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

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 $t$.