Existence and smoothness of the Navier-Stokes equations and semigroups of linear operators

TL;DR

This paper establishes the existence and smoothness of solutions to the Navier-Stokes equations by representing them through linear operators and semigroups, providing explicit solutions under smooth initial conditions.

Contribution

It introduces a linear operator formulation for Leray's Navier-Stokes equations and proves it generates a contraction semigroup, ensuring unique smooth solutions.

Findings

Existence of a unique smooth classical solution in \\mathbb{R}^3

Explicit operator solution for Navier-Stokes equations

Linear operator generates a contraction semigroup

Abstract

Based on Leray's formulation of the Navier-Stokes equations and the conditions of the exact linear representation of the nonlinear problem found in this paper, a compact explicit expression for the exact operator solution of the Navier-Stokes equations is given. It is shown that the introduced linear operator for Leray's equations is the generator of one-parameter contraction semigroup. This semigroup yields the existence of a unique and smooth classical solution of the associated Cauchy problem of Navier-Stokes equations in space $\mathbb{R}^3$ under smooth initial conditions.