Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 86, pp. 1–33. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.86 GLOBAL LOW-ENERGY WEAK SOLUTIONS FOR COMPRESSIBLE MAGNETO-MICROPOLAR FLUIDS WITH DISCONTINUOUS INITIAL DATA IN R3 WANPING WU, YINGHUI ZHANG Abstract. This article concerns the weak solutions of a 3D Cauchy problem of compressible magneto-micropolar fluids with discontinuous initial data. Under the assumption that the initial data are of small energy and the initial density is positive and essentially bounded, we establish the existence of weak solutions that are global-in-time. Moreover, we obtain the large-time behavior of such solutions. 1. Introduction We consider the 3D compressible magneto-micropolar fluid equation ρt + div(ρu) = 0, (ρu)t + div(ρu⊗ u) +∇P (ρ) = (µ+ ζ)∆u+ (µ+ λ− ζ)∇ div(u) + 2ζ∇× w + (∇×H)×H, (ρw)t + div(ρu⊗ w) + 4ζw = µ′∆w + (µ′ + λ′)∇div(w) + 2ζ∇× u, Ht −∇× (u×H) = −∇× (ν∇×H), div(H) = 0, (1.1) where the functions ρ = ρ(x, t) ≥ 0, u = u(x, t), P (ρ) = aργ (a > 0, γ > 1), w = w(x, t) and H = H(x, t) are density, velocity, γ-law pressure, micro-rotational velocity and magnetic field for (x, t) ∈ R3 × R+. Furthermore, the unknown con- stants µ, ζ, λ and ν are the shear viscosity coefficient, dynamics micro-rotation viscosity, bulk viscosity coefficient and resistivity coefficient, respectively. µ′ and λ′ denote the angular viscosities which satisfy the conditions: µ, ν, ζ, µ′ > 0, 2µ′ + 3λ′ ≥ 0, 2µ+ 3λ− 4ζ ≥ 0. 1.1. History of the problem. Before introducing the mathematical theory of the system, we explain the significance of studying magneto-micropolar fluids system. Equation (1.1) is commonly used to model the motion of a compressible conducting micropolar fluid in any magnetic field (see [2]). Because of the great research 2020 Mathematics Subject Classification. 35Q35, 35B40,76N10. Key words and phrases. Magneto-micropolar fluids; weak solutions; global-in-time existence; low-energy. ©2023. This work is licensed under a CC BY 4.0 license. Submitted June 12, 2023. Published December 20, 2023. 1 2 W. P. WU, Y. H. ZHANG EJDE-2023/86 value, the research challenge and the phenomenological significance of magnetic- micropolar fluids in physics and mathematics, more and more researchers in physics and mathematics have devoted themselves to the research of magnetic-micropolar fluid equations. The purpose of this article is to prove the existence weak solutions, that are global-in-time and low energy, for the Cauchy problem of system(1.1) with the initial conditions (ρ(·, 0), u(·, 0), w(·, 0), H(·, 0)) = (ρ0, u0, w0, H0). (1.2) So that ρ0(x) has an upper and lower bounds far from zero, and u0(x), w0(x), H0(x) ∈ Lp(R3) satisfy (1.8) and (1.9) for some p > 6. Since (ρ0(x), u0(x), w0(x), H0(x)) is small in L2(R3), the total initial energy is small. No other smallness or regularity conditions are imposed. As is well known, if there is no micro-rotational velocity, magnetic field and dy- namics micro-rotation viscosity, i.e. w = ζ = H = 0, magnetic-micropolar system (1.1) reduces to the compressible Navier-Stokes equations. For initial data ap- proaching non-vacuum equilibrium in the H3(R3) space, Matsumura-Nishida first established relevant results on global classical solutions in [18, 19]. Furthermore, when the initial density of the system is small L2 and bounded in L∞, and the initial velocity is small in L2 and bounded in L2n (the norm of L2 must be slightly weighted in two dimensions), Hoff [8] proved the global existence of weak solu- tions in two and three dimensions. Moreover, Hoff [9] extended the above results to general initial data. Later, Lions [14](see also Feireisl et al. [6]) made a major breakthrough in the existence of solutions in two and three dimensional space with arbitrary initial data. In the absence of a micro-rotation velocity and dynamics micro-rotation viscosity w = ζ = 0, (1.1) reduces to the Magnetohydrodynamic equations (MHD). Assuming that the initial data of system is small in L2 and the initial density is nonnegative and essentially bounded, Hu-Wang [11] established the global-in-time existence of the weak solutions. In the vacuum case, assuming that the initial energy is suitably small in L2, Liu-Yu-Zhang [16] investigated the global existence of weak solutions in three-dimensional space. Other related results of the MHD system can be found in [25, 26, 31] and references therein. When there is no dynamic