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

Abstract

We consider the initial problem for the Navier-Stokes equations over ${\mathbb{R}}^3\times [0,T]$ with a positive time $T$ over specially constructed scale of function spaces of Bochner-Sobolev type. We prove that the problem induces an open both injective and surjective mapping of each space of the scale. In particular, intersection of these classes gives a uniqueness and existence theorem for smooth solutions to the Navier-Stokes equations for smooth data with a prescribed asymptotic behaviour at the infinity with respect to the time and the space variables.

A significant gap in the proof has been found

References in corpus (1)

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