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

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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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Authors and Affiliations

  1. Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD, 20742, USA

    Hantaek Bae

  2. Department of Mathematics and Statistics, University of North Carolina, Charlotte, NC, 28223, USA

    Animikh Biswas

  3. Department of Mathematics, Center for Scientific Computation and Mathematical Modeling (CSCAMM), Institute for Physical Science & Technology (IPST), University of Maryland, College Park, MD, 20742, USA

    Eitan Tadmor

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  1. Hantaek Bae
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  2. Animikh Biswas
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  3. Eitan Tadmor
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Correspondence to Eitan Tadmor.

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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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