micro-rotational viscosity ζ > 0 and magnetic field, the compressible micropolar fluids become the magneto-micropolar system. The theory of micropolar fluids was first proposed by Eringen [4] and Lukaszewicz [17]. The problems related to one-dimensional micropolar flow can be referred to the literature such as [20, 21, 22]. In addition, in the case of multi-dimensional microp- olar flows, we refer to [23, 27, 15]. In recent years, for weak solutions to equations with discontinuous initial data, Chen has studied the global existence of compress- ible micropolar fluids in the case that the vacuum state may be included and the oscillations of the solutions can be arbitrarily large. For the compressible magneto-micropolar fluids (1.1), [30] analyzes the global existence and optimal convergence rates of the solutions. In the framework of Lions [14], Amirat-Hamdache [3] proved the global existence of finite-energy weak solutions. Recently, for the case of the half-space Rn+(n = 2, 3), Xu-Tan-Wang-Tong [32] established the global low-energy of the weak solutions magnetic-micropolar fluids (1.1) with no slip boundary and discontinuous initial data. Other relevant results can be found in references [28, 24, 30] and their references. EJDE-2023/86 GLOBAL LOW-ENERGY WEAK SOLUTIONS 3 However, to the best of our knowledge, there is no result on the low-energy weak solutions for the 3D compressible magneto-micropolar fluid systems with discon- tinuous initial data in the whole space. Compared to the NS equation [8] and the MHD system [26], the coupling of the micro-rotation velocity w with the momen- tum equation (1.1)2 brings research difficulties and research challenges in proving time-independent global energy estimates. In the proof, the local existence theo- rem of the smooth solution of the system needs to be established first. Compared with [32], our results and methods are very different. The main differences can be outlined as follows. First, [32] established the local existence Theorem by using the results proved by Kagei and Kawashima on the local solvability of an ini- tial boundary value problem for a quasilinear hyperbolic-parabolic system in [12], while we establish local existence Theorem by using Kawashima’s results in [13]. Secondly, during the proof process, we establish more effective viscous flux and en- ergy functionals, and the uniform estimate of the energy functionals is much more complicated and difficult. Thirdly, to obtain our main results, we derive pointwise bound for the density ρ which is independent both of time and initial smoothness. The proof process consists of a maximum-principle arguments applied to integral curves of the velocity field and Hölder-continuity of u(·, t). In two cases, we derive the upper and lower pointwise bounds of density. However, [32] only proved the boundness of the density ρ. Finally, compared with [32], we also prove low-energy estimates of effective viscous flux and vorticity and the large-time behavior of the weak solutions. 1.2. Main results. Before stating the principal result of the paper, we introduce the notation that we need in later sections. For the definition of Hölder seminorms: if u : R3 → Rm and γ ∈ (0, 1], we have 〈u〉γ = sup y1,y2∈R3, y1 6=y2 |u(y2)− u(y1)| |y2 − y1|γ ; (1.3) and if u : S ⊆ R3 × [0,∞)→ R3 and γ1, γ2 ∈ (0, 1], 〈u〉γ1,γ2 Q = sup (y1,t1),(y2,t2)∈S (y1,t1)6=(y2,t2) |u(y2, t2)− u(y1, t1)| |y2 − y1|γ1 + |t2 − t1|γ2 . (1.4) For simplicity, we abbreviate X3 to X, where X is a Banach space. When I ⊂ [0,∞) is an interval, then C1(I;X) will be the elements u ∈ C(I;X) such that the distribution derivative ut ∈ D′(intI;R3) is regarded as an element of C(I;X). As described in [8, 10], the effective viscous flux plays a crucial part in the study of compressible fluid dynamics. In the following G1 and G2 represent the “effective viscosity” of flux, and W1, W2 denote the vorticity of magneto-micropolar fluids: G1 , (2µ+ λ) div(u)− (P (ρ)− P (ρ̃)), W1 , ∇× u, G2 , (2µ′ + λ′) div(w), W2 , ∇× w. (1.5) In addition, for the pressure P (ρ) = aργ(a > 0, γ > 1), we select two positive bounding densities ρ and ρ̄ to fix a positive reference density ρ̃, ρ < ρ̃ < ρ̄, (1.6) then we define the nonnegative number δ = min{ρ̃− ρ, 1 2 (ρ̄− ρ), ρ̄− ρ̃}. (1.7) 4 W. P. WU, Y. H. ZHANG EJDE-2023/86 It should be mentioned that δ need not be “small”. For the parameters µ, ζ, λ, µ′, λ′, and ν, we assume that ν > 0, 0 ≤ λ′ < (−1 2 + √ 21 6 )µ′, 1 4 (µ+ ζ)(p− 2)− [ 1 4 (µ+ λ− ζ)(p− 2)]2 1 3 (µ+ ζ) + (µ+ λ− ζ) > 0. (1.8) Hence, we obtain 1 4 µ′(p− 2)− [ 1 4 (µ′ + λ′)(p− 2)]2 1 3µ ′ + (µ′ + λ′) > 0, (1.9) for p = 6 and thus for some p > 6, which we now fix. For (ρ0, u0, w0, H0), suppose that we have the nonnegative numbers s < δ and N satisfying ‖u0‖Lp + ‖w0‖Lp + ‖H0‖Lp ≤ N, (1.10) ρ+ s < ess inf ρ0 ≤ ess sup ρ0 < ρ̄− s, (1.11) where N > 0 can be arbitrarily large. At the same time, we assume that div(H)0 = 0 in D′(R3), (1.12) E0 , ∫ R3 (1 2 ρ0|u0|2 +D(ρ0) + 1 2 ρ0|w0|2 + 1 2 |H0|2 ) dx. (1.13) where the potential energy density D can be represented by D(ρ) , ρ ∫ ρ ρ̃ P (s)− P (ρ̃) s2 ds. (1.14) It is easy to show that C1(ρ, ρ̃, ρ̄)(ρ− ρ̃)2 ≤ D(ρ) ≤ C2(ρ, ρ̃, ρ̄)(ρ− ρ̃)2, (1.15) where C1 and C2 are nonnegative constants that depend only on ρ̃, ρ̄ and ρ. Definition 1.1. The weak solutions (ρ, u, w,H) of system (1.1) is defined as follows: we assume that (ρ − ρ̃, ρu, w,H) ∈ C([0,∞);H−1(R3)) with (ρ, u, w,H)|t=0 = (ρ0, u0, w0, H0), (∇u,∇w,∇H) ∈ L2(R3 × (0,∞)) and div(H) = 0 in D′(R3) for t > 0. Furthermore, the following equations hold for t2 > t1 ≥ 0 and C1 test functions ψ having uniformly bounded support in x for t ∈ [t1, t2]:∫ R3 ρψ(t, x) dx ∣∣∣t2 t1 = ∫ t2 t1 ∫ R3 (ρψt + ρu · ∇ψ) dxdt, (1.16)∫ R3 ρuψ(t, x) dx ∣∣∣t2 t1 + ∫ t2 t1 ∫ R3 ((µ+ ζ)∇u · ∇ψ + (µ+ λ− ζ)(div(u))∇ψ) dxdt = ∫ t2 t1 ∫ R3 ( ρuψt + ρuu · ∇ψ + P (ρ)divψ + 2ζw rotψ + 1 2 ∇|H|2divψ −HTH∇ψ ) dxdt, (1.17) EJDE-2023/86 GLOBAL LOW-ENERGY WEAK SOLUTIONS 5∫ R3 ρwψ(t, x) dx ∣∣t2 t1 + ∫ t2 t1 ∫ R3 (µ′∇w · ∇ψ + (µ′ + λ′)(div(w))divψ + 4ζwψ) dx dt = ∫ t2 t1 ∫ R3 (ρwψt + ρuw · ∇ψ + 2ζu rotψ) dx dt, (1.18)∫ R3 H(x, t)ψ(x, t) dx ∣∣∣t2 t1 + ν ∫ t2 t1 ∫ R3 ∇H∇ψ dxdt = ∫ t2 t1 ∫ R3 (HTu− uTH)∇ψ dxdt. (1.19) The main results of this article reads as follows. Theorem 1.2. Let the parameters of system (1.1)-(1.2) satisfy (1.6)–(1.9), let the nonnegative numbers N and s < δ be given. Then,depending on N , a nonnegative lower bound for s, the parameters and hypothesis of (1.6)-(1.9), there exist positive numbers ε, C, and τ , such that, if (ρ0, u0, w0, H0) satisfies (1.16)-(1.19) and E0 < ε. (1.20) Then we have a weak solution (ρ, u, w,H) which satisfies Definition 1.1. At the same time, the solution satisfies the following: ρ− ρ̃, ρu, w,H ∈ C([0,∞);H−1(R3)), (1.21) ∇u,∇w,∇H ∈ L2([0,∞);R3), (1.22) u(·, t), w(·, t), H(·, t) ∈ H1(R3), t > 0, (1.23) G1(·, t), G2(·, t),W1(·, t),W2(·, t) ∈ H1(R3), t > 0, (1.24) 〈u〉1/2,1/8R3×[ε,∞), 〈w〉 1/2,1/8 R3×[ε,∞), 〈H〉 1/2,1/8 R3×[ε,∞) ≤ C(ε)Cτ0 , (1.25) where C(ε) may depand on the nonnegative lower bound of ε, ρ ≤ ρ(x, t) ≤ ρ̄ a.e. on R3 × [0,∞), (1.26) and sup t>0 ∫ R3 [ |ρ− ρ̃|2 + |u|2 + |w|2 + |H|2 + ϑ(|∇u|2 + |∇w|2 + |w|2 + |∇H|2) + ϑ5(G2 1 +G2 2 + |∇W1|2 + |∇W2|2) ] dx + ∫ ∞ 0 ∫ R3 [|∇u|2 + |∇w|2 + |∇H|2 + ϑ ( |u̇|2 + |ẇ|2 + |Ht|2 + |∇W1|2 + |∇W2|2 ) + ϑ5(|∇u̇|2 + |∇ẇ|2 + |∇Ht|2)] dx ds ≤ CCτ0 , (1.27) where ϑ(t) = min{1, t}. Moreover, we also have the following large-time behavior: lim t→∞ (‖ρ− ρ̃‖Ll(R3) + ‖u‖W 1,r(R3) + ‖w‖W 1,r(R3) + ‖H‖W 1,r(R3)) = 0, (1.28) holds for l ∈ (2,∞), r ∈ (2, 6). In section 2, we list a number of auxiliary inequalities and fundamental results that play an irreplaceable role in the proof. In section 3, we work on calculating energy estimates independent of time. In section 4, we establish the upper and 6 W. P. WU, Y. H. ZHANG EJDE-2023/86 lower limits of the key points of density. Finally, we complete the proof of the Theorem 1.2 in section 5. 2. Preliminaries First, we introduce the celebrated Gagliardo-Nirenberg inequality [1, 33]. Lemma 2.1. For each α ∈ [2, 6] and f ∈ H1(R3), we can find a constant C(α) such that ‖f‖Lα(R3) ≤ C(α)‖f‖(6−α)/2α L2(R3) ‖∇f‖(3α−6)/2α L2(R3) . (2.1) For any α ∈ (3,∞), q > 1 and f ∈ Lq(R3) ∩W 1,α(R3), there exists a constant C(α, q) such that ‖f‖L∞(R3) ≤ C(α, q)‖f‖q(α−3)/(3α+q(α−3)) Lq(R3) ‖∇f‖3α/(3α+q(α−3)) Lα(R3) , (2.2) 〈f〉βR3 ≤ C(α)‖∇f‖Lα(R3), (2.3) where β = 1− 3 α . The next Lemma can be found in [8]. Lemma 2.2. Given q1 ∈ [1, 3) and q2 ∈ (3,∞], let Γ be the fundamental solution for the Laplace operator in R3, we can find a constant C = C(n, q1, q2) such that ‖Γxj ∗ g‖L∞(R3) ≤ C(n, q1, q2)[‖g‖Lq1 (R3) + ‖g‖Lq2 (R3)]. (2.4) In the process of proving Theorem 1.2, the existence of smooth solutions plays an indispensable role. Therefore, based on a direct generalization of the classical results of the Navier-Stokes equation [29] and Magnetohydrodynamics equation [13], we