Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 58, pp. 1–29. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.58 GLOBAL UNIQUE SOLUTION FOR 3D INCOMPRESSIBLE INHOMOGENEOUS MAGNETO-MICROPOLAR EQUATIONS WITH DISCONTINUOUS DENSITY XIAO SONG, CHENHUA WANG, XIAOJIE WANG, FUYI XU Abstract. This article concerns the Cauchy problem of the incompressible inhomogeneous magneto-micropolar equations in R3. We first prove the global solvability of the model when the initial density is bounded from above and below with positive constants and the initial velocity, angular velocity, and magnetic field in a critical Besov spaces are sufficiently small. Then we obtain the Lipschitz regularity for the fluid velocity, magnetic field, and angular ve- locity by exploiting some extra time-weighted energy estimates. We show the uniqueness of the constructed global solutions by the duality approach. 1. Introduction and main results The magnetohydrodynamic model is often regarded as a reasonable description of the dynamics of a plasma, but it cannot describe fluids with microstructure and such complex fluids may be of different shape. Moreover, they may rotate, independently of the rotation and movement of the fluid. Therefore, it is necessary to refine the fluid models. Ahmadi and Shahinpoor [?] proposed a magneto-micropolar fluid model, which extends the valid domain of MHD equations and accounts for microrotation effect. If the density of the fluid cannot be considered a constant quantity, a consequence of the complex structure of the flow due to, for example a mixture of fluids or pollution. This requires that we look at the density as a nonnegative unknown function which has constant values along the stream line. The simplest model which can capture such a physical property is the so-called incompressible inhomogeneous magneto-micropolar equations [?, ?, ?]: ∂tρ+ div(ρu) = 0, ∂t(ρu) + div(ρu⊗ u)− (µ+ µr)∆u+∇P = curlH ×H + 2µr curlω, ρ∂tω + ρ(u · ∇)ω − (ca + cd)∆ω − (c0 + cd − ca)∇ divω + 4µrω = 2µr curlu, ∂tH − ν∆H = curl(u×H), div u = divH = 0, (ρ, u, ω,H)|t=0 = (ρ0, u0, ω0, H0), (1.1) where ρ = ρ(x, t), u = u(x, t), ω = ω(x, t), H = H(x, t), and P = P (x, t) describe the density, the velocity field, the angular velocity vector of rotation of particles, the magnetic field and the pres- sure, respectively. The positive constants µ, µr, c0, ca, cd and ν characterize isotropic properties of the fluid; µ is the usual Newtonian dynamic viscosity; µr represent the dynamic microrotation viscosity; c0, ca and cd are called coefficients of angular viscosities; ν is the magnetic diffusivity. These new viscosities are related to the asymmetry of the stress tensor and in consequence related to the appearance of the field of internal rotation ω. Furthermore, these positive constants satisfy c0 + cd > ca. Without lose of generality, we take µ = µr = 1 2 , ca + cd = 1, c0 + cd − ca = 1 and 2020 Mathematics Subject Classification. 35Q35, 35A01, 35A02. Key words and phrases. Discontinuous density; global well-posedness; critical Besov space; magneto-micropolar equation. ©2025. This work is licensed under a CC BY 4.0 license. Submitted December 22, 2024. Published June 2, 2025. 1 2 X. SONG, C. WANG, X. WANG, F. XU EJDE-2025/58 ν = 1. The body force curlH ×H = H · ∇H − 1 2∇(|H|2) and curl(u×H) = H · ∇u− u · ∇H if div u = divH = 0. Thus, taking π = P + |H|2 2 and using transport equation and incompressible condition, we first transform the original system (??) into the form ∂tρ+ u · ∇ρ = 0, ρ(∂tu+ u · ∇u)−∆u+∇π = H · ∇H + curlω, ρ(∂tω + u · ∇ω)−∆ω −∇ divω + 2ω = curlu, ∂tH + u · ∇H −∆H = H · ∇u, div u = divH = 0, (ρ, u, ω,H)|t=0 = (ρ0, u0, ω0, H0). (1.2) We would like to point out that the model (??) includes several important models as special cases. In what follows, we briefly review some of the existing results for the system and related models. If we ignore micro-rotational velocity, i.e. ω = 0, the system (??) reduces to the inhomogeneous incompressible MHD equations. When the initial density is away from zero and is close enough to a positive constant, local existence of strong solutions was recently considered by Abidi and Hmidi [?]. They also proved global existence of strong solutions when the initial data are small in some Sobolev spaces and Besov spaces. Global existence of strong solutions with small initial data in critical Besov spaces was considered by Abidi and Paicu [?]. Precisely, [?] allowed variable viscosity and conductivity coefficients, strongly oscillating initial velocity and magnetic field but required an essential assumption that there is no vacuum. The results in [?, ?] have been extended by Zhai, Li and Yan [?] in the critical functional framework to the model with one component of the initial velocity and magnetic being large. We note that all the previously mentioned works [?, ?, ?] assume the density to be at least uniformly continuous. When ρ0 ∈ L∞(R3) is bounded above and below by positive constants, and the initial velocity and magnetic field are sufficiently small in Hs(R3) with 1 2 < s ≤ 1, Chen, Guo, and Zhai [?] demonstrated the global existence and uniqueness of the solution. Notably, the Hs functional setting used here is not critical with respect to scaling. Xu, Qiao and Fu [?] further lowered the regularity assumptions on the initial data from [?] to a critical framework, and established the global existence of solutions in critical Besov spaces. The uniqueness issue of the constructed global solutions was presented in [?]. When H = 0, system (??) becomes the inhomogeneous incompressible asymmetric fluids. Local well-posedness of strong solution to the system was constructed by Lukaszewicz [?] when the initial density is strictly separated from zero. Using a spectral semi-Galerkin method, when the initial density is bounded and away from zero, Boldrini et al. [?] proved the unique local solvability of strong solution and some global existence results for small data. In particular, uniqueness results in [?, ?], though, are available only if one requires much more regularity for the solutions. Recently, Braz e Silva et al. [?] constructed the global existence and uniqueness of the solution to the 3D system when the initial density ρ0 ∈ L∞(R3), 0 < α < ρ0 < β < ∞ and the initial velocities (u0, ω0) ∈ H1(R3)×H1(R3) satisfy a suitable smallness condition. More recently, Qian et al. [?] further improved the result of [?] by relaxing the regularity restriction on the initial (angular) velocity and obtained global existence of solutions of the model when initial density is bounded from above and below by some positive constants and initial (angular) velocity in critical Besov spaces are sufficiently small. In [?], authors proved the uniqueness issue for the constructed global solutions in [?]. When ρ is constant, which means the fluid is homogeneous, the magneto-micropo-lar fluid equations have been extensively studied, such as existence and stability of solutions [?, ?], large time behavior of solutions [?, ?], blow-up criterion of solutions [?, ?] and so on. Because of its wide applicability in physics and mathematical importance, the system (??) has attracted considerable interests. In the 3D case, Zhang and Zhu [?] showed the global existence of strong solution to the initial boundary value problem with vacuum provided that some smallness condition holds. Yang and Zhong [?] constructed the global existence of strong solutions to the Cauchy problem for the system with initial data being of small norm but allowed to have vacuum EJDE-2025/58 GLOBAL SOLUTION FOR MAGNETO-MICROPOLAR EQUATIONS 3 and large oscillations. Zhong [?] proved the existence and exponential decay of global strong solutions to the system with some smallness conditions in a bounded simply connected smooth domain with homogeneous Dirichlet boundary conditions for the velocity and micro-rotational velocity and Navier-slip boundary condition for the magnetic field. In the 2D case, Zhong [?, ?] studied the existence of local