Abstract
In this paper, we establish analyticity of the Navier–Stokes equations with small data in critical Besov spaces \({\dot{B}^{\frac{3}{p}-1}_{p,q}}\) . 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 \({\|e^{\sqrt{t}\Lambda}v(t)\|_{E_p}>\infty}\) holds in \({E_p:=\tilde{L}^{\infty}(0,T;\dot{B}^{\frac{3}{p}-1}_{p,q})\cap \tilde{L}^{1}(0,T;\dot{B}^{\frac{3}{p}+1}_{p,q})}\) , 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.
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Communicated by C. Dafermos
Research was supported in part by NSF grants DMS10-08397 and FRG07-57227 (H. Bae and E. Tadmor) and DMS11-09532 (A. Biswas).
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Bae, H., Biswas, A. & Tadmor, E. Analyticity and Decay Estimates of the Navier–Stokes Equations in Critical Besov Spaces. Arch Rational Mech Anal 205, 963–991 (2012). https://doi.org/10.1007/s00205-012-0532-5
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DOI: https://doi.org/10.1007/s00205-012-0532-5