will give the following formal existence results: Theorem 2.3. Suppose that µ, µ′, ζ and ν are positive constants and the pressure satisfy P ∈ C3((0,∞)). Then given ρ̃ > ρ > 0 and C3 > 0, there exists a positive time T which depends on ρ̃, ρ, and C3 and on the system parameters µ, ζ, µ′, ν, λ, λ′ and P , such that, if the initial data (ρ̄0 − ρ̃, u0, w0, H0) satisfies ‖(ρ0 − ρ̃, u0, w0, H0)‖H3(R3) < C3, (2.5) inf ρ0 ≥ ρ, and div(H)0 = 0, then in R3 × [0, T ] the solution (ρ, u, w,H) of (1.1)- (1.2) satisfying ρ− ρ̃ ∈ C1([0, T ];H2(R3)) ∩ C([0, T ];H3(R3)), (2.6) u,w,H ∈ C1([0, T ];H1(R3)) ∩ C([0, T ];H3(R3)) ∩ L2([0, T ];H4(R3)). (2.7) Then the equations in (1.1) are satisfied in the sense of equality of weak derivatives on R3×(0, T ), and each weak derivative is treated as an element of C([0, T ];H1(R3)), and the weak form (1.16)-(1.19) hold. In addition, there exists a nonnegative number that depends on µ, ζ, µ′, ν, λ, λ′ and P . Then, if the above assumptions hold with C3 < ε, the solution exists on R3 × [0,∞). Lemma 2.4. Under the assumption that (ρ, u, w,H) is the smooth solution of (1.1) on (0, T )× R3. Then, for any 2 ≤ p ≤ 6, there exists a constant C > 0 depending on µ, λ, µ′, λ′ and ζ such that ‖(∇G1,∇G2,∇W1,∇W2)‖Lp ≤ C‖(ρu̇, ρẇ,∇u,∇w,w,H · ∇H,∇|H|2)‖Lp , (2.8) EJDE-2023/86 GLOBAL LOW-ENERGY WEAK SOLUTIONS 7 ‖∇u(·, t)‖Lp ≤ C‖G1(·, t),W1(·, t), (P (ρ)− P̃ )(·, t)‖Lp , (2.9) ‖∇w(·, t)‖Lp ≤ C‖G2(·, t),W2(·, t)‖Lp , (2.10) where the definitions of G1, G2, W1 and W2 can be found (1.5) and ‖(f, g)‖Lp represents ‖f‖Lp + ‖g‖Lp . Proof. According to (1.5), one has ρu̇ = ∇G1 − (µ+ ζ) rotW1 + 2ζ rotw − (∇×H)×H, ρẇ + 4ζw = ∇G2 − µ′ rotW2 + 2ζ rotu, (2.11) and thus we obtain the following equations ∆G1 = div(ρu̇)− div[(∇×H)×H], ∆G2 = div(ρẇ)− 4ζ div(w), (µ+ ζ)∆W1 = ∇× (ρu̇)− 2ζ∇×W2 −∇× [(∇×H)×H], µ′∆W2 − 4ζW2 = ∇× (ρẇ)− 2ζ∇×W1, (2.12) where ġ , gt + u · ∇g. Next, applying standard Lp-estimates for elliptic systems, we obtain (2.8). Meanwhile, by the definition −∆u = ∇×W1 −∇ div u we have ∇u = ∇(−∆)−1∇×W1 −∇(−∆)−1∇ div u. (2.13) On the other hand, employing the Marcinkiewicz multiplier Theorem for the above relation, it holds that ‖∇u(·, t)‖Lp ≤ C[div u+ ‖W1(·, t)‖Lp ] ≤ C‖G1(·, t),W1(·, t), (P − P̃ )(·, t)‖Lp . (2.14) Inequality (2.10) can be obtained by employing the similar method. The proof is complete. � 3. Energy estimates Here we establish several prior bounds of the smooth solution described in Section 2, which can roughly correspond to (1.27). These estimates require quite complex and technical methods. Therefore we omit the identical and analogous parts of the proof as in [8] and [26]. Specifically, we define a new energy functional A(t) = sup 0 1 and 1 ≤ t ≤ T , one has sup 1≤s≤t (‖∇u‖2L2 + ‖w‖2L2 + ‖∇w‖2L2 + ‖∇H‖2L2) + ∫ t 1 ( ‖ρ1/2u̇‖2L2 + ‖ρ1/2ẇ‖2L2 + ‖Ht‖2L2 ) ds ≤M [ C0 + C 3/2 0 Ā1/2 + Q̄ ] +A(1). (3.17) Here 1 ∧ T = min{1, T}. Proof. For the case of 0 ≤ t ≤ 1 ∧ T , multiplying (1.1)2-(1.1)3 by ϑu̇ and ϑẇ, respectively, and integrating the resultant equations over R3 × [0, t], we have∫ t 0 ϑ ( ‖ρ1/2u̇‖2L2 + ‖ρ1/2ẇ‖2L2 ) ds = − ∫ t 0 ∫ R3 ϑu̇ · ∇P (ρ) dx+ (µ+ ζ) ∫ t 0 ∫ R3 ϑu̇ ·∆udx ds + (µ+ λ− ζ) ∫ t 0 ∫ R3 ϑu̇ · ∇ div(u) dxds+ µ′ ∫ t 0 ∫ R3 ϑẇ ·∆w dxds + (µ′ + λ′) ∫ t 0 ∫ R3 ϑẇ · ∇ div(w) dx ds+ 2ζ ∫ t 0 ∫ R3 ϑ rotw · u̇dx ds − 4ζ ∫ t 0 ∫ R3 ϑw · ẇ dxds+ 2ζ ∫ t 0 ∫ R3 ϑ rotu · ẇ dxds + ∫ t 0 ∫ R3 ϑ(H · ∇H − 1 2 ∇|H|2) · u̇dxds. (3.18) From (1.1), we can infer that (P (ρ)− P (ρ̃))t + γP (ρ) div(u) + u · ∇(P (ρ)− P (ρ̃)) = 0, (3.19) Then integrating by parts and (3.19) for the first term, we have − ∫ t 0 ∫ R3 ϑu̇ · ∇P (ρ) dx = − ∫ t 0 ∫ R3 ϑ(ut + u · ∇u)∇P (ρ) dxds = ϑ(t) ∫ R3 div(u)(P (ρ)− P (ρ̃)) dx− ∫ t 0 ∫ R3 ϑt div(u)(P (ρ)− P (ρ̃)) dxds − ∫ t 0 ∫ R3 ϑ((P (ρ)− P (ρ̃))t div(u) +∇P (ρ)u · ∇u) dx ds = ϑ(t) ∫ R3 div(u)(P (ρ)− P (ρ̃)) dx− ∫ t 0 ∫ R3 ϑt div(u)(P (ρ)− P (ρ̃)) dxds + ∫ t 0 ∫ R3 ϑ((γ − 1)(P (ρ)− P (ρ̃))(div(u))2 +∇P (ρ)u · ∇u) dx ds ≤M [ ϑ(t) ∫ R3 |∇u||(ρ− ρ̃)|dx+ ∫ 1∧t 0 ∫ R3 |∇u||ρ− ρ̃|dx + ∫ t 0 ∫ R3 |∇u|2 dxds ] . (3.20) EJDE-2023/86 GLOBAL LOW-ENERGY WEAK SOLUTIONS 11 For the second term in (3.18), we obtain (µ+ ζ) ∫ t 0 ∫ R3 ϑ(ut + u · ∇u)∆udxds = − (µ+ ζ)ϑ(t) 2 ‖∇u‖2L2 dx+ (µ+ ζ) 2 ∫ 1∧t 0 ‖∇u‖2L2 dxds + ∑ 1≤hi,kj≤3 ∫ t 0 ∫ R3 ϑuk1 xh1 uk2 xh2 uk3 xh3 ds, (3.21) Similarly, we have (µ+ λ− ζ) ∫ t 0 ∫ R3 ϑ(ut + u · ∇u) · ∇ div(u) dx ds = − (µ+ λ− ζ)ϑ(t) 2 ‖ div(u)‖2L2 + (µ+ λ− ζ) 2 ∫ 1∧t 0 ‖div(u)‖2L2 ds + ∑ 1≤hi,kj≤3 ∫ t 0 ∫ R3 ϑuk1 xh1 uk2 xh2 