and global strong solutions to the Cauchy problem with the arbitrarily large initial data and vacuum. Here, it should be pointed out that all the previously mentioned works in [?, ?, ?, ?, ?] for the system (??) assume the density to be at least uniformly continuous, excluding cases where the density has discontinuities across a hypersurface. However, in many practical applications, such as modeling mixtures of two fluids, we are often interested in fluids with piecewise constant densities. Motivated by [?, ?, ?, ?, ?], in this paper, we intend to investigate the unique global solvability of the 3D Cauchy problem (??) in more general scenario where the initial density is bounded above and below by positive constants and the initial (angular) velocity and magnetic field in critical Besov spaces are sufficiently small. Here, we first note that the system (??) possesses a scaling invariance. Namely, if (ρ, u, ω,H) is the solution of the system (??) corresponding to the initial data (ρ0, u0, ω0, H0), then (ρλ, uλ, ωλ, Hλ, πλ)(t, x) ≜ (ρ(λ2t, λx), λu(λ2t, λx), λω(λ2t, λx), λH(λ2t, λx), λ2π(λ2t, λx)) is also a solution with initial data (ρ0λ, u0λ, ω0λ, H0λ)(x) ≜ (ρ0(λx), λu0(λx), λω0(λx), λH0(λx)) for all λ > 0, if we discard the terms curlu, curlω and the damping term ω, which is the same as the inhomogeneous incompressible Navier-Stokes system and MHD system. Thus, one can define the critical spaces which is invariant under the above scaling. For the system (??), the incompressibility condition on the convection velocity field in the density transport equation ensures that ∥ρ∥L∞ = ∥ρ0∥L∞ . (1.3) Our first main result on the existence of global strong solution to the system (??) then reads as follows. Theorem 1.1. Given ρ, ρ ∈ (0,∞), assume (ρ0, u0, ω0, H0) satisfy ρ ≤ ρ0 ≤ ρ, (u0, ω0, H0, curlu0 − 2w0) ∈ Ḃ 1/2 2,1 × Ḃ 1/2 2,1 × Ḃ 1/2 2,1 × Ḃ −1/2 2,1 . (1.4) Then there exist a constant ε0 > 0 depending only on ρ, ρ such that if E0 def = ∥(u0, ω0, H0)∥Ḃ1/2 2,1 + ∥curlu0 − 2w0∥Ḃ−1/2 2,1 ≤ ε0, (1.5) system (??) admits a global solution (ρ, u, ω,H) with ρ ∈ L∞ t ( R+;L∞(R3) ) and with (u, ω) in C ( [0,∞); Ḃ 1/2 2,1 ) ∩ L2 ( R+; Ḃ 3/2 2,1 ) ×C ( [0,∞); Ḃ 1/2 2,1 ) ∩ L2 ( R+; Ḃ 3/2 2,1 ) satisfying for (t, x) ∈ R+ ×R3 ρ ≤ ρ ≤ ρ, (1.6) and ∥(u, ω,H)∥ L̃∞ t (Ḃ 1/2 2,1 ) + ∥(u, ω,H)∥ L̃2 t (Ḃ 3/2 2,1 ) + ∥ √ t(u, ω,H)∥ L̃∞ t (Ḃ 3/2 2,1 ) + ∥ √ t∂t(u, ω,H)∥ L̃2 t (Ḃ 1/2 2,1 ) + ∥ √ t∇(u, ω,H)∥ L̃2 t (Ḃ 1/2 6,1 ) + ∥tDt(u, ω,H)∥ L̃∞ t (Ḃ 1/2 2,1 ) + ∥t∇Dt(u, ω,H)∥ L̃2 t (Ḃ 1/2 2,1 ) ≤ CE0, (1.7) where Dt = ∂t + u · ∇ denotes the material derivative. Our second main result on the uniqueness of strong solution is the following theorem. Theorem 1.2. Let (ρ1, u1, H1, π1) and (ρ2, u2, H2, π2) be two solutions of the system (??) on [0, T ]× R3 constructed by Theorem ?? corresponding to the same initial data. Then (ρ1, u1, ω1, H1, π1) ≡ (ρ2, u2, ω,H2, π2) on [0, T ]× R3. (1.8) 4 X. SONG, C. WANG, X. WANG, F. XU EJDE-2025/58 Remark 1.3. Compared with the existing results in [?, ?, ?, ?, ?], we obtain the existence and uniqueness of the global solution to system (??) when the initial density is bounded from above and below by some positive constants. In particular, the initial velocity, angular velocity, and magnetic field have critical regularity indices. The structure of this article is as follows. In the next section, we review fundamental concepts related to Littlewood-Paley decomposition, Besov and Lorentz spaces, product estimates, and relevant propositions. Section 3 is dedicated to the proof of Theorem ??. The last section is focused on proving the uniqueness result stated in Theorem ??. Throughout this paper, we assume C be a positive generic constant that may vary at different places and denote A ≤ CB by A ≲ B. Let f and g be two operators; we denote [f, g] = fg − gf , the commutator between f and g. We always define D Dt = ∂t + u · ∇ and u̇ = ∂tu+ u · ∇u to be the material derivative. 2. Preliminaries Let S(Rd) be the Schwartz class of rapidly decreasing function. Given f ∈ S(Rd), its Fourier transform Ff = f̂ is defined by f̂(ξ) = ∫ Rd e−ix·ξf(x) dx. Let (χ, φ) be a couple of smooth functions valued in [0, 1] such that χ is supported in the ball {ξ ∈ Rd : |ξ| ≤ 4 3}, φ is supported in the shell {ξ ∈ Rd : 3 4 ≤ |ξ| ≤ 8 3}, φ(ξ) := χ(ξ/2)− χ(ξ) and χ(ξ) + ∑ j≥0 φ(2−jξ) = 1, ∀ξ ∈ Rd, ∑ j∈Z φ(2−jξ) = 1, ∀ξ ∈ Rd \ {0}. The homogeneous frequency localization operators ∆̇j and Ṡj are defined by ∆̇jf := F−1(φ(2−j ·)Ff), Ṡjf := ∑ q≤j−1 ∆̇qf for j ∈ Z. We denote the space S ′ h(Rd) by the following subset of the dual space of S ′(Rd) = {f ∈ S(Rd) : Dαf̂(0) = 0,where α is multi-index}, it also can be identified by the quotient space of S ′(Rd)/P with the polynomial space P. The formal equality f = ∑ j∈Z ∆̇jf holds for f ∈ S ′ h(Rd) and is called the homogeneous Littlewood-Paley decomposition. One easily verifies that with our choice of φ, ∆̇j∆̇qf ≡ 0 if |j − q| ≥ 2 and ∆̇j(Ṡq−1f∆̇qf) ≡ 0 if |j − q| ≥ 5. Let us recall the definition of the homogeneous Besov spaces and some properties (see [?, ?, ?, ?, ?]). Definition 2.1. Let S ′ be the space of all tempered distributions. For s ∈ R, 1 ≤ p ≤ ∞, we define the homogeneous Besov space Ḃs p,1 to be Ḃs p,1 = { f ∈ S ′ h : ∥f∥Ḃs p,1 < ∞ } , with S ′ h = { f ∈ S ′ : ∑ j∈Z ∆̇jf = f ∈ S ′} and ∥f∥Ḃs p,1 = ∑ j∈Z 2js∥∆̇jf∥Lp . We introduce the Besov-Chemin-Lerner space L̃q T (Ḃ s p,r) which is defined in [?]. EJDE-2025/58 GLOBAL SOLUTION FOR MAGNETO-MICROPOLAR EQUATIONS 5 Definition 2.2. Let s ≤ d p (respectively s ∈ R), (r, ρ, p) ∈ [1,+∞]3 and T ∈ (0,+∞]. We define L̃ρ T (Ḃ s p,r) as the completion of C([0, T ];S′ h) with the norm ∥f∥L̃ρ T (Ḃs p,r) ≜ ∥∥∥2js∥∆̇jf(t)∥Lρ(0,T ;Lp) ∥∥∥ ℓr ≤ ∞, with the usual change if r = ∞. Obviously, L̃1 T Ḃ s p,1 = L1 T Ḃ s p,1, By a direct application of Minkowski’s inequality, we have the following relations between these spaces Lρ T Ḃ s p,r ↪→ L̃ρ T Ḃ s p,r, r ≥ ρ, L̃ρ T Ḃ s p,r ↪→ Lρ T Ḃ s p,r, ρ ≥ r. The following Bernstein’s inequality will be used frequently. Definition 2.3. Given f a measurable function on a measure space (X,µ) and 1 ≤ p, r ≤ ∞, we define ∥f̃∥Lp,r(X,µ) =  (∫∞ 0 ( t1/pf∗(t) )r dt t )1/r if r < ∞, supt>0 t 1/pf∗(t) if r = ∞, (2.1) where f∗(t) := inf { s ≥ 0 : µ({|f | > s}) ≤ t } . The set of all f with ∥f̃∥Lp,r(X,µ) < ∞ is called the Lorentz space with p and r. Lemma 2.4 ([?]). Let 1 ≤ p ≤ q ≤ +∞. Assume that f ∈ Lp(Rd), then for any γ ∈ (N ∪ {0})d, there exist constants C1, C2 independent of f , j such that supp f̂ ⊆ {|ξ| ≤ A02 j} ⇒ ∥∂γf∥q ≤ C12 j|γ|+jN( 1 p− 1 q )∥f∥p; supp f̂ ⊆ {A12 j ≤ |ξ| ≤ A22 j} ⇒ ∥f∥p ≤ C22 −j|γ| sup |β|=|γ| ∥∂βf∥p. The usual product is continuous in many Besov spaces. The proof of the following lemma can be found in [?, section 4.4] (see in particular inequality (28) page 174). ∥fg∥Ḃs p,r ≤ C∥f∥L∞∥g∥Ḃs p,r + C∥g∥L∞∥f∥Ḃs p,r , if s > 0; ∥fg∥ Ḃ s1+s2− d p p,r ≤ C∥f∥Ḃs1 p,r ∥g∥Ḃs2 p,∞ , if s1, s2 < d p , and s1 + s2 > 0; ∥fg∥Ḃs p,r ≤ C∥f∥Ḃs p,r ∥g∥ Ḃ d/p p,∞∩L∞ , if |s| < d p . Some embedding properties and interpolation inequalities about the Besov spaces can be found in [?] are in order. Lemma 2.5. • For each p ∈ [1,∞] we have the continuous embedding Ḃ0 p,1 ↪→ Lp ↪→ Ḃ0 p,∞. • If s ∈ R, 1 ≤ p1 ≤ p2 ≤ ∞ and 1 ≤ r1 ≤ r2 ≤ ∞, then Ḃs p1,r1 ↪→ Ḃ s−d( 1 p1 − 1 p2 ) p2,r2 . • The space Ḃ d/p p,1 is continuously embedded in the set of bounded continuous functions (going to 0 at infinity if p < ∞). • For 1 ≤ p, r1, r2, r ≤ ∞, σ1 ̸= σ2 and θ ∈ (0, 1), then ∥f∥ Ḃ θσ2+(1−θ)σ1 p,r ≤ C∥f∥1−θ Ḃ σ1 p,r1 ∥f∥θ Ḃ σ2 p,r2 . Remark 2.6. Since Lp,p(X,µ) coincide with the Lebesgue space Lp(X,µ), the Lorentz spaces can be endowed with the quasi-norm ∥f∥Lp,r(X,µ) = p1/r ( ∫∞ 0 ( s µ({|f | > s})1/p )r ds s )1/r if r < ∞, sups>0 s µ({|f | > s})1/p if r = ∞. (2.2) 6 X. SONG, C. WANG, X. WANG, F. XU EJDE-2025/58 The following properties of the Lorentz spaces may be found in [?]