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

We consider the Cauchy problem of Navier-Stokes equations in weak Morrey spaces. We first define a class of weak Morrey type spaces M _p,λ ^* (ℝ ⁿ ) on the basis of Lorentz space L _p,∞ = L _p ^* (ℝ ⁿ ) (in particular, M _p,0 ^* (ℝ ⁿ ) = L _p,∞, if p > 1), and study some fundamental properties of them; Second, we prove that the heat operator U(t) = e ^tΔ. and Calderón-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 M _p,λ ^* (ℝ ⁿ ) (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 M _p,n−p ^* (ℝ ⁿ ) can admit the singular initial data with a self-similar structure. Hence this paper generalizes Kato’s results. This work was partially supported by the National Natural Science Foundation of China (Grant No. 10571016), the China Postdoctoral Science Foundation (Grant No. 20060390530) and the Natural Science Foundation of Henan Province (Grant No. 0611055500)