Hokkaido Mathematical Journal

NAKAI Eiichi, YONEDA Tsuyoshi,

Riesz transforms on generalized Hardy spaces and a uniqueness theorem for the Navier-Stokes equations.

Hokkaido Mathematical Journal, 40 (2011) pp.67-88

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Abstract

The purpose of this paper is twofold. Let Rj (j = 1,2, ... , n) be Riesz transforms on $\mathbb{R}$n. First we prove the convergence of truncated operators of RiRj in generalized Hardy spaces. Our first result is an extension of the convergence in Lp($\mathbb{R}$^n) (1 < p < ∞). Secondly, as an application of the first result, we show a uniqueness theorem for the Navier-Stokes equation. J. Kato (2003) established the uniqueness of solutions of the Navier-Stokes equations in the whole space when the velocity field is bounded and the pressure field is a BMO-valued locally integrable-in-time function for bounded initial data. We extend the part "BMO-valued" in his result to "generalized Campanato space valued". The generalized Campanato spaces include L1, BMO and homogeneous Lipschitz spaces of order α (0 < α < 1).

MSC(Primary)35Q30
MSC(Secondary)76D05, 42B35, 42B30
Uncontrolled KeywordsNavier-Stokes equation, uniqueness, Campanato space, Hardy space