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

TL;DR

This paper establishes existence and uniqueness of smooth solutions to the Navier-Stokes equations in three-dimensional space using specialized function spaces, providing a rigorous mathematical framework for initial value problems.

Contribution

It introduces a novel approach using Bochner-Sobolev type spaces to prove existence and uniqueness of solutions for the Navier-Stokes initial problem in ${\mathbb R}^3$.

Findings

Proves the problem induces an open, injective, and surjective mapping in the constructed function spaces.

Establishes a theorem for smooth solutions with prescribed asymptotic behavior.

Provides a mathematical foundation for the well-posedness of Navier-Stokes in these spaces.

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.