. Proposition 2.7. For 1 < p, p1, p2 < ∞ and 1 ≤ r, r1, r2 ≤ ∞, we have (1) Interpolation. For all 1 ≤ r, q ≤ ∞ and θ ∈ (0, 1), we have( Lp1 ( R+;L q(R3) ) ;Lp2 ( R+;L q(R3) )) θ,r = Lp,r ( R+;L q(R3) ) , where 1 < p1 < p < p2 < ∞ are such that 1 p = (1−θ) p1 + θ p2 . (2) Embedding. Lp,r1 ↪→ Lp,r2 if r1 ≤ r2, and Lp,p = Lp. (3) Hölder’s inequality. If 1 p = 1 p1 + 1 p2 , and 1 r = 1 r1 + 1 r2 , we have ∥fg∥Lp,r ≲ ∥f∥Lp1,r1 ∥g∥Lp2,r2 . This still holds for the pairs (1,1) and (∞,∞) with the convention L1,1 = L1 and L∞,∞ = L∞. (4) For any α > 0 and non-negative measurable function f , we have ∥fα∥Lp,r = ∥f∥αLpα,rα . (5) For any t > 0, we have ∥t−α1R+∥L 1 α ,∞ = 1. 3. Proof of Theorem ?? For a clear presentation, we split the proof of Theorem ?? into three propositions. Proposition 3.1. Under the assumptions of (??) and (??), let (ρ, u, ω,H) be a smooth enough solution of system (??). Then we have (??) and ∥(u, ω,H)∥ L̃∞ t (Ḃ 1/2 2,1 ) + ∥∇(u, ω,H)∥ L̃2 t (Ḃ 1/2 2,1 ) ≤ CE0 for t ∈ R+. (3.1) Proof. Employing the classical theory on transport equation, (??) and (??), we can easily obtain (??) for t > 0. To bound (??), we first consider the following linear coupled system of (uj , ωj , Hj , πj): ρ∂tuj + ρu · ∇uj −∆uj +∇πj = H · ∇Hj + curlωj , ρ∂tωj + ρu · ∇ωj −∆ωj −∇ divωj + 2ωj = curluj , ∂tHj + u · ∇Hj −∆Hj = H · ∇uj , div uj = divHj = 0, (uj , ωj , Hj)|t=0 = (∆ju0,∆jω0,∆jH0), (3.2) where {mj}j∈Z smooth functions satisfying suppmj ⊂ 2jC and ∥{2js∥mj∥Lp}j∈Z∥lr < ∞ with 0 < s < d p (1 ≤ p, r ≤ ∞) or s = d p (r = 1). Then we deduce from the uniqueness of local smooth enough solution to the system (??) on [0, T ∗) that u = ∑ j∈Z uj , ω = ∑ j∈Z ωj , H = ∑ j∈Z Hj , π = ∑ j∈Z πj , (3.3) which together the low-high frequency decomposition and Lemma ?? yields that ∥∆̇j(u, ω,H)∥L∞ t (L2) + ∥∇∆̇j(u, ω,H)∥L2 t (L 2) ≲ ∑ j′>j ( ∥∆̇j(uj′ , ωj′ , Hj′)∥L∞ t (L2) + ∥∇∆̇j(uj′ , ωj′ , Hj′)∥L2 t (L 2) ) + 2−j ∑ j′≤j ( ∥∇∆̇j(uj′ , ωj′ , Hj′)∥L∞ t (L2) + ∥∇2∆̇j(uj′ , ωj′ , Hj′)∥L2 t (L 2) ) ≲ ∑ j′>j ( ∥(uj′ , ωj′ , Hj′)∥L∞ t (L2) + ∥∇(uj′ , ωj′ , Hj′)∥L2 t (L 2) ) + 2−j ∑ j′≤j ( ∥∇(uj′ , ωj′ , Hj′)∥L∞ t (L2) + ∥∇2(uj′ , ωj′ , Hj′)∥L2 t (L 2) ) . (3.4) Now, we first bound the terms ∥(uj , ωj , Hj)∥L∞ t (L2) and ∥∇(uj , ωj , Hj)∥L2 t (L 2) in the above in- equality as follows. To begin with, taking the L2-scalar product of the first equation of the system EJDE-2025/58 GLOBAL SOLUTION FOR MAGNETO-MICROPOLAR EQUATIONS 7 (??) with uj , the second equation with ωj and the third equation with Hj respectively and the combining together, and noting that (ωj , curluj) + (uj , curlωj) ≤ 2∥ωj∥2L2 + 1 2 ∥∇uj∥2L2 , we conclude from the continuity equation (??), that 1 2 d dt ∫ R3 (ρ|uj |2 + ρ|ωj |2 + |Hj |2)dx+ ∫ R3 ( 1 2 |∇uj |2 + |∇ωj |2 + |∇Hj |2)dx+ ∫ R3 |divωj |2dx Integrating the above inequality over [0, t], we have 1 2 ∥(√ρuj , √ ρωj , Hj)∥2L∞ t (L2) + 1 2 ∥∇uj∥2L2 t (L 2) + ∥∇ωj∥2L2 t (L 2) + ∥∇Hj∥2L2 t (L 2) + ∥ divωj∥2L2 t (L 2) ≤ 1 2 ∥(√ρ∆̇ju0, √ ρ∆̇jω0, ∆̇jH0)∥2L2 . By (??) and (??), it holds ∥(uj , ωj , Hj)∥L∞ t (L2) + ∥∇(uj , ωj , Hj)∥L2 t (L 2) ≤ C∥∆̇j(u0, ω0, H0)∥L2 ≤ Cdj2 −j/2E0. (3.5) Next, let us turn to bound the terms ∥∇(uj , ωj , Hj)∥L∞ t (L2) and ∥∇2(uj , ωj , Hj)∥L2 t (L 2) in (??). Taking the L2-scalar product of the first equation of the system (??) with ∂tuj , the second equation with ∂tωj and the third equation with ∂tHj respectively, combining together, and using Hölder’s inequality and the embedding Ḣ1/2(R3) ↪→ L3(R3), we have ∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥2L2 + 1 2 d dt ( ∥∇ωj∥2L2 + ∥ divωj∥2L2 + ∥∇Hj∥2L2 ) + 1 4 d dt ( ∥∇uj∥2L2 + ∥ curluj − 2wj∥2L2 ) = ∫ R3 (H · ∇Hj − ρu · ∇uj) · ∂tuj + (H · ∇uj − u · ∇Hj) · ∂tHj dx− ∫ R3 ρu∇ωj · ∂tωj ≤ C∥(u,H)∥L3∥∇(uj , ωj , Hj)∥L6∥∂t(uj , ωj , Hj)∥L2 ≤ C∥(u,H)∥Ḣ1/2∥∇2(uj , ωj , Hj)∥L2∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥L2 . (3.6) For second order derivatives of (uj , ωj , Hj), it follows from (??), that −∆uj +∇πj = H · ∇Hj + curlωj − ρ∂tuj − ρu · ∇uj , −∆ωj −∇ divωj = curluj − ρ∂tωj − ρu · ∇ωj − 2ωj , −∆Hj = H · ∇uj − ∂tHj − u · ∇Hj . (3.7) Taking the L2 scalar product of equation (??)1 with −∆uj , of equation (??)2 with −∆ωj and of equation (??)3 with −∆Hj , respectively, and then adding the resulting equations, we easily infer that ∥∇2uj∥2L2 + ∥∇2ωj∥2 + ∥∇ divωj∥2 + ∥∇2Hj∥2L2 + 2∥∇ω∥2 = 2(curlωj ,−∆uj) + (ρ∂tuj ,∆uj) + (ρ∂tωj ,∆ωj) + (∂tHj ,∆Hj) + (u · ∇Hj ,∆Hj) + (H · ∇uj ,∆Hj) + (ρu · ∇uj ,∆uj) + (ρu · ∇ωj ,∆ωj), because of (curlu,−∆ω) = (curlω,−∆u). Hence by Cauchy-Schwarz’s and Young’s inequalities, we conclude that ∥∇2(uj , ωj , Hj)∥L2 ≤ C ( ∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥L2 + ∥H · ∇Hj∥L2 + ∥ρu · ∇uj∥L2 + ∥H · ∇uj∥L2 + ∥u · ∇Hj∥L2 + ∥u · ∇ωj∥L2 ) ≤ C ( ∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥L2 + ∥(u,H)∥L3∥∇(uj , ωj , Hj)∥L6 ) ≤ C ( ∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥L2 + ∥(u,H)∥Ḣ1/2∥∇2(uj , ωj , Hj)∥L2 ) . (3.8) We denote T ⋆ 1 ≜ sup { t < T ∗ : ∥(u,H)∥L∞ t (Ḣ1/2) ≤ c1 } . (3.9) 8 X. SONG, C. WANG, X. WANG, F. XU EJDE-2025/58 Then for c1 in (??) being so small that Cc1 ≤ 1 2 , thus we infer for t ≤ T ∗ 1 that ∥∇2(uj , ωj , Hj)∥L2 ≤ C∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥L2 . (3.10) Combining (??) with (??) and Cc1 ≤ 1 2 , for t ≤ T ∗ 1 , we have d dt ( ∥(∇uj , 2∇ωj , 2∇Hj)∥2L2 + ∥(2 divωj , curluj − 2ωj)∥2L2 ) + ∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥2L2 ≤ 0, which together with (??) yields ∥∇(uj , ωj , Hj)∥L∞ t (L2) + ∥∇2(uj , ωj , Hj)∥L2 t (L 2) + ∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥L2 t (L 2) ≤ C∥(∇uj ,∇ωj ,∇Hj , curluj − 2ωj)t=0∥L2 ≤ ∥∇∆̇ju0,∇∆̇jω0,∇∆̇jH0, ∆̇j(curlu0 − 2ω0)∥L2 ≤ Cdj2 j/2E0. (3.11) Substituting (??) and (??) into (??), we have ∥∆̇j(u, ω,H)∥L∞ t (L2) + ∥∇∆̇j(u, ω,H)∥L2 t (L 2) ≤ Cdj2 −j/2E0, (3.12) which implies that ∥(u, ω,H)∥ L̃∞ t (Ḃ 1/2 2,1 ) + ∥∇(u, ω,H)∥ L̃2 t (Ḃ 1/2 2,1 ) ≤ CE0, for t ≤ T ∗ 1 . Using the continuous embedding Ḃ 1/2 2,1 (R3) ↪→ Ḣ1/2(R3), and taking ε0 in (??) so small that CE0 ≤ Cε0 ≤ c1 2 for c1 given by (??). Thus, we deduce by using a continuous argument, that T ∗ 1 determined by (??) equals any number smaller than T ∗. That is, ∥(u, ω,H)∥ L̃∞ t (Ḃ 1/2 2,1 ) + ∥∇(u, ω,H)∥ L̃2 t (Ḃ 1/2 2,1 ) ≤ CE0 for t ≤ T ∗. (3.13) Finally, we shall show T ∗ = +∞. Let c2 be a small enough positive constant, which will be determined later on. We define T ≜ max { t ∈ [0, T ∗) : ∥(u, ω,H)∥ L̃∞ t (Ḃ 1/2 2,1 ) + ∥∇(u, ω,H)∥ L̃2 t (Ḃ 1/2 2,1 ) ≤ c2 } . (3.14) We claim that T = +∞ provided that there holds (??). Indeed, taking c2 = 2Cε0, for t ≤ T , we deduce from (??) that ∥(u, ω,H)∥ L̃∞ t (Ḃ 1/2 2,1 ) + ∥∇(u, ω,H)∥ L̃2 t (Ḃ 1/2 2,1 ) ≤ CE0 ≤ Cε0 ≤ c2 2 , which contradicts (??). Thus, we conclude that T = T ∗ = ∞, and (??) hold. This completes the proof. □ Proposition 3.2. Under the assumptions of Proposition ??, we have ∥ √ t∇(u, ω,H)∥ L̃∞ t (Ḃ 1/2 2,1 ) + ∥ √ t∂t(u, ω,H)∥ L̃2 t (Ḃ 1/2 2,1 ) ≤ CE0 for t ∈ R+. (3.15) Proof. Similar to the derivation of (??), we have ∥ √ t∆̇j∇(u, ω,H)∥L∞ t (L2) + ∥ √ t∆̇j∂t(u, ω,H)∥L2 t (L 2) ≲ ∑ j′>j ( ∥ √ t∇(uj′ , ωj′ , Hj′)∥L∞ t (L2) + ∥ √ t∂t(uj′ , ωj′ , Hj′)∥L2 t (L 2) ) + 2−j ∑ j′≤j ( ∥ √ t∇2(uj′ , ωj′ , Hj′)∥L∞ t (L2) + ∥ √ t∇∂t(uj′ , ωj′ , Hj′)∥L2 t (L 2) ) . (3.16) Let us now bound the last two terms in the above inequality. For the terms ∥ √ t∇(uj , ωj , Hj)∥L∞ t (L2) and ∥ √ t∂t(uj , ωj , Hj)∥L2 t (L 2), it follows from (??) that 1 2 d dt ∥∇(uj , ωj , Hj)∥2L2 + κ 2 d dt ∥ divωj∥2L2 + ∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥2L2 ≤ C∥(u,H)∥L∞∥∇(uj , ωj , Hj)∥L2∥∂t(uj , ωj , Hj)∥L2 ≤ C∥(u,H)∥2L∞∥∇(uj , ωj , Hj)∥2L2 + 1 2 ∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥2L2 . EJDE-2025/58 GLOBAL SOLUTION FOR MAGNETO-MICROPOLAR EQUATIONS 9 Multiplying the above inequality by t and using Ḃ 3/2 2,1 (R3) ↪→ L∞(R3) in Lemma ?? we have d dt (∥ √ t∇(uj , ωj , Hj)∥2L2 + ∥ √ tdivωj∥2L2) + ∥ √ t( √ ρ∂tuj , √ ρ∂tωj , ∂tHj)∥2L2 ≤ C∥∇(uj , ωj , Hj)∥2L2 + C∥(u,H)∥2 Ḃ 3/2 2,1 ∥ √ t∇(uj , ωj , Hj)∥2L2 . Applying Gronwall’s inequality and then using (??) and (??), we have ∥ √ t∇(uj , ωj , Hj)∥2L∞ t (L2) + ∥ √ tdivωj∥2L∞ t (L2) + ∥ √ t( √ ρ∂tuj , √ ρ∂tωj , ∂tHj)∥2L2 t (L 2) ≤ ∥∇(uj , ωj , Hj)∥2L2 t (L 2) exp ( C∥(u,H)∥2 L2 t (Ḃ 3/2 2,1 ) ) ≤ Cd2j2 −jE0, which together with (??) yields that ∥ √ t∇(uj , ωj , Hj)∥L∞ t (L2) + ∥ √ t∇2(uj , Hj , ωj)∥L2 t (L 2) + ∥ √ t( √ ρ∂tuj , √ ρ∂tωj , ∂tHj)∥L2 t (L 2) ≤ Cdj2 −j/2E0. (3.17) On the other hand, in order to bound ∥ √ t∇2(uj , ωj , Hj)∥L∞ t (L2) and ∥ √ t∇∂t(uj , ωj , Hj)∥L2 t (L 2) in (??), applying ∂t to the former three equations of the system (??) yields that ρ∂2 t uj + ρu · ∇∂tuj −∆∂tuj +∇∂tπj = −ρtDtuj − ρut · ∇uj + ∂t(H · ∇Hj) + ∂t curlωj , ρ∂2 t ωj + ρu · ∇∂tωj −∆∂tωj −∇ div ∂tωj + 2∂tωj = −ρtDtωj − ρut · ∇ωj + ∂t curluj , ∂2 tHj + u · ∇∂tHj −∆∂tHj = −ut · ∇Hj + ∂t(H · ∇uj). (3.18) Taking the L2-scalar product of the first equation of the system (??) with ∂tuj , the second equation with ∂tωj and the third equation with ∂tHj respectively, and the combining together, we obtain 1 2 d dt ∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥2L2 + ∥∇∂t(uj , ωj , Hj)∥2L2 = − ∫ R3 ρtDtuj∂tuj dx− ∫ R3 ρut · ∇uj∂tuj dx+ ∫ R3 ∂t(H · ∇Hj)∂tuj dx − ∫ R3 ρtDtωj∂tωj dx− ∫ R3 ρut · ∇ωj∂tωj dx− ∫ R3 ut · ∇Hj∂tHj dx + ∫ R3 ∂t(H · ∇uj)∂tHj dx. (3.19) We bound term by term in the above expression as follows. For the term ∫ R3 ρtDtuj∂tuj dx, we infer using the definition of material derivative that∫ R3 ρtDtuj∂tuj dx = ∫ R3 ρt|∂tuj |2dx+ ∫ R3 ρtu · ∇uj∂tuj dx. (3.20) For the term ∫ R3 ρt|∂tuj |2dx in (??), by virtue of the transport equation of (??), integration by parts and the embedding Ḃ 3/2 2,1 ↪→ L∞ in Lemma ??, Hölder’s and Young’s inequalities, we deduce that ∫ R3 ρt|∂tuj |2dx ≤ ∫ R3 div(ρu)|∂tuj |2dx ≤ C ∫ R3 ρu∇|∂tuj |2dx ≤ C∥u∥L∞∥√ρ∂tuj∥L2∥∇∂tuj∥L2 ≤ C∥u∥2 Ḃ 3/2 2,1 ∥√ρ∂tuj∥2L2 + ε∥∇∂tuj∥2L2 . (3.21) 10 X. SONG, C. WANG, X. WANG, F. XU EJDE-2025/58 For the term ∫ R3 ρtu · ∇uj∂tuj dx in (??), by the transport equation of (??) and using integration by parts, we have∫ R3 ρtu∇uj · ∂tuj dx ≤ ∫ R3 div(ρu)u · ∇uj∂tuj dx ≤ ∫ R3 ρu∇u · ∇uj∂tuj dx+ ∫ R3 ρu · u · ∇2uj∂tuj dx+ ∫ R3 ρu · u · ∇uj · ∇∂tuj dx. By using (??), Hölder’s inequality, embeddings Ḃ 3/2 2,1 (R3) ↪→ L∞(R3) and Ḃ 1/2 2,1 (R3) ↪→ L3(R3) in Lemma ??, we obtain∣∣∣ ∫ R3 ρu∇u · ∇uj∂tuj dx ∣∣∣ ≤ C∥u∥L∞∥∇u∥L3∥∇uj∥L6∥√ρ∂tuj∥L2 ≤ C∥u∥2 Ḃ 3/2 2,1 ∥∇2uj∥L2∥√ρ∂tuj∥L2 ≤ C∥u∥2 Ḃ 3/2 2,1 ∥√ρ∂tuj , √ ρ∂tωj , ∂tHj∥2L2 . Similarly, we can deal with the term ∫ R3 ρu · u · ∇2uj∂tuj dx as follows∣∣∣ ∫ R3 ρu · u · ∇2uj∂tuj dx ∣∣∣ ≤ C∥u∥L∞∥u∥L∞∥∇2uj∥L2∥√ρ∂tuj∥L2 ≤ C∥u∥2 Ḃ 3/2 2,1 ∥√ρ∂tuj , √ ρ∂tωj , ∂tHj∥2L2 . Along the same lines, we also have∣∣∣ ∫ R3 ρu · u · ∇uj∇∂tuj dx ∣∣∣ ≤ C∥u∥L∞∥u∥L3∥∇uj∥L6∥∇∂tuj∥L2 ≤ C∥u∥ Ḃ 3/2 2,1 ∥u∥ Ḃ 1/2 2,1 ∥∇2uj∥L2∥∇∂tuj∥L2 ≤ C∥u∥2 Ḃ 3/2 2,1 ∥u∥2 Ḃ 1/2 2,1 ∥√ρ∂tuj∥2L2 + ε∥∇∂tuj∥2L2 . As a result,∣∣∣ ∫ R3 ρtDtuj∂tuj dx ∣∣∣ ≤ C ( 1+ ∥u∥2 Ḃ 1/2 2,1 ) ∥u∥2 Ḃ 3/2 2,1 ∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥2L2 + ε∥∇∂tuj∥2L2 . (3.22) Similarly,∣∣∣ ∫ R3 ρtDtωj∂tωj dx ∣∣∣ ≤ C ( 1+∥u∥2 Ḃ 1/2 2,1 ) ∥u∥2 Ḃ 3/2 2,1 ∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥2L2 +ε∥∇∂tωj∥2L2 . (3.23) For ∫ R3 ρut · ∇uj∂tuj dx, it follows from (??), Hölder’s and Young’s inequalities that∣∣∣ ∫ R3 ρ(ut · ∇uj)∂tuj dx ∣∣∣ ≤ C∥ut∥L2∥∇uj∥L3∥∂tuj∥L6 ≤ C∥ut∥L2∥∇uj∥1/2L2 ∥∇2uj∥1/2L2 ∥∂tuj∥L6 ≤ C∥ut∥2L2∥∇uj∥L2∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥L2 + ε∥∇∂tuj∥2L2 . (3.24) By a similar derivation, we also have∣∣∣ ∫ R3 ρut · ∇ωj∂tωj dx ∣∣∣ ≤ C∥ut∥L2∥∇ωj∥L3∥∂tωj∥L6 ≤ C∥ut∥2L2∥∇ωj∥L2∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥L2 + ε∥∇∂tωj∥2L2 . (3.25) For ∫ R3 ∂t(H · ∇Hj)∂tuj dx, it follows from Hölder’s and Young’s inequalities, (??), that∣∣∣ ∫ R3 ∂t(H · ∇Hj)∂tuj dx ∣∣∣ ≤ ∣∣∣ ∫ R3 (Ht · ∇Hj)∂tuj dx ∣∣∣+ ∣∣∣ ∫ R3 (H · ∇∂tHj)∂tuj dx ∣∣∣ EJDE-2025/58 GLOBAL SOLUTION FOR MAGNETO-MICROPOLAR EQUATIONS 11 ≤ C∥Ht∥L2∥∇Hj∥1/2L2 ∥∇2Hj∥1/2L2 ∥√ρ∂tuj∥L6 + C∥H∥L∞∥∂tuj∥L2∥∇∂tHj∥L2 ≤ C∥Ht∥2L2∥∇Hj∥L2∥∇2H∥L2 + C∥H∥2 Ḃ 3/2 2,1 ∥√ρ∂tuj∥2L2 + ε∥∇∂t(uj , Hj)∥2L2 ≤ C∥(ut, Ht)∥2L2∥∇Hj∥L2∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥L2 + ε∥∇∂t(uj , Hj)∥2L2 + C∥(u,H)∥2 Ḃ 3/2 2,1 ∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥2L2 . The same estimate holds for − ∫ R3 ut · ∇Hj∂tHj dx+ ∫ R3 ∂t(H · ∇uj)∂tHj dx. Then inserting the above estimates into (??) yields that d dt ∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥2L2 + ∥∇∂t(uj , ωj , Hj)∥2L2 ≤ C ( 1 + ∥(u,H)∥2 Ḃ 1/2 2,1 ) ∥u∥2 Ḃ 3/2 2,1 ∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥2L2 + C∥∂t(u,H)∥2L2∥∇(uj , ωj , Hj)∥L2∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥L2 . (3.26) Multiplying the above inequality by t, it follows from (??) that d dt ∥( √ tρ∂tuj , √ tρ∂tωj , √ t∂tHj)∥2L2 + ∥ √ t∇∂t(uj , ωj , Hj)∥2L2 ≤ ∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥2L2 + C∥ 4 √ t∂t(u,H)∥2L2∥∇(uj , ωj , Hj)∥2L2 + C ( ∥ 4 √ t∂t(u,H)∥2L2 + ∥(u,H)∥2 Ḃ 3/2 2,1 ) ∥( √ tρ∂tuj , √ tρ∂tωj , √ t∂tHj)∥2L2 . (3.27) Thus applying (??) together with Gronwall’s inequality gives rise to ∥( √ tρ∂tuj , √ tρ∂tωj , √ t∂tHj)∥2L∞ t (L2) + ∥ √ t∇∂t(uj , ωj , Hj)∥2L2 t (L 2) ≤ C ( ∥(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥2L2 t (L 2) + ∥ 4 √ t∂t(u,H)∥2L2 t (L 2) × ∥∇(uj , ωj , Hj)∥2L∞ t (L2) ) exp ( C∥ 4 √ t∂t(u,H)∥2L2 t (L 2) + C∥(u,H)∥2 L2 t (Ḃ 3/2 2,1 ) ) . (3.28) In what follows, we deal with the term ∥ 4 √ t∂t(uj , Hj)∥L2 t (L 2) in the above inequality. We deduce from (??) and (??) that ∥ 4 √ t∂t(uj , Hj)∥L2 t (L 2) ≤ ∥∂t(uj , Hj)∥1/2L2 t (L 2) ∥ √ t∂t(uj , Hj)∥1/2L2 t (L 2) ≤ CdjE0, which along with (??) ensures that ∥ 4 √ t∂t(u,H)∥2L2 t (L 2) ≤ ∑ j∈Z ∥ 4 √ t∂t(uj , Hj)∥2L2 t (L 2) ≤ CE0. (3.29) By (??), (??) and (??), we arrive at ∥( √ tρ∂tuj , √ tρ∂tωj , √ t∂tHj)∥L∞ t (L2) + ∥ √ t∇∂t(uj , ωj , Hj)∥L2 t (L 2) ≤ Cdj2 j/2 exp(E0)E0 ≤ Cdj2 j/2E0, (3.30) which together with (??) implies that ∥ √ t∇2(uj , ωj , Hj)∥L∞ t (L2) + ∥ √ t∇∂t(uj , ωj , Hj)∥L2 t (L 2) ≤ Cdj2 j/2E0. (3.31) Then plugging (??) and (??) into (??), we finally conclude that (??) holds for any t > 0. This completes the proof. □ Proposition 3.3. Under the assumptions of Proposition ??, we have ∥tDt(u, ω,H)∥ L̃∞ t (Ḃ 1/2 2,1 ) + ∥t∇Dt(u, ω,H)∥ L̃2 t (Ḃ 1/2 2,1 ) + ∥ √ t∇(u, ω,H)∥ L̃2 t (Ḃ 1/2 6,1 ) ≤ CE0 for t ∈ R+. (3.32) 12 X. SONG, C. WANG, X. WANG, F. XU EJDE-2025/58 Proof. As for the derivation of (??), we have ∥t∆̇jDt(u, ω,H)∥L∞ t (L2) + ∥t∆̇j∇Dt(u, ω,H)∥L2 t (L 2) ≲ ∑ j′>j ( ∥tDt(uj′ , ωj′ , Hj′)∥L∞ t (L2) + ∥t∇Dt(uj′ , ωj′ , Hj′)∥L2 t (L 2) ) + 2−j ∑ j′≤j ( ∥t∇Dt(uj′ , ωj′ , Hj′)∥L∞ t (L2) + ∥t∇2Dt(uj′ , ωj′ , Hj′)∥L2 t (L 2) ) . (3.33) Next, we bound the terms ∥tDt(uj , ωj , Hj)∥L∞ t (L2) and ∥t∇Dt(uj , ωj , Hj)∥L2 t (L 2) in the above inequality as follows. Multiplying (??) by t2 gives rise to d dt ∥t(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥2L2 + ∥t∇∂t(uj , ωj , Hj)∥2L2 ≤ 2∥( √ tρ∂tuj , √ tρ∂tωj , √ t∂tHj)∥2L2 + C∥ 4 √ t∂t(u,H)∥2L2∥ √ t∇(uj , ωj , Hj)∥2L2 + C ( ∥ 4 √ t∂t(u,H)∥2L2 + ∥(u,H)∥2 Ḃ 3/2 2,1 ) ∥t(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥2L2 . (3.34) Applying Gronwall’s inequality yields that ∥t(√ρ∂tuj , √ ρ∂tωj , ∂tHj)∥2L∞ t (L2) + ∥t∇∂t(uj , ωj , Hj)∥2L2 t (L 2) ≤ C ( ∥( √ tρ∂tuj , √ tρ∂tωj , √ t∂tHj)∥2L2 t (L 2) + ∥ 4 √ t∂t(u,H)∥2L2 t (L 2) × ∥ √ t∇(uj , ωj , Hj)∥2L∞ t (L2) ) exp ( C∥ 4 √ t∂t(u,H)∥2L2 t (L 2) + C∥(u,H)∥2 L2 t (Ḃ 3/2 2,1 ) ) , which together with (??), (??) and (??) ensures that ∥t(√ρ∂tuj , √ ρ∂tωj∂tHj)∥L∞ t (L2) + ∥t∇∂t(uj , ωj , Hj)∥L2 t (L 2) ≤ Cdj2 −j/2E0. (3.35) It follows from (??), (??)and (??) that ∥tDt(uj , ωj , Hj)∥L∞ t (L2) + ∥t∇Dt(uj , ωj , Hj)∥L2 t (L 2) ≤ ∥t∂t(uj , ωj , Hj)∥L∞ t (L2) + ∥tu · ∇(uj , ωj , Hj)∥L∞ t (L2) + ∥t∇∂t(uj , ωj , Hj)∥L2 t (L 2) + ∥t∇ ( u · ∇(uj , ωj , Hj) ) ∥L2 t (L 2) ≤ ∥t∂t(uj , ωj , Hj)∥L∞ t (L2) + ∥ √ tu∥L∞ t (L∞)∥ √ t∇(uj , ωj , Hj)∥L∞ t (L2) + ∥t∇∂t(uj , ωj , Hj)∥L2 t (L 2) + ∥∇u∥L2 t (L 3)∥t∇(uj , ωj , Hj)∥L∞ t (L6) + ∥u∥L2 t (L ∞)∥t∇2(uj , ωj , Hj)∥L∞ t (L2) ≤ ∥t∂t(uj , ωj , Hj)∥L∞ t (L2) + ∥ √ tu∥ L∞ t (Ḃ 3/2 2,1 ) ∥ √ t∇(uj , ωj , Hj)∥L∞ t (L2) + ∥t∇∂t(uj , ωj , Hj)∥L2 t (L 2) + C∥u∥ L2 t (Ḃ 3/2 2,1 ) ∥t∂t(uj , ωj , Hj)∥L∞ t (L2) ≤ Cdj2 −j/2E0. (3.36) Now we turn to the terms ∥t∇Dt(uj , ωj , Hj)∥L∞ t (L2) and ∥t∇2Dt(uj , ωj , Hj)∥L2 