Discretely self-similar solutions to the Navier-Stokes equations with Besov space data

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 ${\dot{B}}_{p,\infty }^{-1+3/p}$ where $3<p<6$. We also construct discretely self-similar solutions for divergence free initial data in ${\dot{B}}_{p,\infty }^{-1+3/p}$ for $3<p<6$ that is discretely self-similar for some scaling factor $\lambda >1$. These results extend those of \cite{BT1} which dealt with initial data in $L_w^3$ since $L_w^3⊊{\dot{B}}_{p,\infty }^{-1+3/p}$ for $p>3$. We also provide several concrete examples of vector fields in the relevant function spaces.

References in corpus (2)

• Existence of discretely self-similar solutions to the Navier-Stokes equations for initial value in $L_{loc}^2({\mathbb{R}}^3)$
• Rotationally corrected scaling invariant solutions to the Navier-Stokes equations

Cited by in corpus (6)

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• An $\epsilon$-regularity criterion and estimates of the regular set for Navier-Stokes flows in terms of initial data
• Spatial decay of discretely self-similar solutions to the Navier-Stokes equations
• Self-similar solutions to the Navier-Stokes equations: a survey of recent results
• Existence and Weak* Stability for the Navier-Stokes System with Initial Values in Critical Besov Spaces