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Discretely Self-Similar Solutions to the Navier–Stokes Equations with Besov Space Data

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Abstract

We construct self-similar solutions to the three dimensional Navier–Stokes equations for divergence free, self-similar initial data that can be large in the critical Besov space \({\dot{B}_{p,\infty}^{3/p-1}}\) where 3 < p < 6. We also construct discretely self-similar solutions for divergence free initial data in \({\dot{B}_{p,\infty}^{3/p-1}}\) for 3 < p < 6 that is discretely self-similar for some scaling factor λ > 1. These results extend those of Bradshaw and Tsai (Ann Henri Poincaré 2016. https://doi.org/10.1007/s00023-016-0519-0) which dealt with initial data in L 3 w since \({L^3_w\subsetneq \dot{B}_{p,\infty}^{3/p-1}}\) for p > 3. We also provide several concrete examples of vector fields in the relevant function spaces.

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Author notes

  1. Zachary Bradshaw

    Present address: Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR, 72701, USA

Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Zachary Bradshaw & Tai-Peng Tsai

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  1. Zachary Bradshaw
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  2. Tai-Peng Tsai
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Correspondence to Zachary Bradshaw.

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Communicated by N. Masmoudi

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Bradshaw, Z., Tsai, TP. Discretely Self-Similar Solutions to the Navier–Stokes Equations with Besov Space Data. Arch Rational Mech Anal 229, 53–77 (2018). https://doi.org/10.1007/s00205-017-1213-1

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