t (L 2). Applying the operator Dt to the former three equations of system (??) yields that ρD2 t uj −∆Dtuj +∇Dtπj = [Dt; ∆]uj − [Dt;∇]πj +Dt(H · ∇Hj) +Dt curlωj ≜ fj +Dt curlωj , ρD2 tωj −∆Dtωj −∇ divDtωj + 2Dtωj = [Dt; ∆]ωj + [Dt;∇ div]ωj +Dt curluj ≜ gj +Dt curluj , D2 tHj −∆DtHj = [Dt; ∆]Hj +Dt(H · ∇uj) ≜ hj , (3.37) where [Dt; ∆] = −∆f · ∇f − 2 ∑3 i=1 ∂iu · ∇∂if and [Dt;∇]f = −∇u · ∇f . Now, taking L2 inner product of (??)1 with D2 t uj , we obtain ∥√ρD2 t uj∥2L2 − ∫ R3 ∆DtujD 2 t uj dx+ ∫ R3 ∇DtπjD 2 t uj dx = ∫ R3 Dt curlwjD 2 t uj dx+ ∫ R3 fjD 2 t uj dx, EJDE-2025/58 GLOBAL SOLUTION FOR MAGNETO-MICROPOLAR EQUATIONS 13 − ∫ R3 ∆Dtuj ·D2 t uj dx = 1 2 d dt ∫ R3 |∇Dtuj |2dx− ∫ R3 ∇Dtuj · [Dt;∇]Dtuj dx, It then follows that 1 2 d dt ∥∇Dtuj∥2L2 + ∥√ρD2 t uj∥2L2 = ∫ R3 Dt curlwjD 2 t uj dx− ∫ R3 ∇DtπjD 2 t uj dx+ ∫ R3 fjD 2 t uj dx+ ∫ R3 ∇Dtuj [Dt;∇]Dtuj dx. Taking the L2 inner product of (??)2 with D2 tωj , we obtain ∥√ρD2 twj∥2L2 + 1 2 d dt ( ∥∇Dtwj∥2L2 + ∥divDtwj∥2L2 + 2∥Dtwj∥2L2 ) = ∫ R3 Dt curlujD 2 twj dx+ ∫ R3 gjD 2 twj dx− ∫ R3 divDtwj [div;Dt]Dtwj dx + ∫ R3 ∇Dtwj [Dt;∇]Dtwj dx, and by testing (??)3 by D2 tHj , we obtain 1 2 d dt ∥∇DtHj∥2L2 + ∥D2 tHj∥2L2 = ∫ R3 hjD 2 tHj dx+ ∫ R3 ∇DtHj [Dt;∇]DtHj dx. Moreover, we have∫ R3 Dt curlujD 2 twj dx+ ∫ R3 Dt curlwjD 2 t uj dx = ∫ R3 [Dt; curl]ujD 2 twj dx+ ∫ R3 [Dt; curl]wjD 2 t uj dx + ∫ R3 curlDtujD 2 twj dx+ ∫ R3 curlDtwjD 2 t uj dx = ∫ R3 [Dt; curl]wjD 2 t uj dx+ ∫ R3 [Dt; curl]ujD 2 twj dx+ ∫ R3 Dtwj curl(D 2 t uj) dx + d dt ∫ R3 curlDtujDtwj dx− ∫ R3 Dt(curlDtuj)Dtwj dx = ∫ R3 [Dt; curl]wjD 2 t uj dx+ ∫ R3 [Dt; curl]ujD 2 twj dx + d dt ∫ R3 curlDtujDtwj dx− ∫ R3 [Dt; curl]DtujDtwj dx. Combining the above several equalities and applying the same argument as in (??), we infer that 1 2 d dt ( ∥∇Dtwj∥2L2 + ∥divDtwj∥2L2 + ∥∇DtHj∥2L2 ) + ∥(√ρD2 t uj , √ ρD2 twj , D 2 tHj)∥2L2 + 1 4 d dt ( ∥∇Dtuj∥2L2 + ∥ divDtuj∥2L2 + ∥ curlDtuj − 2Dtwj∥2L2 ) = ∫ R3 ∇Dtuj [Dt;∇]Dtuj dx+ ∫ R3 ∇DtπjD 2 t uj dx+ ∫ R3 fjD 2 t uj dx + ∫ R3 gjD 2 twj dx+ ∫ R3 ∇Dtwj [Dt;∇]Dtwj dx− ∫ R3 divDtwj [div;Dt]Dtwj dx + ∫ R3 hjD 2 tHj dx+ ∫ R3 ∇DtHj [Dt;∇]DtHj dx+ ∫ R3 [Dt; curl]ujD 2 twj dx + ∫ R3 [Dt; curl]wjD 2 t uj dx− ∫ R3 [Dt; curl]DtujDtwj dx. (3.38) 14 X. SONG, C. WANG, X. WANG, F. XU EJDE-2025/58 Multiplying (??) by t2 and then integrating over [0, t] give rise to 2∥t∇Dt(ωj , Hj)∥2L∞ t (L2) + 2∥tdivDtωj∥2L∞ t (L2) + ∥t(√ρD2 t uj , √ ρD2 tωj , D 2 tHj)∥2L2 t (L 2) + ∥t∇Dtuj∥2L∞ t (L2) + ∥tdivDtuj∥2L∞ t (L2) + ∥t(curlDtuj − 2Dtwj)∥2L∞ t (L2) ≲ ∥ √ t∇Dt(uj , ωj , Hj)∥2L2 t (L 2) + ∥ √ tdivDtωj , Dtωj∥2L2 t (L 2) + ∫ t 0 τ2 ∫ R3 ∇DtπjD 2 t uj dx dτ + ∫ t 0 τ2 ∫ R3 ∇DtHj · [Dt;∇]DtHj dx dτ + ∫ t 0 τ2 ∫ R3 ∇Dtuj · [Dt;∇]Dtuj dx dτ + ∫ t 0 τ2 ∫ R3 ∇Dtωj · [Dt;∇]Dtωj dx dτ + ∫ t 0 τ2 ∫ R3 ( fjD 2 t uj + gjD 2 tωj + hjD 2 tHj ) dx dτ − ∫ t 0 τ2 ∫ R3 divDtwj [div;Dt]Dtwj dx dτ + ∫ t 0 τ2 ∫ R3 [Dt; curl]ujD 2 twj dx dτ + ∫ t 0 τ2 ∫ R3 [Dt; curl]wjD 2 t uj dx dτ − ∫ t 0 τ2 ∫ R3 [Dt; curl]DtujDtwj dx dτ. (3.39) In what follows, we shall bound term by term from the above inequality. For ∫ t 0 τ2 ∫ R3 ( fjD 2 t uj + gjD 2 tωj + hjD 2 tHj ) dx dτ , it follows from the definitions of fj , gj and hj , that ∥t[Dt;∇]πj∥L2 t (L 2) = ∥t∇u · ∇πj∥L2 t (L 2) ≤ ∥ √ t∇u∥ L∞ t (Ḃ 1/2 2,1 ) ∥ √ t∇πj∥L2 t (L 6), ∥t[Dt; ∆]uj∥L2 t (L 2) ≤ C ( ∥t∆u · ∇uj∥L2 t (L 2) + ∥ 3∑ i t∂iu · ∇∂iuj∥L2 t (L 2) ) ≤ C ( ∥ √ t∇2u∥L2 t (L 3)∥ √ t∇uj∥L∞ t (L6) + ∥ √ t∇u∥L∞ t (L3)∥ √ t∇2uj∥L2 t (L 6) ) ≤ C ( ∥ √ t∇2u∥L2 t (L 3)∥ √ t∇2uj∥L∞ t (L2) + ∥ √ t∇u∥ L∞ t (Ḃ 1/2 2,1 ) ∥ √ t∇2uj∥L2 t (L 6) ) . Similarly, ∥t[Dt;∇ div]ωj∥L2 t (L 2) ≤ C ( ∥ √ t∇2u∥L2 t (L 3)∥ √ t∇2ωj∥L∞ t (L2) + ∥ √ t∇u∥ L∞ t (Ḃ 1/2 2,1 ) ∥ √ t∇2ωj∥L2 t (L 6) ) , ∥t[Dt; ∆]ωj∥L2 t (L 2) ≤ C ( ∥ √ t∇2u∥L2 t (L 3)∥ √ t∇2ωj∥L∞ t (L2) + ∥ √ t∇u∥ L∞ t (Ḃ 1/2 2,1 ) ∥ √ t∇2ωj∥L2 t (L 6) ) , ∥t[Dt; ∆]Hj∥L2 t (L 2) ≤ C ( ∥ √ t∇2u∥L2 t (L 3)∥ √ t∇2Hj∥L∞ t (L2) + ∥ √ t∇u∥ L∞ t (Ḃ 1/2 2,1 ) ∥ √ t∇2Hj∥L2 t (L 6) ) . From (??) we deduce that ∥ √ t(∇2uj ,∇2ωj ,∇2Hj ,∇πj)∥L2 t (L 6) ≤ C ( ∥ √ t∂t(uj , ωj , Hj)∥L2 t (L 6) + ∥ √ tu · ∇uj∥L2 t (L 6) + ∥ √ tH · ∇Hj∥L2 t (L 6) + ∥ √ tu · ∇Hj∥L2 t (L 6) + ∥ √ tH · ∇uj∥L2 t (L 6) ) ≤ C ( ∥ √ t∇∂t(uj , ωj , Hj)∥L2 t (L 2) + ∥(u,H)∥ L2 t (Ḃ 3/2 2,1 ) ∥ √ t∇2(uj , ωj , Hj)∥L∞ t (L2) ) , which together with (??) and (??) ensures that ∥ √ t(∇2uj ,∇2ωj ,∇2Hj ,∇πj)∥L2 t (L 6) ≤ Cdj2 j/2E0. (3.40) On the other hand, from (??), we have −∆u+∇π = H · ∇H + curlω − ρ∂tu− ρu · ∇u, −∆ω −∇ divω = curlu− ρ∂tω − ρu · ∇ω − 2ω, −∆H = H · ∇u− ∂tH − u · ∇H. EJDE-2025/58 GLOBAL SOLUTION FOR MAGNETO-MICROPOLAR EQUATIONS 15 Now, let us take the L2 scalar product of the the first equation of the above system by −∆u, the second equation with −∆ω and the third equation by −∆H respectively and then add the resulting equations, we easily deduce that ∥ √ t∇2(u, ω,H)∥L2 t (L 3) ≤ C ( ∥ √ t∂t(u, ω,H)∥ L2 t (Ḃ 1/2 2,1 ) + ∥(u,H)∥ L2 t (Ḃ 3/2 2,1 ) ∥ √ t∇(u, ω,H)∥ L∞ t (Ḃ 1/2 2,1 ) ) ≤ CE0. (3.41) Combining this with (??), (??) and (??), yields ∥t([Dt; ∆]uj , [Dt; ∆]Hj , [Dt;∇]πj , [Dt; ∆]ωj , [Dt;∇div]ωj)∥L2 t (L 2) ≤ Cdj2 j/2E0. (3.42) It follows from (??), (??), (??) and (??), that ∥tDt(H · ∇Hj)∥L2 t (L 2) ≤ ∥t∂t(H · ∇Hj)∥L2 t (L 2) + ∥tu · ∇(H · ∇Hj)∥L2 t (L 2) ≤ ∥tHt · ∇Hj∥L2 t (L 2) + ∥tH · ∇∂tHj∥L2 t (L 2) + ∥t(u · ∇H) · ∇Hj∥L2 t (L 2) + ∥tu ·H · ∇2Hj∥L2 t (L 2) ≤ C ( ∥ √ tHt∥L2 t (L 3)∥ √ t∇Hj∥L∞ t (L6) + ∥u∥L2 t (L ∞)∥ √ t∇H∥L∞ t (L3)∥ √ t∇Hj∥L∞ t (L6) + ∥ √ tH∥L∞ t (L∞)∥ √ t∂t∇Hj∥L2 t (L 2) + ∥u∥L∞ t (L3)∥ √ tH∥L∞ t (L∞)∥ √ t∇2Hj∥L2 t (L 6) ) ≤ C ( ∥ √ tHt∥L2 t (Ḃ 1/2 2,1 ) ∥ √ t∇2Hj∥L∞ t (L2) + ∥u∥ L2 t (Ḃ 3/2 2,1 ) ∥ √ t∇H∥ L∞ t (Ḃ 1/2 2,1 ) ∥ √ t∇2Hj∥L∞ t (L2) + ∥ √ tH∥ L∞ t ((Ḃ 3/2 2,1 )) ∥ √ t∂t∇Hj∥L2 t (L 2) + ∥u∥ L∞ t (Ḃ 1/2 2,1 ) ∥ √ tH∥ L∞ t (Ḃ 3/2 2,1 ) ∥ √ t∇2Hj∥L2 t (L 6) ) ≤ Cdj2 j/2E0. The term ∥(tDt(H ·∇uj))∥L2 t (L 2) may be treated along the same lines, we omit the process. Thus, we have ∥t(fj , gj , hj)∥L2 t (L 2) ≤ Cdj2 j/2E0, (3.43) which implies ∣∣∣ ∫ t 0 τ2 ∫ R3 fjD 2 t uj + gjD 2 tωj + hjD 2 tHj dx dτ ∣∣∣ ≤ Cd2j2 jE2 0 + ε∥t(√ρD2 t uj , √ ρD2 tωj , D 2 tHj)∥2L2 t (L 2). (3.44) For the term ∫ t 0 τ2 ∫ R3 ∇Dtuj · [Dt;∇]Dtuj dx dτ , it follows from Hölder’s inequality that∣∣∣ ∫ t 0 τ2 ∫ R3 ∇Dtuj · [Dt;∇]Dtuj dx dτ ∣∣∣ ≤ ∥∇u∥L2 t (L 3)∥t∇Dtuj∥L2 t (L 6)∥t∇Dtuj∥L∞ t (L2) ≤ ∥u∥ L2 t (Ḃ 3/2 2,1 ) ∥t∇2Dtuj∥L2 t (L 2)∥t∇Dtuj∥L∞ t (L2). To bound ∥t∇2Dtuj∥L2 t (L 2) in the above inequality, we apply the operator div to the first equation in (??); thus ∆Dtπj = 3∑ i=1 ∆(∂iu · ∇ui j)− div(ρD2 t uj) + divfj + div(Dt curlwj), (3.45) where we have used that Dtuj = ∑3 i=1 ∂iu ·∇ui j . Multiplying (??) by −Dtπj and then integrating, we have ∥∇Dtπj∥2L2 = − ∫ R3 ( 3∑ i=1 ∆(∂iu · ∇ui j)− div ( ρD2 t uj ) + divfj ) Dtπj dx − ∫ R3 div(Dt curlwj)Dtπj dx, 16 X. SONG, C. WANG, X. WANG, F. XU EJDE-2025/58 where the last term on the right-hand side of the above equation reduces to − ∫ R3 div (Dt curlwj)Dtπj dx = ∫ R3 Dt curlwj∇Dtπj dx = ∫ R3 [Dt, curl]wj∇Dtπj dx+ ∫ R3 curlDtwj∇Dtπj dx = ∫ R3 [Dt, curl]wj∇Dtπj dx. From the above calculations, we infer that ∥∇Dtπj∥L2 ≲ (∥∇Tr(∇u∇uj)∥L2 + ∥ρD2 t uj , fj∥L2 + ∥[Dt, curl]wj∥L2). (3.46) Next, taking the L2 inner product of the equation (??)1 by −∆Dtuj , of the equation (??)2 by −∆Dtωj and of the equation (??)3 by −∆DtHj respectively, and then adding them together, we conclude that ∥∇2Dtuj∥2L2 + ∥∇2Dtwj∥2L2 + ∥∇divDtwj∥2L2 + 2∥∇Dtwj∥2L2 + ∥∇2DtHj∥2L2 = − ∫ R3 Dt curlwj∆Dtuj dx− ∫ R3 Dt curluj∆Dtwj dx+ ∫ R3 ∇Dtπj∆Dtuj dx + ∫ R3 ρD2 t uj∆Dtuj dx+ ∫ R3 ρD2 twj∆Dtwj dx+ ∫ R3 D2 tHj∆DtHj dx − ∫ R3 fj∆Dtuj dx− ∫ R3 gj∆Dtwj dx− ∫ R3 hj∆DtHj dx. (3.47) For the first term on the right-hand side of (??), noting that [Dt; curl]h = −∇uk × ∂kh, (3.48) and employing Cauchy-Schwarz’s and Young’s inequalities, we have − ∫ R3 Dt curlwj∆Dtuj dx = − ∫ R3 [Dt, curl]wj∆Dtuj dx− ∫ R3 curlDtwj∆Dtuj dx = − ∫ R3 ∇uk × ∂kwj∆Dtuj dx− ∫ R3 curlDtwj∆Dtuj dx ≤ ∥∇u∥L3∥∇wj∥L6∥∇2Dtuj∥L2 + ∥∇2Dtuj∥L2∥ curlDtwj∥L2 ≤ C∥∇u∥2L3∥∇wj∥2L6 + 1 2 ∥∇2Dtuj∥2L2 + ∥∇Dtwj∥2L2 , (3.49) where we have used that ∆F = ∇divF − curl curlF for vector filed F , and