Existence and regularizing rate estimates of solutions to the 3‐D generalized micropolar system in Fourier‐Besov spaces

Abstract

In this paper, we study global existence and asymptotic stability of solutions for the initial value problem of the three‐dimensional (3‐D) generalized incompressible micropolar system in Fourier‐Besov spaces. Besides, we also establish some regularizing rate estimates of the higher‐order spatial derivatives of solutions, which particularly imply the spatial analyticity and the temporal decay of global solutions.

Sharp well-posedness and ill-posedness for the 3-D micropolar fluid system in Fourier–Besov spaces

“…Moreover, if r = 1, the range of p for the existence can be extended to[1, ∞). Recently, in a collaborating work of the first author of the present paper with Zhao[30], well-posedness of the problem (1.1) in the Fourier-Besov spaces Ḟ B…”

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We study the Cauchy problem of the incompressible micropolar fluid system in R 3 . In a recent work of the first author and Jihong Zhao [30], it is proved that the Cauchy problem of the incompressible micropolar fluid system is locally well-posed in the Fourier-Besov spacesḞ B 2− 3 p p,r for 1 < p ≤ ∞ and 1 ≤ r < ∞, and globally well-posed in these spaces with small initial data. In this work we consider the critical case p = 1.

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“…We study the Cauchy problem of the incompressible micropolar fluid system in R 3 . In a recent work of the first author and Jihong Zhao [30], it is proved that the Cauchy problem of the incompressible micropolar fluid system is locally well-posed…”

A class large solution of 3D incompressible micropolar fluid system

“…In [3], Chen and Miao established global well-posedness of the system (1.1) for small initial data in the Besov 6). In [23], Zhu and Zhao proved that the micropolar fluid system is globally well-posed in the Fourier-Besov spaces Ḟ B 2− 3 p p,r (R 3 ) for p ∈ (1, ∞] and r ∈ [1, ∞) with small initial data. Recently, the corresponding author of the present paper [22] showed that this problem is well-posed in Ḟ B −1 1,r (R 3 ) for 1 r 2, while ill-posed for 2 < r ∞.…”