Analyticity and Decay Estimates of the Navier–Stokes Equations in Critical Besov Spaces
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Bae, Hantaek, Animikh Biswas, and Eitan Tadmor. “Analyticity and Decay Estimates of the Navier–Stokes Equations in Critical Besov Spaces.” Archive for Rational Mechanics and Analysis 205, no. 3 (September 1, 2012): 963–91. https://doi.org/10.1007/s00205-012-0532-5.
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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.