# Global existence, uniqueness and estimates of the solution to the Navier-Stokes equations

## TL;DR

This paper establishes conditions for the global existence and uniqueness of solutions to the Navier-Stokes equations, providing explicit integral representations and energy estimates within a specific function class.

## Contribution

It introduces a novel integral equation formulation for Navier-Stokes solutions and proves their uniqueness and existence in a defined function space.

## Key findings

- Uniqueness of solutions in a class with finite norm N_1(v)
- Existence of solutions under certain assumptions
- Energy estimates for the solutions

## Abstract

The Navier-Stokes (NS) problem consists of finding a vector-function $v$ from the Navier-Stokes equations. The solution $v$ to NS problem is defined in this paper as the solution to an integral equation. The kernel $G$ of this equation solves a linear problem which is obtained from the NS problem by dropping the nonlinear term $(v \cdot \nabla)v$. The kernel $G$ is found in closed form. Uniqueness of the solution to the integral equation is proved in a class of solutions $v$ with finite norm $N_1(v)=\sup_{\xi\in \mathbb{R}^3, t\in [0, T]}(1+|\xi|)(|v|+|\nabla v|)\le c (*)$, where $T>0$ and $C>0$ are arbitrary large fixed constants. In the same class of solutions existence of the solution is proved under some assumption. Estimate of the energy of the solution is given.