uk3 xh3 dx ds. (3.22) Later, to deal with the terms 2ζ ∫ t 0 ∫ R3 ϑ rotw · u̇dx ds and 2ζ ∫ t 0 ∫ R3 ϑ rotu ·ẇ dxds, we use integration by parts to obtain 2ζ ∫ t 0 ∫ R3 ϑ rotw · u̇dxds+ 2ζ ∫ t 0 ∫ R3 ϑ rotu · ẇ dxds = 2ζ ∫ t 0 ∫ R3 ϑ(w · rotut + wt · rotu) dxds + 2ζ ∫ t 0 ∫ R3 ϑ(rotw · (u · ∇u) + rotu · (u · ∇w)) dxds ≤ 2ζϑ(t) ∫ R3 w · rotudx− 2ζ ∫ 1∧t 0 ∫ R3 w · rotudxds+ 2ζ ∫ t 0 ∫ R3 ϑ|∇w|2 dxds + 2ζ ∫ t 0 ∫ R3 ϑ|u|2|∇u|2 dxds+ 2ζ ∫ t 0 ∫ R3 ϑ|∇u|2 dxds + 2ζ ∫ t 0 ∫ R3 ϑ|u|2|∇w|2 dxds. For the terms µ′ ∫ t 0 ∫ R3 ϑẇ · ∆w dxds and (µ′ + λ′) ∫ t 0 ∫ R3 ϑẇ · ∇div(w) dx ds, we take the similar proof of (3.21) and (3.22). At the same time, it is obvious that − 4ζ ∫ t 0 ∫ R3 ϑw · ẇ dxds = −2ζϑ(t) ∫ R3 w2 dx+ 2ζ ∫ 1∧t 0 ∫ R3 w2 dxds− 4ζ ∫ t 0 ∫ R3 ϑw · (u · ∇w) dx ≤ −2ζϑ(t) ∫ R3 w2 dx+ 2ζ ∫ 1∧t 0 ∫ R3 w2 dxds+ 2ζ ∫ t 0 ∫ R3 ϑ|w|2 dx ds + 2ζ ∫ t 0 ∫ R3 ϑ|u|2|∇w|2 dxds. (3.23) 12 W. P. WU, Y. H. ZHANG EJDE-2023/86 Then, for magnetic field H, multiplying the magnetic field equation by ϑHt and integrating, we have∫ t 0 ϑ‖Ht‖2L2 ds+ 1 2 ν ∫ t 0 ϑ‖∇H‖2L2 ds = 1 2 ν ∫ 1∧t 0 ‖∇H‖2L2 ds+ ∫ t 0 ∫ R3 ϑHt[∇× (u×H)] dxds. (3.24) Plugging (3.20)-(3.24) into (3.18) and employing Cauchy inequality, we obtain sup 0 1 and 1 ≤ t ≤ T , it holds that sup 1≤s≤t ( ‖u̇‖2L2 + ‖Ht‖2L2 + ‖ẇ‖2L2 ) + ∫ t 1 ( ‖∇u̇‖2L2 + ‖∇Ht‖2L2 + ‖∇ẇ‖2L2 ) ds ≤M { C0 + C0ĀQ̄+ C 2(q−3) 3(q−2) 0 Ḡ 2 3(q−2) q Ā+ Q̄ } +A(1). (3.29) Proof. For the case of 0 ≤ t ≤ 1 ∧ T , (1.1)2 and (1.1)3 can be rewritten as ρu̇+∇P = (µ+ ζ)∆u+ 2ζ∇× w + (µ+ λ− ζ)∇div(u) + (∇×H)×H, ρẇ + 4ζw = µ′∆w + 2ζ∇× u+ (µ′ + λ′)∇div(w). (3.30) Next, for the magnetic field H, taking the derivative of (1.1)4 with respect to t, multiplying the result by ϑ5Ht and integrating over R3 × [0, t], we have 1 2 ϑ5‖Ht‖2L2 + ν ∫ t 0 ϑ5‖∇Ht‖2L2 ds = 5 2 ∫ t 0 ϑ4ϑ′‖Ht‖2L2 ds+ ∫ t 0 ∫ R3 ϑ5[∇× (u×H)]tHt dxds. (3.31) By applying the derivative (3.30) with respect to t and (1.1)1, we can infer that ρu̇t + ρu · ∇u̇+∇(P (ρ)t)− (2ζ∇× w + (∇×H)×H)t = (µ+ ζ)∆u̇+ (µ+ λ− ζ)∇ div u̇− [(µ+ ζ)∆(u · ∇u) + (µ+ λ− ζ)∇div(u · ∇u)] + div[((µ+ ζ)∆u+ (µ+ λ− ζ)∇div(u))⊗ u −∇P (ρ)⊗ u+ (2ζ∇× w + (∇×H)×H)⊗ u], (3.32) and ρẇt + ρw · ∇ẇ + (4ζw)t − (2ζ∇× u)t = µ′∆u̇+ (µ′ + λ′)∇div ẇ − [µ′∆(u · ∇w) + (µ′ + λ′)∇ div(u · ∇w)] + div[(µ′∆w + (µ′ + λ′)∇ div(w))⊗ u− (4ζw − 2ζ∇× u)⊗ u]. (3.33) 14 W. P. WU, Y. H. ZHANG EJDE-2023/86 Multiplying (3.32) and (3.33) by ϑ5u̇ and ϑ5ẇ, respectively and integrating over R3 × [0, t], we obtain sup 0 1, one has sup 1≤s≤t (‖∇G1|2 + ‖∇G2|2 + ‖∇W1|2 + ‖∇W2|2) + ∫ t 1 (‖∇G1|2 + ‖∇G2|2 + ‖∇W1|2 + ‖∇W2|2) ds ≤M ( Ā+ Q̄+ sup 1≤s≤t ∫ R3 |u̇|2 dx+ ∫ t 1 ∫ R3 |u̇|2 + sup 1≤s≤t ∫ R3 |ẇ|2 dx+ ∫ t 1 ∫ R3 |ẇ|2 dxds ) . (3.48) 18 W. P. WU, Y. H. ZHANG EJDE-2023/86 Proof. From the definition of G1 and G2, we obtain ∆G1 = div(ρu̇) − div[(∇ × H)×H] and ∆G2 = div(ρẇ)− 4ζ div(w). Then∫ t 0 ∫ R3 ϑ(|∇G1|2 + |∇G2|2 + |∇W1|2 + |∇W2|2) dxds ≤M ∫ t 0 ∫ R3 ϑ(|u̇|2 + |ẇ|2 + |∇u|2 + |∇w|2 + |w|2 + |∇H|2|H|2) dx ds ≤M [ A+ (∫ t 0 ∫ R3 |H|6 dx ds )1/3(∫ t 0 ∫ R3 ϑ3|∇H|2 dxds )2/3] + ∫ t 0 ∫ R3 ϑ|u̇|2 dxds+ ∫ t 0 ∫ R3 ϑ|ẇ|2 dx ds ≤M [ A+Q2/3C q−6 3q−6 0 O 4 3q−6 q ] + ∫ t 0 ∫ R3 ϑ|u̇|2 dxds+ ∫ t 0 ∫ R3 ϑ|ẇ|2 dxds, and ϑ5 ∫ R3 (|∇G1|2 + |∇G2|2 + |∇W1|2 + |∇W2|2) dxds ≤M [ sup 0≤s≤t ∫ R3 ϑ5(|u̇|2 + |ẇ|2 + |∇u|2 + |∇w|2 + |w|2 + |∇H|2|H|2) dx ds ] ≤M(A+ P ) + sup 0 1 and 1 ≤ t ≤ T , one has P̄ ≤M [ C 1/4 0 Ā7/4 + C0Ā 4 + Ā2 + Ā5 ] . (3.50) Proof. We only focus on the proof of (3.49), since the proof of (3.50) is similar. Using Gagliardo-Nirenberg inequality and Cauchy inequality, we can bound the first term of P as∫ R3 ϑ5|∇u|2|H|2 dx ≤ ϑ4‖H(·, t)‖2L∞ [ϑ‖∇u(·, t)‖2L2 ] ≤MA [ ϑ4‖H(·, t)‖2L4 + ϑ4‖∇H(·, t)‖2L4 ] ≤MA [ C q−4 2q−4 0 O 2 2q−4 q + ‖ϑ1/2∇H(·, t)‖1/2L2 ‖ϑ5/2∇2H(·, t)‖3/2L2 ] . (3.51) From (1.1)4, we have ‖ϑ5/2∇2H(·, t)‖L2 ≤M [ ∫ R3 ϑ5(|Ht|2 + |∇H · u|2 + |∇u ·H|2) dx ]1/2 ≤M(A+ P )1/2, (3.52) EJDE-2023/86 GLOBAL LOW-ENERGY WEAK SOLUTIONS 19 which implies∫ R3 ϑ5|∇u|2|H|2 dx ≤MA [ C q−4 2q−4 0 O 2 2q−4 q +A1/4(A+ P )3/4 ] . (3.53) The terms ∫ R3 ϑ 5|∇H|2|H|2 dx and ∫ R3 ϑ 5|∇H|2|u|2 dx in P can be estimated in the same method, and we obtain (3.49). � Now we estimate the energy functional Oq. Lemma 3.7. Under the assumptions of Proposition 3.1, for 0 < t ≤ T one has Oq ≤M [ C p−q p−2 0 N q−2 p−2 + C q−6 6q−12 0 O 2 3q−6 +1 q + C q+3 3(q−2) 0 O 3q−11 3(q−2) q + C q 6q−12 0 O 3q−7 3q−6 q ] . (3.54) Proof. Multiplying (1.1)2 and (1.1)3 by |u|q−2u and |w|q−2w, respectively and in- tegrating by parts, we