Well-Posedness in Variable-Exponent Function Spaces for the Three-Dimensional Micropolar Fluid Equations

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Introduction
In this paper, our aim is to study the Cauchy problem of the three-dimensional incompressible micropolar fuid equations: where the ω(y, t) is the linear velocity of the fuid, the scalar π(y, t) represents the pressure, and θ(y, t) is the feld of microrotation and represents the angular velocity of the rotation of particles in the fuid.Te symbols ∇ • ω and ∇ × ω represent the divergence and rotational of the feld ω, respectively.Te Newton kinematics viscosity is denoted by ] and χ represents the microrotational viscosity.Te symbols κ and μ denote the angular viscosities.For convenience, we assume χ � ] � 1/2 and κ � μ � 1.
Te standard micropolar fuid equations, e.g., equation (1), were frst investigated by Eringen [1].Tis is a kind of fuid that possesses microrotational efects and microrotational inertia, which is thus classifed as a non-Newtonian fuid.In the context of physical importance, a micropolar fuid could be a model of fuids where the particles are rigid and foating in a viscous medium without consideration for their deformation.It is signifcant to the researchers dealing with MHD fuid problems and other related phenomena because it can relate to many behaviors that occur in a wide variety of complicated fuids such as liquid crystals, animal blood, and suspensions that cannot be described properly by the Navier-Stokes (NS) equation.For more details related to its physical applications, we suggest the readers read [2] and the references therein.
Its signifcance in both physics and mathematics has contributed to extensive work in the area of mathematical analysis.Existence of the weak solution was established by Lukaszewicz [2].Similarly, in 1997, Galdi and Rionero [3] published a note in which they considered equation (1) and achieved for the frst time that the solution exists and is unique.Moreover, in their study, they have further proved that the corresponding solution is globally well-posed if the initial data are small and local well-posed for large initial data.Regarding the smooth solutions to the twodimensional case of equation (1) with partial viscosity and full viscosity, Chen [4] worked on the global wellposedness and its uniqueness.Te large-time phenomenon of the solution and global in time regularity were established by Dong et al. [5].For more results related to the two-dimensional case, we refer to [6,7].Regarding the regularity criterion of the weak solution and the blown-up criterion, for the smooth solution, see [8,9] and the references therein.Te main difculty of equation ( 1) is to deal with the linear coupling terms ∇ × θ and ∇ × u.Tese terms involve the curl operations on the microrotation feld θ and velocity u, which make the equations more complicated and complex.Te nonlinearity makes it harder to fgure out how the system will behave and investigate it, which could lead to instability or chaotic behavior.Fereirra and Villamizar-Roa [10] overcame this difculty and established the global existence of singular solution to the fractional case of equation ( 1) in the pseudomeasure space PM a .
Te PM a spaces were studied in [11] to obtain the singular solutions to NS equations.Te global existence for the solution of equation ( 1) in Besov spaces B .− 1+(3/p) p,∞ with 1 ≤ p < 6 was established by Chen and Miao [12].Tis work extended the result of Cannone-related NS equations, such as [13].Zhu and Zhao recently obtained the decay results for equation (1) in critical Besov spaces [14] and Fourier -Besov spaces [15].More recently, Nie and Zheng [16] have investigated the existence of global solution of equation (1) in Fourier -Herz spaces.Tese results motivate us to study the global well-posedness of equation (1) in the framework of variable-exponent Fourier -Besov spaces.
When θ � χ � 0, equation (1) corresponds to the threedimensional incompressible NS equations.Te scaling invariance is an important feature of the NS equations.Te global existence of solutions for NS equations in the critical homogeneous Sobolev spaces _ H 1/2 was proven by Fujita and Kato [17].Chemin [18] established its existence and uniqueness in homogeneous Besov spaces for 1 ≤ p < ∞ and 1 ≤ c ≤ ∞.Te global unique solution in a more general space, BMO − 1 , is obtained by Koch and Tataru [19].Recently, for the fractional case of NS equations, Ru and Abidin [20] obtained the global well-posedness result under the smallness condition on initial data in variable-exponent Fourier -Besov spaces.Te main purpose of this paper is to extend this result to the solution of equation (1).In this research article, we investigate the existence and analyticity of the global solution of the threedimensional micropolar fuid equations in the framework of variable-exponent Fourier -Besov spaces.Tere exist some signifcant diferences between variable-exponent Fourier-Besov spaces and Fourier-Besov spaces.Certain classical results, such as Young's inequality and the multiplier theorem, are not applicable within the context of variable-exponent Fourier-Besov spaces.Due to this setting, it becomes difcult to determine the well-posedness of equations within these spaces.Tis work primarily utilizes the tools outlined in Sections 2 and 3, together with Banach's contraction mapping concept, to study the global wellposedness of the micropolar fuid equations in variableexponent Fourier-Besov spaces.
Te constant exponent Fourier-Besov space can be traced back to the research of Konieczny and Yoneda [21], which focuses on the study of dispersion efect of Coriolis for the NS equations.Moreover, Iwabuchi [22] introduced Fourier-Herz spaces to study Keller-Segel system, which is a particular case of Fourier-Besov spaces.
Te origin of the variable-exponent Lebesgue space L q(•) can be followed back to Orlicz [23].Te researchers, such as Musielak, Nakano, and Zhikov, contributed to its further development.Section 1.2 of [24] provides a brief overview of the early development of the theory of variable-exponent function spaces.Kovaik and Rákosik [25], Cruz-Uribe [26], Diening [27], and Fan and Zhao [28] are considered the pioneers of modern theory of variable-exponent function space.For other details related to the variable-exponent Besov space and variableexponent Fourier-Besov space, see [29][30][31][32] and the references therein.One of the primary motivations for the development of variable-exponent theory is the mathematical modelling of electrorheological fuids, which can be seen in [33].In Section 3, we will give the defnitions of various variable-exponent function spaces.
Te following symbols are introduced for the convenience of description.Let Y, Z be Banach spaces; in this part and afterwards, we write For the sake of convenience, we assume that κ � μ � 1 and χ � ] � 1/2.Tis work is arranged in the following pattern: basic concepts of variable-exponent function space are given in Section 2. Te equivalent integral form of equation ( 1) is presented in Section 3. In Section 4, we give the linear estimate, and in the last section, Section 5, we present the proof of Teorems 9 and 11.

