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# Communications in Mathematical Sciences

## Volume 18 (2020)

### Number 8

### Lower bounds of blow up solutions in $\dot{H}^1_p (\mathbb{R}^3)$ of the Navier–Stokes equations and the quasi-geostrophic equation

Pages: 2263 – 2270

DOI: https://dx.doi.org/10.4310/CMS.2020.v18.n8.a8

#### Authors

#### Abstract

In this paper, we derive some new lower bounds of possible blow up solutions in $\dot{H}^1_p (\mathbb{R}^3)$ with $3 / 2 \lt p \lt \infty$ to the 3D Navier–Stokes equations, which provides a new proof of the corresponding recent results involving blow up rates in $\dot{H}^s$ with $1 \leq s \lt 5/2$ in [A. Cheskidov and M. Zaya, *J. Math. Phys.*, 57:023101, 2016; J.C. Cortissoz and J.A. Montero,* J. Math. Fluid Mech.*, 20:1–5, 2018; D.S. McCormick, E.J. Olson, J.C. Robinson, J.L. Rodrigo, A. Vidal-López, and Y. Zhou, *SIAM J. Math. Anal.*, 48:2119–2132, 2016; J.C. Robinson, W. Sadowski, and R.P. Silva, 53:115618, 2012]. We apply this to study the upper box dimension of the set of singular times of weak solutions. In addition, blow up rates of solutions to the 2D supercritical surface quasi-geostrophic equation in $\dot{H}^1_p (\mathbb{R}^2)$ are established.

#### Keywords

Navier–Stokes equations, blow up, regularity

#### 2010 Mathematics Subject Classification

35B33, 35Q35, 76D03, 76D05

Guoliang He was partially supported by National Natural Science Foundation of China under Grant 11871232 and the Training Plan of Young Key Teachers in Universities of Henan Province.

Yanqing Wang was partially supported by the National Natural Science Foundation of China under grants (No. 11971446 and No. 11601492).

Daoguo Zhou was partially supported by the National Natural Science Foundation of China under grant (No. 12071113) and Doctor Fund of Henan Polytechnic University (No. B2012-110).

Received 6 January 2020

Accepted 4 July 2020

Published 22 December 2020