obtain q−1 ∫ R3 ρ|u|q dx ∣∣t 0 + q−1 ∫ R3 ρ|w|q dx ∣∣t 0 + ∫ t 0 ∫ R3 µ′|w|q−2|∇w|2 + (µ+ ζ)|u|q−2|∇u|2 dxds + ∫ t 0 ∫ R3 [1 4 (µ+ ζ)(q − 2)|u|q−4|∇(|u|2)|2 + (µ+ λ− ζ)|u|q−2(div u)2 ] dxds + ∫ t 0 ∫ R3 [1 4 µ′(q − 2)|∇(|w|2)|2|w|q−4 + (µ′ + λ′)|w|q−2(divw)2 ] dxds = ∫ t 0 ∫ R3 [ div(|u|q−2u)(P − P̃ ) + |u|q−2u · ((∇×H)×H) + 2ζ|u|q−2u · (∇× w) ] dxds − ∫ t 0 ∫ R3 1 2 (µ+ λ− ζ)(q − 2)|u|q−4(div u)u · ∇(|u|2) dxds − ∫ t 0 ∫ R3 1 2 (µ′ + λ′)(q − 2)|w|q−4(divw)w · ∇(|w|2) dxds + ∫ t 0 ∫ R3 [2ζ|w|q−2w · (∇× u)] dxds := 4∑ i=1 Ji. (3.55) Firstly, we estimate the second integral of (3.55). For % > 0 we have J2 ≤ 1 2 (µ+ λ− ζ)(q − 2) ∫ t 0 ∫ R3 |u| q−2 2 |div u||u| q−4 2 |∇(|u|2)|dx ds ≤ 1 4 (µ+ λ− ζ)(q − 2) [ % ∫ t 0 ∫ R3 |u|q−2|div u|2 dxds + %−1 ∫ t 0 ∫ R3 |u|q−4|∇(|u|2)|2 dxds ] . 20 W. P. WU, Y. H. ZHANG EJDE-2023/86 If we select 1 4 (µ+ λ− ζ)(q − 2)% = β(µ+ ζ) + (µ+ λ− ζ), (3.56) for a fixed β > 0, then J2 ≤ 3β(µ+ ζ) ∫ t 0 ∫ R3 |u|q−2|∇u|2 dxds+ (µ+ λ− ζ) ∫ t 0 ∫ R3 |u|q−2(div u)2 dxds + [ 1 4 (µ+ λ− ζ)(q − 2)]2 β(µ+ ζ) + (µ+ λ− ζ) ∫ t 0 ∫ R3 |u|q−4|∇(|u|2)|2 dxds. (3.57) Similarly, for any %′ > 0, we have J3 ≤ 1 2 (µ′ + λ′)(q − 2) ∫ t 0 ∫ R3 |w| q−2 2 |divw||w| q−4 2 |∇(|w|2)|dxds ≤ 1 4 (µ′ + λ′)(q − 2) [ %′ ∫ t 0 ∫ R3 |w|q−2|divw|2 dx ds + %′ −1 ∫ t 0 ∫ R3 |w|q−4|∇(|w|2)|2 dx ds ] . If we select 1 4 (µ′ + λ′)(q − 2)%′ = β′µ′ + (µ′ + λ′), (3.58) for a fixed β′ > 0, J3 ≤ 3β′µ′ ∫ t 0 ∫ R3 |w|q−2|∇w|2 dxds+ (µ′ + λ′) ∫ t 0 ∫ R3 |w|q−2(divw)2 dxds + [ 1 4 (µ′ + λ′)(q − 2)]2 β′µ′ + (µ′ + λ′) ∫ t 0 ∫ R3 |w|q−4|∇(|w|2)|2 dxds. (3.59) Substituting (3.57) and (3.59) into (3.55), we obtain q−1 ∫ R3 ρ|u|q dx ∣∣t 0 + (µ+ ζ)(1− 3β) ∫ t 0 ∫ R3 |∇u|2|u|q−2 dxds + q−1 ∫ R3 ρ|w|q dx ∣∣t 0 + µ′(1− 3β′) ∫ t 0 ∫ R3 |w|q−2|∇w|2 dxds + [1 4 (µ+ ζ)(q − 2)− [ 1 4 (µ+ λ− ζ)(q − 2)]2 β(µ+ ζ) + (µ+ λ− ζ) ] ∫ t 0 ∫ R3 |∇(|u|2)|2|u|q−4 dxds + [1 4 µ′(q − 2)− [ 1 4 (µ′ + λ′)(q − 2)]2 β′µ′ + (µ′ + λ′) ] ∫ t 0 ∫ R3 |∇(|w|2)|2|w|q−4 dxds ≤ ∣∣ ∫ t 0 ∫ R3 (P − P̃ ) div(|u|q−2u) dxds ∣∣+ ∣∣ ∫ t 0 ∫ R3 2ζ|u|q−2u · (∇× w) dxds ∣∣ + ∣∣ ∫ t 0 ∫ R3 |u|q−2u · (H · ∇H) dx ds ∣∣+ ∣∣ ∫ t 0 ∫ R3 |u|q−2u · ∇( 1 2 |H|2) dxds ∣∣ + ∣∣ ∫ t 0 ∫ R3 2ζ|w|q−2w · (∇× u) dxds ∣∣. Because q ∈ [6, p), inequalities (1.8) and (1.9) remain unchanged when p is replaced by q. When β = 1/3, the condition int the left bracket is nonnegative. Then for EJDE-2023/86 GLOBAL LOW-ENERGY WEAK SOLUTIONS 21 some β ∈ (0, 1 3 ), this term which we now fix is positive. Next, we have q−1 ∫ R3 ρ|u|q dx ∣∣t 0 + q−1 ∫ R3 ρ|w|q dx ∣∣t 0 + ∫ t 0 ∫ R3 |∇(|u|2)|2|u|q−4 dx ds+ ∫ t 0 ∫ R3 |w|q−4|∇(|w|2)|2 dxds ≤M [∣∣ ∫ t 0 ∫ R3 div(|u|q−2u)(P − P̃ ) dx ds ∣∣+ ∣∣ ∫ t 0 ∫ R3 2ζ|u|q−2u · (∇× w) dxds ∣∣ + ∣∣ ∫ t 0 ∫ R3 |u|q−2u · (H · ∇H) dxds ∣∣+ ∣∣ ∫ t 0 ∫ R3 |u|q−2u · ∇( 1 2 |H|2) dx ds ∣∣ + ∣∣ ∫ t 0 ∫ R3 2ζ|w|q−2w · (∇× u) dxds ∣∣]. (3.60) For the magnetic field H, in a similar way as for (3.55), one has q−1 ∫ R3 |H|q dx ∣∣t 0 + ∫ t 0 ∫ R3 ν|H|q−2|∇H|2 + 1 4 v(q − 2)|H|q−4|∇(|H|2)|2 dx ds = − ∫ t 0 ∫ R3 |H|q−2H · (∇× (u×H)) dx ds. (3.61) By adding (3.60) and (3.61), and using the Cauchy-Schwartz inequality, we obtain∫ R3 (|u|q + |w|q + |H|q) dx + ∫ t 0 ∫ R3 (|u|q−2|∇u|2 + |w|q−2|∇w|2 + |H|q−2|∇H|2) dxds + ∫ t 0 ∫ R3 [ |∇(|u|2)|2|u|q−4 + |w|q−4|∇(|w|2)|2 + |H|q−4|∇(|H|2)|2) ] dxds ≤M [ ∫ R3 (|u0|q + |w0|q + |H0|q) dx+ ∫ t 0 ∫ R3 |ρ− ρ̃| |div(|u|q−2u)|dx ds + ∣∣ ∫ t 0 ∫ R3 |u|q−2u · (H · ∇H) dxds ∣∣+ ∣∣ ∫ t 0 ∫ R3 |u|q−2u · ∇( 1 2 |H|2) dx ds ∣∣ + ∣∣ ∫ t 0 ∫ R3 |H|q−2H · (∇× (u×H)) dx ds ∣∣ + ∣∣ ∫ t 0 ∫ R3 2ζ|w|q−2w · (∇× u) dxds ∣∣ + ∣∣ ∫ t 0 ∫ R3 2ζ|u|q−2u · (∇× w) dx ds ∣∣] := 7∑ i=1 Ji. (3.62) By Hölder’s and Sobolev’s inequalities, one has J1 ≤ (∫ R3 |u0|2 + |w0|2 + |H0|2 dx ) p−q p−2 (∫ R3 |u0|p + |w0|p + |H0|p dx ) q−2 p−2 ≤MC p−q p−2 0 N q−2 p−2 , (3.63) 22 W. P. WU, Y. H. ZHANG EJDE-2023/86 Employing Gagliardo-Nirenberg inequality, one has J2 ≤ [ ∫ t 0 ∫ R3 |u|2q−4 dxds ]1/2[ ∫ t 0 ∫ R3 |∇u|2 dxds ]1/2 ≤ C1/2 0 [ ∫ t 0 (∫ R3 |u|3q dx )1/3(∫ R3 |u| 32 (q−4) dx )2/3 ds ]1/2 ≤ C1/2 0 [ ∫ t 0 (∫ R3 |u|q−2|∇u|2 dx )(∫ R3 |u| 32 (q−4) dx )2/3 ds ]1/2 ≤ C1/2 0 O1/2 q sup 0 1, it holds that Q̄ ≤M [ϕ1(C0 +A(1)) + ϕ2(Ā)]. (3.69) The polynomial ϕ1 contains no constant term, and the degrees of the monomials in ϕ2 are strictly greater than 1. Proof. We only give the proof of (3.68), and we only need to estimate the term∫ t 0 ∫ R3 [ϑ3/2(|∇u|3 + |∇w|3 + |∇H|3) + ϑ5(|∇u|4 + |∇w|4 + |∇H|4)] dxds occurring in the definition (3.3) of P . The term∑ 1≤ki,jm≤3 ∣∣ ∫ t 0 ∫ R3 ϑuj1xk1 