Regularizing rate estimates for the 3-D incompressible micropolar fluid system in critical Besov spaces

ABSTRACT

In this paper, we are concerned with regularizing rates of the higher order derivatives of solutions for the initial value problem of a three-dimensional incompressible micropolar fluid system with initial data $(u_0,{\omega }_0)\in \dot{\phantom{a}}{B}_{p,\mathrm{\infty }}^{-1+3/p}\left({\mathbb{R}}^3\right)$ for $1\le p<6$. More precisely, we prove that there exist two positive constants $K_1$ and $K_2$ such that for any $\beta \in {\mathbb{N}}_0^3$, the ${\dot{B}}_{q,\mathrm{\infty }}^s$-norm of β-order derivatives of solutions obeys the following regularizing rates: $\parallel \left({\mathrm{\partial }}_x^{\beta }u\right(\cdot ,t),{\mathrm{\partial }}_x^{\beta }\omega (\cdot ,t\left)\right){\parallel }_{{\dot{B}}_{q,\mathrm{\infty }}^s}\le K_1(K_2|\beta |{)}^{|\beta |}{t}^{-|\beta |/2-(1/2\left)\right(s+1-3/q)}$ for some suitable choices of s and q. These estimates particularly imply the spatial analyticity and the temporal decay of global solutions.

COMMUNICATED BY:

• Bo Zhang

KEYWORDS:

• Micropolar fluid system
• regularizing rate
• spatial analyticity
• temporal decay
• Besov spaces

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35B65
• 35K45
• 35Q35
• 76D03

Acknowledgments

The authors sincerely acknowledge their gratefulness to the referees for the valuable comments and suggestions, and to Professor Shangbin Cui for his warm guidance.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

W. Zhu is partially supported by the National Natural Science Foundation of China (11571381). J. Zhao is partially supported by the National Natural Science Foundation of China (11501453) and the Natural Science Foundation of Shaanxi Province (2018JM1004, 2017JM1016).