stokes Equations

Abstract

We consider the initial value problem for the Navier-Stokes equations over $R^3\times [0,T]$ with a positive time $T$ in the spatially periodic setting. Identifying periodic vector-valued functions on $R^3$ with functions on the three-dimensional torus $T^3$, we prove that the problem induces an open both injective and surjective mapping of specially constructed function spaces of Bochner-Sobolev type. This gives a uniqueness and existence theorem for regular solutions to the Navier-Stokes equations. Our techniques consist in proving the closedness of the image by estimating all possible divergent sequences in the preimage and matching the asymptotics.

A significant gap in the proof has been found

Cited by in corpus (1)

• Existence theorems for regular solutions to the Cauchy problem for the Navier-Stokes equations in ${\mathbb{R}}^3$