Well-posedness of Navier-Stokes equations established by the decaying speed of single norm

TL;DR

This paper establishes the well-posedness of the Navier-Stokes equations by analyzing the decay rate of a single norm of solutions, using wavelet techniques to avoid traditional integral norms over time.

Contribution

It introduces a new solution space for Navier-Stokes equations based on the decay speed of a single norm, avoiding the intersection of multiple norm spaces.

Findings

Established well-posedness using decay speed of a single norm

Developed a new solution space not contained in $L^{inity}(X)$

Applied parametric Meyer wavelets for analysis

Abstract

The decaying speed of a single norm more truly reflects the intrinsic harmonic analysis structure of the solution of the classical incompressible Navier-Stokes equations. No previous work has been able to establish the well-posedness under the decaying speed of a single norm with respect to time, and the previous solution space is contained in the intersection of two spaces defined by different norms. In this paper, for some separable initial space $X$, we find some new solution space which is not the subspace of $L^{\infty}(X)$. We use parametric Meyer wavelets to establish the well-posedness via the decaying speed of a single norm only, without integral norm to $t$.