Preliminaries
In this section, we give some basic defnitions related to the variable-exponent function space [30] and some important propositions that are helpful to prove our main theorems.Defnition 1.We defne the Lebesgue space in the framework of variable exponent L q(y) by the set with the following Luxemburg-Nakano norm: where q(y) ∈ P 0 and P 0 is the set of all measurable function q: R n ⟶ R + such that 2 Journal of Mathematics 0 < q − � essinf y∈R n q(y), esssup ) is a Banach space.To diferentiate between the variable exponent and the constant exponents, we indicate the variable exponents with q(•) and constant exponents with p. where Notice that For the sake of guarantee that the Hardy-Littlewood maximal operator M is bounded by L q(•) (R n ), we suppose that the variable-exponent function q(•) satisfes the following conditions: (1) If there exists a positive constant c log , such that Ten, q(•) is said to be locally log-Hölder continuous.
(2) If there exists some constant p ∞ independent of y, such that Ten, q(•) is said to be globally log-Hölder continuous.Consider a set, consisting of all functions satisfying conditions 1 and 2, which is termed as the K log (R n ) space.
Proposition 8 (See [34]).Let g be a continuous bilinear mapping from A × A to A, where A is a Banach space and ε > 0 such that For y ∈ g(0, ε) ⊆ A, where g is the ball with origin as a centre and radius ε, then where y obeys the property of uniqueness in the ball g(0, 2ε).

Equivalent Integral Form of Equation (1)
Following [10], the corresponding linear equations to equation ( 25) are as follows: Let G(t) denote the solution operator to the above linear equation ( 21), then where with Applying P, the Leray projector, to equation ( 1) to remove the pressure term π, we have the following: Let Furthermore, we defne According to Duhamel principle, the solution to equation (25) can be deduced to the solution W of the following integral equation:

Linear Estimate
For the sake of the proofs of the main theorems, we have to make a priori estimate by considering equation (28).Terefore, the following lemma is constructed.

Main Results
In this section, we give the main results related to the global well-posedness and Gevrey class regularity of the solution to equation (1).
Ten, there exists a positive constant ε 0 such that for any initial data (ω 0 , θ 0 ) in FB there is a unique global solution (ω, θ) in such that Proof.Defne the set and consider the following mapping: We need to prove that the mapping shown above is a contraction mapping.By the above mapping, we obviously have To estimate I 1 , assume the hypothesis q(•) ≥ 2, and using Lemma 8, we have Hence, we have To estimate I 2 , utilizing Proposition 6, we have Using Proposition 7, the above inequality yields Next, we also have Terefore, 

Journal of Mathematics
Similarly, In this way, Terefore, we have we can obtain where By using Proposition 8, one can easily obtain the global solution for under the condition of small initial data.
Next, we adopt the Gevrey class technique, which has its origin in the investigation of Foias and Temam [35] related to the NS equations to ensure that the solution to equation ( 1) is spatially analytic.Tis method has the obvious beneft of making it easier to get rid of cumbersome estimates of higher-order derivatives.For further details on this issue, see [35][36][37][38] and the references therein.In this work, we use the idea of Bae et al. [37] to establish the Gevrey class regularities for equation (1).
First, we give the following auxiliary lemma; we can easily obtain the spatial analyticity for the solution to (1).□ Lemma 10 (see [10]).Let 0 < s ≤ t < ∞ and for any x, y ∈ R n , we have the following: Ten, there exists a constant ε 0 ′ > 0 such that for the initial data We can easily obtain that ) Te rest of the proof follows a similar pattern established in the preceding proof of Teorem 9. Terefore, the remaining part is omitted.