it satisfies ∥∇F∥2L2 = ∥ divF∥2L2 + ∥ curlF∥2L2 . For the second term, it follows from Cauchy-Schwarz’s and Young’s inequalities again, that − ∫ R3 Dt curluj∆Dtwj dx = − ∫ R3 [Dt, curl]uj∆Dtwj dx− ∫ R3 curlDtuj∆Dtwj dx = − ∫ R3 ∇(∇uk × ∂kuj) · ∇Dtwj dx− ∫ R3 ∆Dtuj curlDtwj dx ≤ ( ∥∇2u∥L3∥∇uj∥L6 + ∥∇u∥L3∥∇2uj∥L6 ) ∥∇Dtwj∥L2 + 1 4 ∥∇2Dtuj∥2L2 + ∥∇Dtwj∥2L2 . (3.50) Hence, applying Cauchy-Schwarz’s and Young’s inequalities again, and then absorbing the small terms, we deduce from (??) and (??), that ∥(∇2Dtuj ,∇2Dtwj ,∇2DtHj)∥2L2 ≲ ∥(√ρD2 t uj , √ ρD2 twj , D 2 tHj)∥2L2 + ∥(fj , gj , hj)∥2L2 + ∥∇u∥2L3∥∇wj∥2L6 + ( ∥∇2u∥L3∥∇uj∥L6 + ∥∇u∥L3∥∇2uj∥L6 ) ∥∇Dtwj∥L2 + ∥∇2u∥2L3∥∇2uj∥2L2 + ∥∇u∥2L3∥∇2uj∥2L6 . (3.51) EJDE-2025/58 GLOBAL SOLUTION FOR MAGNETO-MICROPOLAR EQUATIONS 17 Now, we shall present a new estimate for the term ∫ R3 curlDtuj∆Dtwj dx in (??) and (??) as follows ∫ R3 curlDtuj∆Dtwj dx ≤ C(ε)∥∇2Dtuj∥2L2 + ε∥∇Dtwj∥2L2 . By choosing ε small enough and inserting (??) into the above inequality, and then applying (??), it gives rise to ∥(∇2Dtuj ,∇2Dtwj ,∇Dtwj ,∇2DtHj)∥2L2 ≲ ∥(√ρD2 t uj , √ ρD2 twj , D 2 tHj)∥2L2 + ∥(fj , gj , hj)∥2L2 + ∥∇u∥2L3∥∇wj∥2L6 + (∥∇2u∥L3∥∇uj∥L6 + ∥∇u∥L3∥∇2uj∥L6)∥∇Dtwj∥L2 + ∥∇2u∥2L3∥∇2uj∥2L2 + ∥∇u∥2L3∥∇2uj∥2L6 , (3.52) which together with Young’s inequality implies ∥(∇2Dtuj ,∇2Dtwj ,∇Dtwj ,∇2DtHj)∥2L2 ≲ ∥(√ρD2 t uj , √ ρD2 twj , D 2 tHj)∥2L2 + ∥(fj , gj , hj)∥2L2 + ∥∇u∥2L3∥∇wj∥2L6 + ∥∇2u∥2L3∥∇2uj∥2L2 + ∥∇u∥2L3∥∇2uj∥2L6 . (3.53) Combining (??) with (??), (??), (??), (??), (??) and (??) yields ∥t(∇2Dtuj ,∇2Dtwj ,∇Dtwj ,∇2DtHj)∥L2 t (L 2) + ∥t∇Dtπj∥L2 t (L 2) ≲ ∥t(√ρD2 t uj , √ ρD2 twj , D 2 tHj)∥L2 t (L 2) + ∥t(fj , gj , hj)∥L2 t (L 2) + ∥ √ t∇u∥L∞ t (L3)∥ √ t∇wj∥L2 t (L 6) + ∥ √ t∇2u∥L2 t (L 3)∥ √ t∇2uj∥L∞ t (L2) + ∥ √ t∇u∥L∞ t (L3)∥ √ t∇2uj∥L2 t (L 6) ≲ dj2 j 2E0 + ∥t(√ρD2 t uj , √ ρD2 twj , D 2 tHj)∥L2 t (L 2). (3.54) Thus it follows from Young’s inequality that∣∣∣ ∫ t 0 τ2 ∫ R3 ∇Dtuj · [Dt;∇]Dtuj dx dτ ∣∣∣ ≤ C ( dj2 j/2E0 + ∥t(√ρD2 t uj , √ ρD2 tωj , D 2 tHj)∥L2 t (L 2) ) ∥u∥ L2 t (Ḃ 3/2 2,1 ) ∥t∇Dtuj∥L∞ t (L2) ≤ C ( d2j2 jE2 0 + ∥u∥2 L2 t (Ḃ 3/2 2,1 ) ∥t∇Dtuj∥2L∞ t (L2) ) + ε∥t(√ρD2 t uj , √ ρD2 tωj , D 2 tHj)∥2L2 t (L 2). (3.55) Similarly, we have∣∣∣ ∫ t 0 τ2 ∫ R3 ∇Dtωj · [Dt;∇]Dtωj dx dτ ∣∣∣+ ∣∣∣ ∫ t 0 τ2 ∫ R3 ∇DtHj · [Dt;∇]DtHj dx dτ ∣∣∣ + ∣∣∣ ∫ t 0 τ2 ∫ R3 divDtωj · [div;Dt]Dtωj dx dτ ∣∣∣ ≤ C ( d2j2 jE2 0 + ∥u∥2 L2 t (Ḃ 3/2 2,1 ) ∥t∇Dt(ωj , Hj)∥2L∞ t (L2) ) + ε∥t(√ρD2 t uj , √ ρD2 tωj , D 2 tHj)∥2L2 t (L 2). (3.56) We shall deal with the term ∫ t 0 τ2 ∫ R3 ∇DtπjD 2 t uj dx dτ as follows. First, from the definition of Dt, we have∫ R3 ∇Dtπj ·D2 t uj dx = ∫ R3 ∇Dtπj · (∂2 t uj + ∂tu · ∇uj + u · ∂t∇uj + u · ∇Dtuj)dx = ∫ R3 ∇Dtπj · (∂tu · ∇uj + ∂tuj · ∇u+ u · ∇Dtuj)dx. 18 X. SONG, C. WANG, X. WANG, F. XU EJDE-2025/58 It follows from (??), (??) and (??), that∣∣∣ ∫ t 0 τ2 ∫ R3 ∇Dtπj · (∂tu · ∇uj) dx dτ ∣∣∣ ≤ ∥t∇Dtπj∥L2 t (L 2)∥ √ t∂tu∥L2 t (L 3)∥ √ t∇uj∥L∞ t (L6) ≤ C∥t∇Dtπj∥L2 t (L 2)∥ √ t∂tu∥L2 t (Ḃ 1/2 2,1 ) ∥ √ t∇2uj∥L∞ t (L2) ≤ Cd2j2 jE2 0 + ε∥t(√ρD2 t uj , √ ρD2 tωj , D 2 tHj)∥2L2 t (L 2),∣∣∣ ∫ t 0 τ2 ∫ R3 ∇Dtπj · (∂tuj · ∇u) dx dτ ∣∣∣ ≤ C∥t∇Dtπj∥L2 t (L 2)∥ √ t∇u∥ L∞ t (Ḃ 1/2 2,1 ) ∥ √ t∂t∇uj∥L2 t (L 2) ≤ Cd2j2 jE2 0 + ε∥t(√ρD2 t uj , √ ρD2 tωj , D 2 tHj)∥2L2 t (L 2),∣∣∣ ∫ t 0 τ2 ∫ R3 ∇Dtπj · (u · ∇Dtuj) dx dτ ∣∣∣ ≤ C∥t∇Dtπj∥L2 t (L 2)∥u∥L2 t (Ḃ 3/2 2,1 ) ∥t∇Dtuj∥L∞ t (L2) ≤ C ( d2j2 jE2 0 + ∥u∥2 L2 t (Ḃ 3/2 2,1 ) ∥t∇Dtuj∥2L∞ t (L2) ) + ε∥t(√ρD2 t uj , √ ρD2 tωj , D 2 tHj)∥2L2 t (L 2). Then ∣∣∣ ∫ t 0 τ2 ∫ R3 ∇Dtπj ·D2 t uj dx dτ ∣∣∣ ≤ C ( djj2 jE2 0 + ∥u∥2 L2 t (Ḃ 3/2 2,1 ) ∥t∇Dtuj∥2L∞ t (L2) ) + ε∥t(√ρD2 t uj , √ ρD2 tωj , D 2 tHj)∥2L2 t (L 2). (3.57) For the term ∫ t 0 τ2 ∫ R3 [Dt; curl]ujD 2 twj dx dτ+ ∫ t 0 τ2 ∫ R3 [Dt; curl]wjD 2 t uj dxdτ , recalling (??), (??) and (??), we obtain∫ t 0 τ2 ∫ R3 [Dt; curl]wjD 2 t uj dx dτ = − ∫ t 0 τ2 ∫ R3 (∇uk × ∂kwj)D 2 t uj dx dτ ≤ ∥ √ t∇u∥L∞ t (L3)∥ √ t∇wj∥L2 t (L 6)∥tD2 t uj∥L2 t (L 2) ≤ ∥ √ t∇u∥2L∞ t (L3)∥ √ t∇wj∥2L2 t (L 6) + ε∥tD2 t uj∥2L2 t (L 2) ≤ Cd2j2 jE2 0 + 1 32 ∥tD2 t uj∥2L2 t (L 2). (3.58) For the other term, we observe that∫ t 0 τ2 ∫ R3 [Dt; curl]ujD 2 twj dxdτ = − ∫ t 0 τ2 d dt ∫ R3 (∇uk × ∂kuj)Dtwj dx dτ + ∫ t 0 τ2 ∫ R3 Dt(∇uk × ∂kuj)Dtwj dxdτ. (3.59) Applying integration by parts, it follows from (??), (??), (??), (??) and (??) for the first term on the right-hand side of (??) that − ∫ ι 0 τ2 d dt ∫ R3 (∇uk × ∂kuj)Dtwj dx dτ = − ∫ R3 t2(∇uk × ∂kuj)Dtwj dx+ 2 ∫ t 0 τ ∫ R3 (∇uk × ∂kuj)Dtwj dx dτ = −1 2 ∫ R3 t2(∇uk × ∂kuj)(curlDtuj − 2Dtwj) dx + 1 2 ∫ R3 t2(∇uk × ∂kuj) curlDtuj dx+ 2 ∫ t 0 τ ∫ R3 (∇uk × ∂kuj)Dtwj dx dτ ≲ ∥ √ t∇u∥L∞ t (L3)∥ √ t∇uj∥L∞ t (L6) ( ∥t (curlDtuj + 2Dtwj)∥L∞ t (L2) + ∥t curlDtuj∥L∞ t (L2) ) + ∥∇u∥L2 t (L 3)∥∇uj∥L∞ t (L2)∥tDtwj∥L2 t (L 6) EJDE-2025/58 GLOBAL SOLUTION FOR MAGNETO-MICROPOLAR EQUATIONS 19 ≤ Cd2j2 jE2 0 + 1 4 ( ∥t(curl Dtuj + 2Dtwj)∥2L∞ t (L2) + ∥t curlDtuj∥2L∞ t (L2) ) + 1 32 ∥t(√ρD2 t uj , √ ρD2 twj , D 2 tHj)∥2L2 t (L 2). As for the second term, we see that∫ t 0 τ2 ∫ R3 Dt(∇uk × ∂kuj)Dtwj dx dτ = ∫ t 0 τ2 ∫ R3 Dt∇uk × ∂kujDtwj dx dτ + ∫ t 0 τ2 ∫ R3 ∇uk ×Dt∂kujDtwj dx dτ = − ∫ t 0 τ2 ∫ R3 (∇u · ∇2uk)× ∂kujDtwj dx dτ + ∫ t 0 τ2 ∫ R3 ∇Dtuk × ∂kujDtwj dx dτ − ∫ t 0 τ2 ∫ R3 ∇uk × (∇u · ∇∂kuj)Dtwj dx dτ + ∫ t 0 τ2 ∫ R3 ∇uk × ∂kDtujDtwj dx dτ := I1 + I2 + I3 + I4. (3.60) For I1, applying (??), (??) and (??), we infer that I1 ≤ ∥ √ t∇u∥L∞ t (L3)∥ √ t∇2u∥L2 t (L 3)∥ √ t∇uj∥L∞ t (L6)∥ √ tDtwj∥L2 t (L 6) ≲ ∥ √ t∇uj∥L∞ t (L6)(∥ √ t∂twj∥L2 t (L 6) + ∥ √ tu · ∇wj∥L2 t (L 6)) ≲ ∥ √ t∇uj∥L∞ t (L6)(∥ √ t∇∂twj∥L2 t (L 2) + ∥ √ tu∥L∞ t (L∞)∥∇2wj∥L2 t (L 2)) ≲ d2j2 jE2 0 . For I2, it follows from the integration by parts and (??), (??), (??), that I2 = ∫ t 0 τ2 ∫ R3 ∇Dtuk × ∂kujDtwj dx dτ ≤ ∫ t 0 ∥ √ tDtu∥L3∥ √ t∇2uj∥L6 ( ∥t (curlDtuj − 2Dtwj) ∥L∞ t (L2) + ∥t curlDtuj∥L∞ t (L2) ) dτ + ∫ t 0 ∥ √ tDtu∥L3∥ √ t∇uj∥L6∥t∇Dtwj∥L2 dτ ≤ C ∫ t 0 ∥ √ tDtu∥2L3 ( ∥t(curl Dtuj − 2Dtwj)∥2L∞ t (L2) + ∥t curlDtuj∥2L∞ t (L2) ) dτ + C∥ √ t∇2uj∥2L2 t (L 6) + ε∥t∇Dtwj∥2L2 t (L 2) + C∥ √ tDtu∥2L2 t (L 3)∥ √ t∇2uj∥2L∞ t (L2) ≤ C ∫ t 0 ∥ √ tDtu∥2L3 ( ∥t(curlDtuj − 2Dtwj)∥2L∞ t (L2) + ∥t curlDtuj∥2L∞ t (L2) ) dτ + Cd2j2 jE2 0 + 1 32 ∥t(√ρD2 t uj , √ ρD2 twj , D 2 tHj)∥2L2 t (L 2), where we deduce from (??) and (??) that ∥ √ tDtu∥2L2 t (L 3) ≤ ∥ √ t∂tu∥2L2 t (L 3) + ∥ √ tu · ∇u∥2L2 t (L 3) ≤ ∥ √ t∂tu∥2L2 t (L 3) + ∥u∥2L2 t (L ∞)∥ √ t∇u∥2L∞ t (L3) ≲ E2 0 . (3.61) For I3, applying (??) and (??) again, we conclude that I3 = − ∫ t 0 τ2 ∫ R3 ∇u · ∇2uj∇uDtwj dx dτ ≲ ∥ √ t∇u∥2L∞ t (L3)∥ √ t∇2uj∥L2 t (L 6)∥ √ tDtwj∥L2 t (L 6) ≲ ∥ √ t∇u∥2L∞ t (L3)∥ √ t∇2uj∥L2 t (L 6)∥ √ tDtwj∥L2 t (L 6) ≲ d2j2 jE2 0 . 20 X. SONG, C. WANG, X. WANG, F. XU EJDE-2025/58 As before, one has the estimate for I4 that I4 = − ∫ t 0 τ2 ∫ R3 ∇Dtuj∇uDtwj dx dτ ≤ ∫ t 0 ∥t∇Dtuj∥L6∥∇u∥L3 (∥t(curlDtuj − 2Dtwj)∥L2 + ∥t curlDtuj∥L2) dτ ≤ C(ε) ∫ t 0 ∥∇u∥2L3 ( ∥t(curlDtuj − 2Dtwj)∥2L2 + ∥t curlDtuj∥2L2 ) dτ + ε∥t∇2Dtuj∥2L2 t (L 2) ≤ Cd2j2 jE2 0 + 1 32 ∥t(√ρD2 t uj , √ ρD2 twj , D 2 tHj)∥2L2 t (L 2) + C ∫ t 0 ∥∇u∥2L3 ( ∥t(curlDtuj − 2Dtwj)∥2L2 + ∥t curlDtuj∥2L2 ) dτ. Finally we deal with the term ∫ t 0 τ2 ∫ R3 [Dt; curl]DtujDtwj dx dτ . Applying (??), in view of I4, we also have ∣∣∣ ∫ t 0 τ2 ∫ R3 [Dt; curl]DtujDtwj dx dτ ∣∣∣ ≤ Cd2j2 jE2 0 + 1 32 ∥t(√ρD2 t uj , √ ρD2 twj , D 2 tHj)∥2L2 t (L 2) + C ∫ t 0 ∥∇u∥2L3 ( ∥t(curlDtuj − 2Dtwj)∥2L2 + ∥t curlDtuj∥2L2 ) dτ. Inserting the above estimates into (??) and absorbing some suitable small terms, we have ∥t∇Dt(uj , ωj , Hj)∥2L∞ t (L2) + ∥t(√ρD2 t uj , √ ρD2 tωj , D 2 tHj)∥2L2 t (L 2) + ∥t curlDtuj∥2L∞ t (L2) + ∥t(curlDtuj − 2Dtwj)∥2L∞ t (L2) ≤ C ( ∥ √ t(∇Dtuj ,∇Dtwj ,∇DtHj , Dtwj)∥2L2 t (L 2) ) + Cd2j2 jE2 0 + ∫ t 0 ( ∥∇u∥2B1/2 + ∥u∥2L∞ ) ∥t∇Dt(uj , wj , Hj)∥2L2 dτ + C ∫ t 0 (∥ √ tDtu∥2L3 + ∥∇u∥2L3) ( ∥t(curlDtuj − 2Dtwj)∥2L2 + ∥t curlDtuj∥2L2 ) dτ. (3.62) Furthermore, we deduce from (??), (??) and (??) that ∥ √ t∇Dt(uj , ωj , Hj)∥L2 t (L 2) ≤ ∥ √ t∇∂t(uj , ωj , Hj)∥L2 t (L 2) + ∥u∥L2 t (L ∞)∥ √ t∇2(uj , ωj , Hj)∥L∞ t (L2) + ∥∇u∥L2 t (L 3)| √ t∇(uj , ωj , Hj)∥L∞ t (L6) ≤ ∥ √ t∇∂t(uj , ωj , Hj)∥L2 t (L 2) + ∥u∥ L2 t (Ḃ 3/2 2,1 ) ∥ √ t∇2(uj , ωj , Hj)∥L∞ t (L2) ≤ Cdj2 j 2E0, ∥ √ tDtwj∥L2 t (L 2) ≤ ∥ √ t∂twj∥L2 t (L 2) + ∥ √ tu · ∇wj∥L2 t (L 2) ≤ ∥ √ t∂twj∥L2 t (L 2) + ∥ √ tu∥L∞ t (L∞)∥∇wj∥L2 t (L 2) ≲ dj2 j 2E0. EJDE-2025/58 GLOBAL SOLUTION FOR MAGNETO-MICROPOLAR EQUATIONS 21 Then (??) becomes ∥t∇Dt(uj , ωj , Hj)∥2L∞ t (L2) + ∥t(√ρD2 t uj , √ ρD2 tωj , D 2 tHj)∥2L2 t (L 2) + ∥t curlDtuj∥2L∞ t (L2) + ∥t(curlDtuj − 2Dtwj)∥2L∞ t (L2) ≲ d2j2 jE2 0 + ∫ t 0 ( ∥∇u∥2B1/2 + ∥u∥2L∞ ) ∥t(∇Dtuj ,∇Dtwj ,∇DtHj)∥2L2 dτ + ∫ t 0 ( ∥ √ tDtu∥2L3 + ∥∇u∥2L3 )( ∥t(curlDtuj − 2Dtwj)∥2L2 + ∥t curlDtuj∥2L2 ) dτ. (3.63) Applying Gronwall’s inequality and using (??) gives rise to ∥t∇Dt(uj , ωj , Hj)∥2L∞ t (L2) + ∥t(√ρD2 t uj , √ ρD2 twj , D 2 tHj)∥2L2 t (L 2) ≲ d2j2 jE2 0 exp(C∥u∥2L2 t (B 3/2) + C∥ √ tDtu∥2L2 t (L 3)) ≲ d2j2 jE2 0 , (3.64) which together with (??) and (??), yields ∥t∇Dt(uj , ωj , Hj)∥2L∞ + ∥t∇2Dt(uj , ωj , Hj)∥2L2 t (L 2) ≤ Cd2j2 jE2 0 . (3.65) Putting (??) and (??) into (??), we conclude that (??) holds for t > 0. This completes the proof of Proposition ??. □ Proof of Theorem ??. Let jδ be the standard Friedrich’s mollifier and define ρδ0 = jδ ∗ ρ0, ρ ≤ ρδ0 ≤ ρ, uδ 0 = jδ ∗ u0, ωδ 0 = jδ ∗ ω0, Hδ 0 = jδ ∗H0. By using the similar method as in [?], one can prove that the 3D incompressible inhomogeneous magneto-micropolar equations (??) has a unique local smooth solution (ρδ, uδ, ωδ, Hδ) on [0, T δ). Moreover, we can show that (??), (??) and (??) hold for (ρδ, uδ, ωδ, Hδ). In particular, Proposition ?? implies that Tδ = ∞. As in [?, ?], we can complete the existence part of Theorem ?? based one the uniform estimates (??), (??) and (??) for (ρδ, uδ, ωδ, Hδ) and a standard compactness argument, which we omit details here. To prove (??), it remains to show that ∥t∂t(u, ω,H)∥ L̃∞ t (Ḃ 1/2 2,1 ) ≤ CE0, It follows from the law of product in Besov spaces and (??), (??) that ∥t∂t(u, ω,H)∥ L̃∞ t (Ḃ 1/2 2,1 ) ≤ ∥tDt(u, ω,H)∥ L̃∞ t (Ḃ 1/2 2,1 ) + ∥ √ t∇(u, ω,H)∥ L̃∞ t (Ḃ 1/2 2,1 ) ∥ √ tu∥ L̃∞ t (Ḃ 3/2 2,1 ) ≤ CE0. □ 4. Proof of Theorem ?? First, we develop some extra time-weighted energy estimates for proving Theorem ??. Proposition 4.1. Let (ρ, u, ω,H, π) be a global solution of (??) constructed by Theorem ??. Then (t∇Dtu, t∇Dtω, t∇DtH) ∈ L2 ( R+;L 3(R3) ) , (tDtu, tDtω, tDtH) ∈ L2 ( R+;L ∞(R3) ) . (4.1) Proof. Thanks to the embeddings Ḃ 1/2 2,1 (R 3) ↪→ L3(R3) and Ḃ 3/2 2,1 (R 3) ↪→ L∞(R3), we conclude from (??), that (??) holds. This completes the proof. □ Proposition 4.2. Let (ρ, u,H, π) be a global solution of (??) constructed by Theorem ??. Then t∇2(u, ω,H) ∈ L4,1 ( 0, T ;L6(R3) ) , √ t(∇2u,∇2ω,∇2H,∇π) ∈ L4,1 ( 0, T ;L2(R3) ) . (4.2) 22 X. SONG, C. WANG, X. WANG, F. XU EJDE-2025/58 Proof. Assume (ρ, u,H,∇π) is a solution to (??) on [0, T ] × R3. As in [?], let us observe the coupled system (??). Combing (??), (??), (??), (??), (??) and (??), ∥ √ t(∇2uj ,∇2ωj ,∇2Hj ,∇πj)∥L2(0,T ;R3) + ∥t∇∂t(uj , ωj , Hj)∥L2(0,T ;R3) + ∥t∇2(uj , ωj , Hj)∥L∞(0,T ;L2(R3)) ≤ d1j2 −j/2E0 with ∑ j∈Z d1j = 1 (4.3) and ∥∇2(uj , ωj , Hj)∥L2(0,T ;R3) + ∥ √ t(∇2uj ,∇2ωj ,∇2Hj ,∇πj)∥L∞(0,T ;L2(R3)) + ∥t∇∂t(uj , ωj , Hj)∥L∞(0,T ;L2(R3)) ≤ d2j2 j/2E0 with ∑ j∈Z d2j = 1. (4.4) Using the following interpolation of the Lorentz space in Proposition ??(1), we have( L2 ( 0, T ;L2(R3) ) , L∞( 0, T ;L2(R3) )) 1/2,1 = L4,1 ( 0, T ;L2(R3) ) and (??)-(??), we infer that ∥ √ t(∇2uj ,∇2ωj ,∇2Hj ,∇πj)∥L4,1(0,T ;L2(R3)) ≤ CdjE0. (4.5) It follows from the first three equations in (??), that −∆uj +∇πj = H · ∇Hj − ρ∂tuj − ρu · ∇uj + curlωj , −∆ωj −∇ divωj + 2ωj = curluj − ρ∂tωj − ρu · ∇ωj , −∆Hj = H · ∇uj − ∂tHj − u · ∇Hj . (4.6) From the classical regularity theory of elliptic equation and the standard Lp estimate of elliptic operator, for any p ∈ (1,∞), it follows that ∥∆uj∥Lp + ∥∇πj∥Lp ≤ C ( ∥ρ∂tuj∥Lp + ∥ρu · ∇uj∥Lp + ∥H · ∇Hj∥Lp + ∥ curlωj∥Lp ) , ∥∆ω∥p ≤ C ( ∥ρ∂tωj∥p + ∥ρu · ∇ωj∥p + ∥ curluj∥p ) , ∥∆Hj∥Lp ≤ C ( ∥H · ∇uj∥Lp + ∥∂tHj∥Lp + ∥u · ∇Hj∥Lp ) , which together with (??), Hölder’s inequality and the embeddings Ḃ 3/2 2,1 (R 3) ↪→ L∞(R3) and Ḣ1(R3) ↪→ L6(R3), yields ∥t(∇2uj , t∇πj)∥L2(0,T ;L6(R3)) + ∥t∇2ωj∥L2(0,T ;L6(R3)) + ∥t∇2Hj∥L2(0,T ;L6(R3)) ≤ C ( ∥t∇∂t(uj , ωj , Hj)∥L2(0,T ;L2(R3)) + ∥H∥L2(0,T ;L∞(R3))∥t∇2(uj , Hj)∥L∞(0,T ;L2(R3)) + ∥u∥L2(0,T ;L∞(R3))∥t∇2(uj , ωj , Hj)∥L∞(0,T ;L2(R3)) + ∥t∇2(uj , ωj)∥L2(0,T ;L2(R3)) ) ≤ C ( ∥t∇∂t(uj , ωj , Hj)∥L2(0,T ;L2(R3)) + ∥H∥L2(0,T ;L∞(R3))∥t∇2(uj , Hj)∥L∞(0,T ;L2(R3)) + ∥u∥L2(0,T ;L∞(R3))∥t∇2(uj , ωj , Hj)∥L∞(0,T ;L2(R3)) + t1/2∥ √ t∇2(uj , ωj)∥L2(0,T ;L2(R3)) ) ≤ Cd1j2 −j/2E0. (4.7) Similarly, we have ∥t(∇2uj ,∇πj)∥L∞(0,T ;L6(R3)) + ∥t∇2Hj∥L∞(0,T ;L6(R3)) + ∥t∇2ωj∥L∞(0,T ;L6(R3)) ≤ C ( ∥t∇∂t(uj , ωj , Hj)∥L∞(0,T ;L2(R3)) + t1/2∥ √ t∇2(uj , ωj)∥L∞(0,T ;L2(R3)) + ∥ √ tH∥L∞(0,T ;L∞(R3))∥ √ t∇2(uj , Hj)∥L∞(0,T ;L2(R3)) + ∥ √ tu∥L∞(0,T ;L∞(R3))∥ √ t∇2(uj , ωj , Hj)∥L∞(0,T ;L2(R3)) ) ≤ Cd2j2 j/2E0. (4.8) EJDE-2025/58 GLOBAL SOLUTION FOR MAGNETO-MICROPOLAR EQUATIONS 23 Using the following interpolation property again,( L2 ( 0, T ;L6(R3) ) , L∞( 0, T ;L6(R3) )) 1 2 ,1 = L4,1 ( 0, T ;L6(R3) ) , we conclude from (??) and (??) that ∥t∇2uj∥L4,1(0,T ;L6(R3)) + ∥t∇2ωj∥L4,1(0,T ;L6(R3)) + ∥t∇2Hj∥L4,1(0,T ;L6(R3)) ≤ CdjE0 with ∑ j∈Z dj = 1. (4.9) Combining this with (??) and (??) for all j ∈ Z, we deduce that ∥t∇2(u, ω,H)∥L4,1(0,T ;L6(R3)) + ∥ √ t(∇2u,∇2ω,∇2H,∇π)∥L4,1(0,T ;L2(R3)) ≤ CE0. (4.10) This completes the proof. □ Proposition 4.3. Let (ρ, u, ω,H, π) be a global solution of (??) constructed by Theorem ??. Then (∇u,∇ω,∇H) ∈ L1 ( 0, T ;L∞(R3) ) . (4.11) Proof. It follows from Gagliardo-Nirenberg inequality, Proposition ??(5) and (??), that∫ T 0 ∥∇(u, ω,H)∥L∞(R3)dt ≤ C ∫ T 0 ∥∇2(u, ω,H)∥1/2L2(R3)∥∇ 2(u, ω,H)∥1/2L6(R3)dt ≤ C∥t− 3 4 ∥ L 4 3 ,∞ ∥∥√t∥∇2(u, ω,H)∥1/2L6(R3) ∥∥ L8,2 ∥∥t1/4∥∇2(u, ω,H)∥1/2L2(R3) ∥∥ L8,2 ≤ C∥t− 3 4 ∥ L 4 3 ,∞∥t∇2(u, ω,H)∥1/2L4,1(0,T ;L6(R3))∥ √ t∇2(u, ω,H)∥1/2L4,1(0,T ;L2(R3)) ≤ CE0. (4.12) This completes the proof. □ Proof of Theorem??. Denoting δρ ≜ ρ1 − ρ2, δu ≜ u1 − u2, δω ≜ ω1 − ω2, δH ≜ H1 − H2, and δπ ≜ π1 − π2, we obtain (δρ)t + δu · ∇ρ1 + u2 · ∇δρ = 0, ρ1(δu)t + ρ1u1 · ∇δu−∆δu+∇δπ = −δρDtu2 − ρ1δu · ∇u2 + δH∇H2 +H1∇δH + curl δω, ρ1(δω)t + ρ1u1 · ∇δω −∆δω −∇ div δω = −δρDtω2 − ρ1δu · ∇ω2 + curl δu− 2δω, (δH)t + u1∇δH −∆δH = δu∇H2 + δH∇u2 +H1∇δu, (δρ, δu, δω, δH)|t=0 = (0, 0, 0, 0). (4.13) In our framework where the density is only in L∞(R3), following the duality approach initiated by Hoff in [?], we shall actually prove the uniqueness for the density in Ḣ−1. In what follows, we need to explain the fact that (δu, δω, δH) is in the energy space. Obviously, it follows form the original system (??)2 that ρut = −ρu · ∇u+∆u−∇π +H · ∇H + curlω. 24 X. SONG, C. WANG, X. WANG, F. XU EJDE-2025/58 Employing Hölder’s inequality in the Lorentz spaces (see Proposition ??(3)), (??), (??), Proposi- tion ??(5) and the embedding Ḃ 1/2 2,1 (R3) ↪→ L3(R3), we have ∥ρut∥ L 4 3 ,1(0,T ;L2(R3)) ≤ ∥ρu · ∇u∥ L 4 3 ,1(0,T ;L2(R3)) + ∥∆u∥ L 4 3 ,1(0,T ;L2(R3)) + ∥∇π∥ L 4 3 ,1(0,T ;L2(R3)) + ∥H · ∇H∥ L 4 3 ,1(0,T ;L2(R3)) + ∥ curlω∥ L 4 3 ,1(0,T ;L2(R3)) ≤ ∥(u,H)∥L∞(0,T ;L3(R3))∥ √ t∇2(u,H)∥L4,1(0,T ;L2(R3))∥t−1/2∥L2,∞ + ∥ √ t∇2(u,H)∥L4,1(0,T ;L2(R3))∥t−1/2∥L2,∞ + ∥ √ t∇π∥L4,1(0,T ;L2(R3))∥t−1/2∥L2,∞ + ∥ √ t∇ω∥L4,1(0,T ;L2(R3))∥t−1/2∥L2,∞ < ∞, (4.14) which together with (??) implies that ut ∈ L4/3,1 ( 0, T ;L2(R3) ) . (4.15) Similarly, we have Ht ∈ L4/3,1 ( 0, T ;L2(R3) ) . (4.16) From (??)3, we have ρωt = −ρu · ∇ω +∆ω +∇divω − 2ω + curlu. For the estimation of ω, introducing ωj = Pωj +Qωj , where P = Id+∇(−∆)−1 div which denote the Leray projection operator. Using the second equation in (??), we obtain −∆Qωj −∇ divQωj + 2Qωj = −Q(ρ∂tωj)−Q(ρu · ∇ωj). From (??), (??)Êœ, Hölder’s inequality and embedding relation Ḃ 1/2 2,1 (R3) ↪→ L3(R3), we obtain ∥ √ t∇2Qωj∥L∞(0,T ;L2(R3)) + ∥ √ tQωj∥L∞(0,T ;L2(R3)) ≤ C ( ∥ √ t∂tωj∥L∞(0,T ;L2(R3)) + ∥ √ t(u · ∇ωj)∥L∞(0,T ;L2(R3)) ) ≤ C ( ∥ √ t∂tωj∥L∞(0,T ;L2(R3)) + ∥u∥L∞(0,T ;L3(R3))∥ √ t∇ωj∥L∞(0,T ;L3(R3)) ) ≤ Cdj2 j/2E0. Similarly, from (??), (??), Hölder’s inequality and embedding relation Ḃ 3/2 2,1 (R3) ↪→ L∞(R3), we obtain ∥ √ t∇2Qωj∥L2(0,T ;L2(R3)) + ∥ √ tQωj∥L2(0,T ;L2(R3)) ≤ C ( ∥ √ t∂tωj∥L2(0,T ;L2(R3)) + ∥ √ t(u · ∇ωj)∥L2(0,T ;L2(R3)) ) ≤ C ( ∥ √ t∂tωj∥L2(0,T ;L2(R3)) + ∥u∥L2(0,T ;L∞(R3))∥ √ t∇ωj∥L∞(0,T ;L2(R3)) ) ≤ Cdj2 −j/2E0. Employing the following interpolation of the Lorentz space in Proposition ??