# Navier-Stokes equations, symmetric and uniform analytic solutions in phase space

## TL;DR

This paper constructs symmetric and uniform analytic solutions to the incompressible Navier-Stokes equations in Fourier-Herz spaces, demonstrating their properties and equivalence to convolution inequalities, advancing understanding of solution structures.

## Contribution

It introduces methods to find symmetric and uniform analytic solutions in Fourier-Herz spaces and links these solutions to convolution inequalities, extending previous theoretical results.

## Key findings

- Established global symmetric and uniform analytic solutions.
- Proved solutions inherit initial data symmetry properties.
- Showed equivalence between uniform analyticity and convolution inequalities.

## Abstract

For incompressible Navier-Stokes equations, Necas-Ruzicka-Sverak proved that self-similar solution has to be zero in 1996. Further, Yang-Yang-Wu find symmetry property plays an important role in the study of ill-posedness. In this paper, we consider two types of symmetry property. We search special symmetric and uniform analytic functions to approach the solution and establish global uniform analytic and symmetric solution with initial value in general symmetric Fourier-Herz space. For two kinds of symmetry of initial data, we prove that the solution has also the same symmetric structure. Further, we prove that the uniform analyticity is equivalent to the convolution inequality on Herz spaces. By these ways, we can use symmetric and uniform analytic functions to approximate the solution.