uj2xk2 uj3xk3 dxds ∣∣ has been bounded in [8]. First, from Lemma 2.4, we have∫ t 0 ∫ R3 ϑ5|∇u|4 dx ds ≤M [ ∫ t 0 ∫ R3 ϑ5(|ρ− ρ̃|4 + |G1|4 + |W1|4) dx ds ] . (3.70) 24 W. P. WU, Y. H. ZHANG EJDE-2023/86 The first term in (3.70) can be bounded by MC0. For the second term, applying Gagliardo-Nirenberg inequality, we have∫ t 0 ∫ R3 ϑ5|G1|4 dx ds ≤ ( sup 0≤s ∫ R3 ϑ|G1|2 dx )1/2( ϑ5 ∫ R3 |∇G1|2 dx )1/2(∫ t 0 ∫ R3 ϑ|∇G1|2 dxds ) . (3.71) From the definition of G1 and Lemma 3.2, we have( sup 0≤s≤t ∫ R3 ϑ|G1|2 dx )1/2 ≤M(C0 +A)1/2. (3.72) Meanwhile, from the defintion ∆G1 = div(ρu̇)− div[(∇×H)×H], we obtain∫ t 0 ∫ R3 ϑ|∇G1|2 dxds ≤M ∫ t 0 ∫ R3 ϑ|g|2 dxds ≤M [ ∫ t 0 ∫ R3 ϑ(|u̇|2 + |∇H|2|H|2) dxds ] ≤M [ A+ (∫ t 0 ∫ R3 ϑ3/2|∇H|2 dx ds )2/3(∫ t 0 ∫ R3 |H|6 dx ds )1/3] ≤M [ A+ P 2/3C q−6 3q−6 0 O 4 3q−6 q ] , (3.73) and ϑ5 ∫ R3 |∇G1|2 dx ≤M [ sup 0≤s≤t ∫ R3 ϑ5(|u̇|2 + |∇H|2|H|2) dx ] ≤M(A+Q). (3.74) Thus∫ t 0 ∫ R3 ϑ5|G1|4 ≤M(C0 +A)1/2(A+Q)1/2 ( A+ P 2/3C q−6 3q−6 0 O 4 3q−6 q ) . (3.75) Similarly, using Lemma 3.5, we obtain∫ t 0 ∫ R3 ϑ5|W1|4 ≤ ( sup 0 1, it holds that Ā ≤M [ϕ1(A(1) + C0) + ϕ2(Ā)]. (3.81) 26 W. P. WU, Y. H. ZHANG EJDE-2023/86 3.1. Proof of Proposition 3.1. The statement can be derived directly from the bounds (3.80) and (3.81) and the fact that the functions A, Ā,Oq are continuous in time. 4. Point wise bounds for the density We establish pointwise estimates for the density ρ, which are independent both of time and of initial smoothness. At the same time, this will close the estimates of Proposition 3.1, and provide an uncontingent estimate for the energy functional A defined in (3.1). We next list two auxiliary lemmas. The first lemma in Hoff [8] is a maximum-principle argument applied to integral curves of the velocity field. Lemma 4.1. Suppose that (ρ, u, w,H) is the solution satisfying Proposition 3.1 and 0 < C1 ≤ ρ ≤ C2 on R3 × [0, T ]. Then, we fix t0 ≥ 0 and define the particle trajectories x : [0,∞)× R3 → R3 by ẋ(t, y) = u(x(t, y), t), x(t0, y) = y. (4.1) Therefore, when f ∈ L1(R3) is non-negative and t ∈ [0, T ], there exists a constant C only depending on C1 and C2 such that each of the integrals ∫ R3 f(x(t, y)) dy and∫ R3 f(x) dx is bounded by C times the other. In the second lemma, we derive a result concerning Hölder-continuity of u(·, t) to various norms appearing in the definition (3.1) of the functional A: Lemma 4.2. Let (ρ, u, w,H) be the solution satisfying Proposition 3.1. If t ∈ (0, T ] and α ∈ (0, 1/2], one has 〈u(·, t)〉α ≤M [ (‖∇H ·H(·, t)‖2L2 + ‖u̇(·, t)‖2L2) 1+2α 4 (C0 + ‖∇u(·, t)‖2L2) 1−2α 4 + ‖∇u(·, t)‖ 1−2α 2 L2 ‖∇W1(·, t)‖ 1+2α 2 L2 + C 1−α 3 0 ] , (4.2) 〈w(·, t)〉α ≤M [ (‖∇w(·, t)‖2L2) 1−2α 4 (‖ẇ(·, t)‖2L2 + ‖w(·, t)‖2L2) 1+2α 4 + ‖∇w(·, t)‖ 1−2α 2 L2 ‖∇W2(·, t)‖ 1+2α 2 L2 ] . (4.3) Proof. Let α ∈ (0, 1/2] and define r ∈ (3, 6] by r = 3/(1− α). Using 2.3 and 2.10, we have 〈u(·, t)〉α ≤M [‖G1(·, t)‖Lr + ‖W1(·, t)‖Lr + ‖(ρ− ρ̃)(·, t)‖Lr ], 〈w(·, t)〉α ≤M [‖G2(·, t)‖Lr + ‖W2(·, t)‖Lr ]. (4.4) Applying Gagliardo-Nirenberg inequality, we have ‖G1(·, t)‖Lr ≤M ( ‖∇G1(·, t)‖(3r−6)/2r L2 ‖G1(·, t)‖(6−r)/2rL2 ) ≤M(‖(ρ− ρ̃)(·, t)‖2L2 + ‖∇u(·, t)‖2L2) 1−2a 4 (‖u̇(·, t)‖2L2 + ‖∇H ·H(·, t)‖2L2) 1+2a 4 ≤M(C0 + ‖∇u(·, t)‖2L2(R3)) 1−2α 4 (‖u̇(·, t)‖2L2(R3) + ‖∇H ·H(·, t)‖2L2(R3)) 1+2α 4 , (4.5) ‖G2(·, t)‖Lr ≤M(‖∇G2(·, t)‖(3r−6)/2r L2 ‖G2(·, t)‖(6−r)/2rL2 ) ≤M(‖∇w(·, t)‖2L2) 1−2a 4 (‖ẇ(·, t)‖2L2 + ‖w(·, t)‖2L2) 1+2a 4 , (4.6) EJDE-2023/86 GLOBAL LOW-ENERGY WEAK SOLUTIONS 27 ‖W1(·, t)‖Lr ≤M(‖W1(·, t)‖(6−r)/2rL2(R3) ‖∇W1(·, t)‖(3r−6)/2r L2(R3) ) ≤M(‖∇u(·, t)‖ 1−2a 2 L2 ‖∇W1(·, t)‖ 1+2a 2 L2 ), (4.7) ‖W2(·, t)‖Lr ≤M(‖W2(·, t)‖(6−r)/2rL2(R3) ‖∇W2(·, t)‖(3r−6)/2r L2(R3) ) ≤M(‖∇w(·, t)‖ 1−2a 2 L2 ‖∇W2(·, t)‖ 1+2a 2 L2 ). (4.8) Consequently, substituting (4.5)-(4.8) into (4.4), we can complete the proof. � Proof of Proposition 3.1. First, we select two positive numbers Φ and Φ′ satis- fying ρ < Φ < ρ+ s < ρ̄− s < Φ′ < ρ̄. Then recall the definition of ρ0 that takes values in [ρ+ s, ρ̄− s]. Thus according to the time regularity (2.6), for some positive ε, we have ρ ∈ [ρ, ρ̄] on R3× [0, ε]. Then, from the proposition 3.1, we obtain A(ε) ≤ MCτ0 , where M is now fixed. If C0 is further restricted, we have Φ < ρ < Φ′ on R3 × [0, T ], and therefore A(T ) ≤MCτ0 . We will establish the required upper bound, since the proof of the lower bound is parallel. For y ∈ R3, the corresponding particle path x(t) can be defined by ẋ(t, y) = u(x(t, y), t), x(t0, y) = y. (4.9) Suppose that