(1),( L2 ( 0, T ;L2(R3) ) , L∞( 0, T ;L2(R3) )) 1 2 ,1 = L4,1 ( 0, T ;L2(R3) ) , we can get √ tω ∈ L4,1 ( 0, T ;L2(R3) ) such that ∥ρωt∥ L 4 3 ,1(0,T ;L2(R3)) ≤ ∥ρu · ∇ω∥ L 4 3 ,1(0,T ;L2(R3)) + ∥∆ω∥ L 4 3 ,1(0,T ;L2(R3)) + ∥∇ divω∥ L 4 3 ,1(0,T ;L2(R3)) + ∥ curlu∥ L 4 3 ,1(0,T ;L2(R3)) + ∥2ω∥ L 4 3 ,1(0,T ;L2(R3)) ≲ ∥u∥L∞(0,T ;L3(R3))∥ √ t∇2ω∥L4,1(0,T ;L2(R3))∥t−1/2∥L2,∞ + ∥ √ t∇2ω∥L4,1(0,T ;L2(R3))∥t−1/2∥L2,∞ + ∥ √ t∇u∥L4,1(0,T ;L2(R3))∥t−1/2∥L2,∞ + ∥ √ tω∥L4,1(0,T ;L2(R3))∥t−1/2∥L2,∞ < ∞. (4.17) EJDE-2025/58 GLOBAL SOLUTION FOR MAGNETO-MICROPOLAR EQUATIONS 25 Thus, we can deduce from (ut, ωt, Ht) ∈ L 4 3 ,1 ( 0, T ;L2(R3) ) and (u, ω,H) ∈ Cb ( 0, T ; Ḃ 1/2 2,1 (R3) ) , which implies that ( u(t) − u0, ωt − ω0, H(t) − H0 ) ∈ C ( 0, T ;B 1/2 2,1 (R3) ) (nonhomogeneous Besov space). Owing to the classical embedding B 1/2 2,1 (R3) ↪→ H1/2(R3), we obtain (u(t)− u0, ωt − ω0, H(t)−H0) ∈ C ( 0, T ;L2(R3) ) . Now, we claim that (∇u,∇ω,∇H) ∈ L4,1(0, T ;L2(R3)). Indeed, one takes two constants q0 and q1 such that 1 < q0 < 4 3 < q1 < ∞ and 1 q0 + 1 q1 = 3 2 . For all γ ∈ (0, 1) and i = 0, 1, by using the mixed derivative theorem, we obtain Ẇ 2,1 2,qi (0, T × R3) ≜ Ẇ 2,1 2,(qi,qi) (0, T × R3) ↪→ Ẇ γ qi ( 0, T ; Ẇ 2−2γ 2 (R3) ) , (4.18) where the space Ẇ 2,1 p,(q,r)(0, T × R3) is defined by Ẇ 2,1 p,(q,r)(0, T × R3) ≜ { f ∈ C ( 0, T ; Ḃ 2− 2 q p,r (R3) ) ; ft,∇2f ∈ Lq,r ( 0, T ;Lp(R3) )} . Taking γ = 1 2 , it follows from the Sobolev embedding, that Ẇ 1/2 qi ( 0, T ; Ẇ 1 2 (R3) ) ↪→ Lsi ( 0, T ; Ẇ 1 2 (R3) ) with 1 si = 1 qi − 1 2 . (4.19) On the other hand, from the proof of [?, Prop.2.1], we find that Ẇ 2,1 2,( 4 3 ,1) (0, T × R3) = ( Ẇ 2,1 2,q0 (0, T × R3); Ẇ 2,1 2,q1 (0, T × R3) ) 1 2 ,1 . By using (??) and (??) with i = 0 and i = 1, we have Ẇ 2,1 2,(4/3,1)(0, T × R3) ↪→ ( Ls0 ( 0, T ; Ẇ 1 2 (R3) ) ;Ls1 ( 0, T ; Ẇ 1 2 (R3) )) 1 2 ,1 . (4.20) We notice that, the definition of γ, si and qi ensure that 1 2 ( 1 s0 + 1 s1 ) = 1 2 ( 1 q0 + 1 q1 )− γ = 1 2 ( 1 q0 + 1 q1 )− 1 2 = 1 4 . Hence, employing Proposition ??(1), we see that the interpolation space in the right of (??) is L4,1 ( 0, T ; Ẇ 1 2 (R3) ) . That is, Ẇ 2,1 2,( 4 3 ,1) (0, T × R3) ↪→ L4,1 ( 0, T ; Ẇ 1 2 (R3) ) , (4.21) which together with (??), (??), (??) and (??) implies (∇u,∇ω,∇H) ∈ L4,1(0, T ;L2(R3)). (4.22) This along with Proposition ??(2)-(3) yields (∇u,∇ω,∇H) ∈ L2(0, T ;L2(R3)). Therefore, (δu, δω, δH) ∈ L∞( 0, T ;L2(R3) ) ∩ L2 ( 0, T ; Ḣ1(R3) ) . Let us denote ϕ ≜ −(−∆)−1δρ (such that ∥δρ∥Ḣ−1(R3) = ∥∇ϕ∥L2(R3)). Taking the L2-scalar product of the first equation of the system (??) with ϕ, and using div u1 = div u2 = 0, we obtain 1 2 d dt ∥∇ϕ∥2L2(R3) ≤ ∫ R3 ∇u2 : (∇ϕ⊗∇ϕ)dx− ∫ R3 ρ1δu · ∇ϕdx ≤ ∥∇u2∥L∞(R3)∥∇ϕ⊗∇ϕ∥L1(R3) + ∥ρ1∥1/2L∞(R3)∥ √ ρ1δu∥L2(R3)∥∇ϕ∥L2(R3). Through time integration, we find that for all t ∈ [0, T ], ∥∇ϕ∥L2(R3) ≤ ∫ t 0 ∥∇u2∥L∞(R3)∥∇ϕ∥L2(R3)dτ + ∫ t 0 ∥ρ1∥1/2L∞(R3)∥ √ ρ1δu∥L2(R3)dτ. (4.23) 26 X. SONG, C. WANG, X. WANG, F. XU EJDE-2025/58 For convenience, for all t ∈ [0, T ], we denote X(t) ≜ sup τ∈(0,t] τ−1∥δρ(τ)∥Ḣ−1(R3), Y (t) ≜ ( sup τ∈[0,t] ∥√ρ1δu(τ), √ ρ1δω(τ), (δH)(τ)∥2L2(R3) + ∥∇(δu, δω, δH)∥2L2(0,t×R3) )1/2 . It follows from the mass conservation and (??), that X(t) ≤ ∫ t 0 gXdτ +R 1/2 0 Y, where g(t) ≜ ∥∇(u2, ω2, H2)∥L∞(R3), R0 ≜ supx∈R3 ρ0(x). Then applying Gronwall’s inequality gives rise to X(t) ≤ R 1/2 0 Y exp (∫ t 0 gdτ ) . (4.24) Next, we need to bound Y (t). Multiplying (??)(2), (??)(3) and (??)(4) by δu, δω and δH respec- tively, we conclude that 1 2 d dt ∫ Rd ρ1|δu|2dx+ ∫ Rd |∇δu|2dx = − ∫ Rd δρDtu2 · δu dx− ∫ Rd ρ1(δu · ∇u2) · δu dx+ ∫ Rd δH∇H2 · δu dx + ∫ Rd H1∇δH · δu dx+ ∫ Rd curl δω · δu dx, (4.25) 1 2 d dt ∫ Rd ρ1|δω|2dx+ ∫ Rd |∇δω|2dx+ ∫ Rd |div δω|2dx = − ∫ Rd δρDtω2 · δω dx− ∫ Rd ρ1(δu · ∇ω2) · δω dx + ∫ Rd curl δu · δω dx− 2 ∫ Rd δω · δω dx, (4.26) 1 2 d dt ∫ Rd |δH|2dx+ ∫ Rd |∇δH|2dx = − ∫ Rd δu∇H2 · δH dx+ δH∇u2 · δH dx+ ∫ Rd H1∇δu · δH dx. (4.27) In particular, usingDo you mean 1/(8∥ · · · ∥)? ∫ Rd curl δω · δu dx+ ∫ Rd curl δu · δω dx ≤ 2∥δω∥2L2(R3) + 1 8 ∥∇δu∥2L2(R3), and combining (??) and (??), we deduce that 1 2 d dt ∫ Rd ρ1|δu|2dx+ ∫ Rd |∇δu|2dx+ 1 2 d dt ∫ Rd ρ1|δω|2dx + ∫ Rd |∇δω|2dx+ ∫ Rd |div δω|2dx ≤ − ∫ Rd δρDtu2 · δu dx− ∫ Rd ρ1(δu · ∇u2) · δu dx+ ∫ Rd δH∇H2 · δu dx + ∫ Rd H1∇δH · δu dx− ∫ Rd δρDtω2 · δω dx− ∫ Rd ρ1(δu · ∇ω2) · δω dx. (4.28) In what follows, we bound term by term in (??) and (??). Employing Hölder’s and Young’s inequalities and Sobolev embedding Ḣ1(R3) ↪→ L6(R3), we have − ∫ Rd δρDtu2 · δu dx ≤ ∥δρ∥Ḣ−1(R3)∥Dtu2 · δu∥Ḣ1(R3) ≤ X(∥τ∇Dtu2 · δu∥L2(R3) + ∥τDtu2 · ∇δu∥L2(R3)) EJDE-2025/58 GLOBAL SOLUTION FOR MAGNETO-MICROPOLAR EQUATIONS 27 ≤ X(∥τ∇Dtu2∥L3(R3)∥δu∥L6(R3) + ∥τDtu2∥L∞(R3)∥∇δu∥L2(R3)) ≤ X∥∇δu∥L2(R3)(∥τ∇Dtu2∥L3(R3) + ∥τDtu2∥L∞(R3)) ≤ 1 2 ∥∇δu∥2L2(R3) + C2X2 2 (∥τ∇Dtu2∥L3(R3) + ∥τDtu2∥L∞(R3)) 2 ≤ 1 2 ∥∇δu∥2L2(R3) + C2X2 2 f2, where f(t) ≜ ∥t(Dtu2, Dtω2, DtH2)∥L∞(R3) + ∥t∇(Dtu2, Dtω2, DtH2)∥L3(R3). It follows from Hölder’s inequality, that − ∫ Rd ρ1(δu · ∇u2) · δu dx ≤ ∥∇u2∥L∞(R3)∥ √ ρ1δu∥2L2(R3). Similarly, we obtain the estimate for ∫ Rd δρDtω2 ·δω dx and ∫ Rd ρ1(δu ·∇ω2) ·δω dx. Using Hölder’s and Young’s inequalities, we have∫ Rd δH∇H2 · δu dx ≤ C∥δH∥L2(R3)∥δu∥L2(R3)∥∇H2∥L∞(R3) ≤ C∥δH∥2L2(R3)∥∇H2∥L∞(R3) + C∥δu∥2L2(R3)∥∇H2∥L∞(R3),∫ Rd H1∇δH · δu dx ≤ C∥δu∥L2(R3)∥∇δH∥L2(R3)∥H1∥L∞(R3) ≤ C∥δu∥2L2(R3)∥H1∥2L∞(R3) + 1 2 ∥∇δH∥2L2(R3), − ∫ Rd δu∇H2 · δH dx ≤ C∥∇H2∥L∞(R3)∥δu∥L2(R3)∥δH∥L2(R3) ≤ C∥δu∥2L2(R3)∥∇H2∥L∞(R3) + C∥δH∥2L2(R3)∥∇H2∥L∞(R3), − ∫ Rd δH∇u2 · δH dx ≤ C∥δH∥2L2(R3)∥∇u2∥L∞(R3) + C∥δH∥2L2(R3)∥∇u2∥L∞(R3),∫ Rd H1∇δu · δH dx ≤ ∥H1∥L∞(R3)∥∇δu∥L2(R3)∥δH∥L2(R3) ≤ C∥H1∥2L∞(R3)∥δH∥2L2(R3) + 1 2 ∥∇δu∥2L2(R3). Plugging the above estimate into (??) and (??), and integrating on [0, t], we have Y 2 ≤ Y 2(0) + C ∫ t 0 r−1 0 ( g + ∥H1∥2L∞(R3) ) Y 2dτ + C ∫ t 0 f2X2dτ, where r0 ≜ infx∈R3 ρ0(x). It follows from Gronwall’s inequality that Y 2 ≤ exp ( C ∫ t 0 r−1 0 ( g + ∥H1∥2L∞(R3) ) dτ )( Y 2(0) + C ∫ t 0 f2X2 dτ ) , which together with (??) implies that Y 2 ≤ exp ( C ∫ t 0 r−1 0 (g + ∥H1∥2L∞(R3)) dτ )( Y 2(0) + CR0 ∫ t 0 exp ( 2 ∫ τ 0 gdτ ) f2Y 2dτ ) . As a result, applying Gronwall’s inequality again, yields Y 2 ≤ eCR0( ∫ t 0 f2dτ) exp(2 ∫ t 0 gdτ)e C ∫ t 0 r−1 0 (g+∥H1∥2 L∞(R3) ) dτ Y 2(0). (4.29) 28 X. SONG, C. WANG, X. WANG, F. XU EJDE-2025/58 Finally, we conclude from (??) and (??), that sup τ∈[0,t] τ−1∥δρ(τ)∥Ḣ−1(R3) ≤ R 1/2 0 ∥(√ρ0δu0, √ ρ0δω0, δH0)∥L2(R3) exp (C 2 R0 ∫ t 0 f2dτ exp ( 2 ∫ t 0 gdτ )) × exp (C 2 ∫ t 0 r−1 0 (g + ∥H1∥2L∞(R3)) dτ ) (4.30) and ∥( √ ρ1(t)δu(t), √ ρ1(t)δω(t), δH(t))∥2L2(R3) + ∫ t 0 ∥(∇δu,∇δω,∇δH)∥2L2(R3)dτ ≤ ∥(√ρ0δu0, √ ρ0δω0, δH0)∥2L2(R3) × exp ( CR0 ∫ t 0 f2dτ exp ( 2 ∫ t 0 gdτ )) × exp ( C ∫ t 0 r−1 0 (g + ∥H1∥2L∞(R3)) dτ ) , (4.31) which together with (??), (??) and (??) implies that (ρ1, u1, H1,∇π1) = (ρ2, u2, H2,∇π2) on [0, T ]× R3. This completes the proof of Theorem ??. □ Acknowledgments. This research supported by the National Natural Science Foundation of China (12326430), and by the Natural Science Foundation of Shandong Province (ZR2021MA017). References [1] H. Abidi and T. 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Xiao Song School of Mathematics and Statistics, Shandong University of Technology, Zibo Shandong 255049, China Email address: ssxnsong@163.com Chenhua Wang School of Mathematics and Statistics, Shandong University of Technology, Zibo Shandong, 255049, China Email address: wangchenhua 2024@163.com Xiaojie Wang School of Mathematics and Statistics, Shandong University of Technology, Zibo Shandong 255049, China Email address: wxj15615637750@163.com Fuyi Xu (corresponding author) School of Mathematics and Statistics, Shandong University of Technology, Zibo Shandong 255049, China Email address: zbxufuyi@163.com