Analyticity and Decay Estimates of the Navier–Stokes Equations in Critical Besov Spaces

Abstract

In this paper, we establish analyticity of the Navier–Stokes equations with small data in critical Besov spaces Ḃ³/ᵖ ⁻¹ₚ,q. The main method is Gevrey estimates, the choice of which is motivated by the work of Foias and Temam (Contemp Math 208:151–180, 1997). We show that mild solutions are Gevrey regular, that is, the energy bound ∥e√ᵗΛv(t)∥Eₚ < ∞ holds in Eₚ := L˜ ∞(0, T; Ḃ³/ᵖ ⁻¹ₚ,q) ∩ L˜¹ (0, T; Ḃ³/ᵖ ⁺¹ₚ,q), globally in time for p < ∞. We extend these results for the intricate limiting case p = ∞ in a suitably designed E∞ space. As a consequence of analyticity, we obtain decay estimates of weak solutions in Besov spaces. Finally, we provide a regularity criterion in Besov spaces.