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On the Regularity Set and Angular Integrability for the Navier–Stokes Equation

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Abstract

We investigate the size of the regular set for suitable weak solutions of the Navier–Stokes equation, in the sense of Caffarelli–Kohn–Nirenberg (Commun Pure Appl Math 35:771–831, 1982). We consider initial data in weighted Lebesgue spaces with mixed radial-angular integrability, and we prove that the regular set increases if the data have higher angular integrability, invading the whole half space \({\{t > 0\}}\) in an appropriate limit. In particular, we obtain that if the \({L^{2}}\) norm with weight \({|x|^{-\frac12}}\) of the data tends to 0, the regular set invades \({\{t > 0\}}\); this result improves Theorem D of Caffarelli et al. (Commun Pure Appl Math 35:771–831, 1982).

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Authors and Affiliations

  1. Dipartimento di Matematica, SAPIENZA, Università di Roma, Piazzale A. Moro 2, 00185, Rome, Italy

    Piero D’Ancona

  2. Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Cientificas, C. Nicolás Cabrera 13-15, 28049, Madrid, Spain

    Renato Lucà

Authors
  1. Piero D’Ancona
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  2. Renato Lucà
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Correspondence to Renato Lucà.

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Communicated by V. Šverák

The authors are partially supported by the Italian Project FIRB 2012 “Dispersive dynamics: Fourier Analysis and Variational Methods”. Renato Lucà is supported by the ERC Grant 277778 and MINECO Grant SEV-2011-0087 (Spain).

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D’Ancona, P., Lucà, R. On the Regularity Set and Angular Integrability for the Navier–Stokes Equation. Arch Rational Mech Anal 221, 1255–1284 (2016). https://doi.org/10.1007/s00205-016-0982-2

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