Weak Morrey spaces and strong solutions to the Navier-Stokes equations

Abstract

We consider the Cauchy problem of Navier-Stokes equations in weak Morrey spaces. We first define a class of weak Morrey type spaces Mp,λ*(ℝn) on the basis of Lorentz space Lp,∞ = Lp*(ℝn) (in particular, Mp,0*(ℝn) = Lp,∞, if p > 1), and study some fundamental properties of them; Second, we prove that the heat operator U(t) = etΔ. and Calderon-Zygmund singular integral operators are bounded linear operators on weak Morrey spaces, and establish the bilinear estimate in weak Morrey spaces. Finally, by means of Kato’s method and the contraction mapping principle, we prove that the Cauchy problem of Navier-Stokes equations in weak Morrey spaces Mp,λ*(ℝn) (1 < p ⩽ n) is time-global well-posed, provided that the initial data are sufficiently small. Moreover, we also obtain the existence and uniqueness of the self-similar solution for Navier-Stokes equations in these spaces, because the weak Morrey space Mp,n−p*(ℝn) can admit the singular initial data with a self-similar structure. Hence this paper generalizes Kato’s results.