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# Dynamics of Partial Differential Equations

## Volume 16 (2019)

### Number 4

### Predual forms, harmonic maps and liquid crystals of $(BMO-Q)$ and $(BMO-Q)^{-1}$

Pages: 359 – 382

DOI: https://dx.doi.org/10.4310/DPDE.2019.v16.n4.a3

#### Authors

#### Abstract

Under $( \alpha , n - 1) \in [0, 1) \times N$ this paper explores the fractional Sobolev type inclusion and the Fefferman–Stein type decomposition of the predual forms (unifying ones in [5] under $\alpha = 0$ and in [2] under $\alpha \in (0, 1)$) of the so-called $(BMO-Q)$ and $(BMO-Q)^{-1}$ spaces\[(-\Delta)^{-{\frac{\alpha}{2}}} \mathscr{L}^{2, 2 \alpha} (\mathbb{R}^n) \: \& \: (-\Delta)^{-{\frac{\alpha}{2}}} \mathscr{L}^{2, 2 \alpha} (\mathbb{R}^n)^{-1} = \mathrm{div} (-\Delta)^{-{\frac{\alpha}{2}}} \mathscr{L}^{2, 2 \alpha} (\mathbb{R}^n)^n\]and their natural actions on revealing\[(-\Delta)^{-{\frac{\alpha}{2}}} \mathscr{L}^{2, 2 \alpha} (\mathbb{R}^n) \: \& \: (-\Delta)^{-{\frac{\alpha}{2}}} \mathscr{L}^{2, 2 \alpha} (\mathbb{R}^n)^{-1}\]analogues of the global results in [26] about the heat flow of harmonic maps and the hydrodynamic flow of nematic liquid crystals.

#### Keywords

$BMO-Q$, heat flow of harmonic maps, hydrodynamic flow of liquid crystals

#### 2010 Mathematics Subject Classification

Primary 35Q30, 42B37, 46E35. Secondary 30H25.

Jie Xiao was supported by NSERC of Canada (#20171864).

Junjie Zhang was supported by the Science Foundation of Hebei Normal University of China (L2019B02), by the Hebei Natural Science Foundation of China (A2019205218), and by NSERC of Canada (#20171864).

Received 25 June 2019

Published 30 August 2019