there exists a time t1 ≤ ε such that ρ(x(t1), t1) = Φ′. Then we take t1 minimal and select t0 < t1 maximal such that ρ(x(t0), t0) = ρ̄− s. Therefore, for t ∈ [t0, t1], ρ(x(t), t) ∈ [ρ̄− s,Φ′]. We consider the following two cases: Case 1: For t0 < t1 ≤ T ∧ 1, according to the mass equation and the definition (1.5), we obtain (2µ+ λ) d dt [log ρ(x(t), t)− log(ρ̃)] + P (ρ(x(t), t))− P̃ = −G1(x(t), t), (2µ′ + λ′) d dt [log ρ(x(t), t)− log(ρ̃)] = −G2(x(t), t). Integrating over [t0, t1] and simplifying ρ(x(t), t) to ρ(t), then we have (2µ+ λ)[log ρ(s)− log(ρ̃)] ∣∣t1 t0 + ∫ t1 t0 [P (s)− P̃ ] ds = − ∫ t1 t0 G1(s) ds, (4.10) (2µ′ + λ′)[log ρ(s)− log(ρ̃)]|t1t0 = − ∫ t1 t0 G2(s) ds. (4.11) We will show that∫ t1 t0 G1(s) ds ≤ M̃Cτ0 , ∫ t1 t0 G2(s) ds ≤ M̃Cτ0 , (4.12) for a constant M̃ which depends on the same quantities as M defined by Proposition 3.1. If so, using (4.10) and (4.11), we have (2µ+ λ)[log Φ′ − log(ρ̄− s)] ≤ ∫ t1 t0 [P (s)− P̃ ] ds+ M̃Cτ0 ≤ M̃Cτ0 , (4.13) (2µ′ + λ′)[log Φ′ − log(ρ̄− s)] ≤ M̃Cτ0 . (4.14) Because ρ(t) takes values in [ρ̄ − s,Φ′] ⊂ [ρ̃, ρ̄], and P is an increasing function on [ρ̃, ρ̄], (4.13)-(4.14) hold. But, if C0 is small depending on M̃ , Φ′, and ρ̄ − s, 28 W. P. WU, Y. H. ZHANG EJDE-2023/86 then (4.13) and (4.14) cannot hold. Specifying the smallness condition, therefore we can infer that there is no time t1 such that ρ(t1) = ρ(x(t1), t1) = Φ′. Due to the arbitrariness of y ∈ R3, we have ρ < Φ′ on R3 × [0, ε]. The method of proving ρ > Φ is analogous. To prove (4.12), supposing that Γ is the fundamental solution of the Laplace operator in R3 and employing (2.12), we have∫ t1 t0 G1(s) ds = ∫ t1 t0 ∫ R3 ρu̇(y, s)(∇xΓ(x(s)− y)) dy ds + ∫ t1 t0 ∫ R3 (∇xΓ(x(s)− y))((∇×H)×H)(y, s) dy ds, (4.15) ∫ t1 t0 G2(s) ds = ∫ t1 t0 ∫ R3 ρẇ(y, s)(∇xΓ(x(s)− y)) dy ds + ∫ t1 t0 ∫ R3 (∇xΓ(x(s)− y))w(y, s) dy ds. (4.16) We notice that the first term in (4.15) is identical to Hoff [8, Lemma 4.2]. Thus, we can arrive at∣∣ ∫ t1 t0 ∫ R3 ρu̇(y, s)(∇xΓ(x(s)− y)) dy ds ∣∣ ≤ ‖∇Γ ∗ (ρu)(·, t1)‖L∞(R3) + ‖∇Γ ∗ (ρu)(·, t2)‖L∞(R3) + ∫ t 0 ∫ R3 [uk(x(s), s)− uk(y, s)]Γxjxk(x(s)− y)(ρuj)(y, s) dy ds ≤ M̃Cτ0 + M̃Cτ0 ∫ 1 0 〈u(·, s)〉α ds ≤ M̃Cτ0 . (4.17) The last inequality is derived from Proposition 3.1 and Lemma 4.2. Note that (3.54) holds for q = 6, therefore if 2 < r < 3q q+3 , the second integral of (4.15) can be estimated by∣∣ ∫ t1 0 ∫ R3 (∇xΓ(x(s)− y))((∇×H)×H)(y, s) dy ds ∣∣ ≤ M̃ ∫ 1 0 ‖((∇×H)×H)(s)‖L2(R3) + ‖((∇×H)×H)(s)‖Lr(R3) ds ≤ M̃ ∫ 1 0 ‖(|H|4|∇H|2)(s)‖1/2L2(R3)‖|∇H| 2(s)‖1/2L2(R3) ds + ∫ 1 0 ‖∇H(s)‖L3(R3)‖H(s)‖ L 3r 3−r (R3) ds ≤ M̃Cτ0 . (4.18) Similarly, the terms of (4.16) can be bounded. Thus we achieve the proof of (4.12). Case 2: For 1 ≤ t0 < t1, similar to case 1, we obtain d dt (ρ(t)− ρ̃) + (2µ+ λ)−1ρ(t)(P (t)− P̃ ) = −(2µ+ λ)−1ρ(t)G1(t), (4.19) d dt (ρ(t)− ρ̃) = −(2µ′ + λ′)−1ρ(t)G2(t). (4.20) EJDE-2023/86 GLOBAL LOW-ENERGY WEAK SOLUTIONS 29 We multiply the above equations by (ρ(t)− ρ̃) to obtain 1 2 d dt (ρ(t)− ρ̃)2 + (2µ+ λ)−1f(t)ρ(t)(ρ(t)− ρ̃)2 = −(2µ+ λ)−1ρ(t)(ρ(t)− ρ̃)G1(t), (4.21) 1 2 d dt (ρ(t)− ρ̃)2 = −(2µ′ + λ′)−1ρ(t)(ρ(t)− ρ̃)G2(t). (4.22) Here f(t) = (P (t)− P̃ )(ρ(t)− ρ̃)−1. Since f(t) ≥ f(t0) > 0 on [t0, t1], it is easy to deduce that (2µ+ λ)−1f(t)ρ(t)(ρ(t)− ρ̃)2 ≥ (2µ+ λ)−1f(t0)ρ(ρ(t)− ρ̃)2 for t ∈ [t0, t1]. Thus integrating (4.21) and (4.22) over [t0, t1], we arrive at (ρ(t1)− ρ̃)2 − (ρ(t0)− ρ̃)2 ≤ M̃ ∫ t1 t0 ‖G1(·, s)‖2L∞(R3) ds, (4.23) (ρ(t1)− ρ̃)2 − (ρ(t0)− ρ̃)2 ≤ M̃ ∫ t1 t0 ‖G2(·, s)‖2L∞(R3) ds. (4.24) We shall show that∫ t1 t0 ‖G1(·, s)‖2L∞ ds ≤ M̄Cτ0 , ∫ t1 t0 ‖G2(·, s)‖2L∞ ds ≤ M̄Cτ0 . (4.25) Therefore, from (4.23) and (4.24), we have |Φ′ − ρ̃|2 − |ρ̄− s− ρ̃|2 ≤ M̃ ∫ t1 t0 ‖G1(·, s)‖2L∞(R3) ds, (4.26) |Φ′ − ρ̃|2 − |ρ̄− s− ρ̃|2 ≤ M̃ ∫ t1 t0 ‖G2(·, s)‖2L∞(R3) ds. (4.27) By employing similar argument used in Case 1, (4.26) and (4.27) cannot hold if C0 is sufficiently small. Because y ∈ R3 is arbitrary, we obtain ρ < Φ′ on R3 × [0, ε]. To prove (4.25), applying (2.12) and Lemma 2.2, we have∫ t1 t0 ‖G1(·, s)‖2L∞ ds ≤ ∫ t1 t0 [‖ρu̇(·, s)‖2L2 + ‖∇H ·H(·, s)‖2L2 ] ds + ∫ t1 t0 [‖ρu̇(·, s)‖2L4 + ‖∇H ·H(·, s)‖2L4 ] ds ≤ ∫ t1 t0 [‖ρu̇(·, s)‖2L2 + ‖∇H ·H(·, s)‖2L2 ] ds + ∫ t1 t0 [‖u̇(·, s)‖1/2L2 ‖∇u̇(·, s)‖3/2L2 + ‖∇H ·H(·, s)‖1/2L2 ‖∇(∇H ·H)(·, s)‖3/2L2 ] ds ≤ M̃Cτ0 + (∫ t1 1 ∫ R3 |u̇|2 dx ds )1/4(∫ t1 1 ∫ R3 |∇u̇|2 dxds )3/4 + (∫ t1 1 ∫ R3 |∇H ·H|2 dxds )1/4(∫ t1 1 ∫ R3 |∇(∇H ·H)|2 dx ds )3/4 ≤ M̃Cτ0 , 30 W. P. WU, Y. H. ZHANG EJDE-2023/86 and ∫ t1 t0 ‖G2(·, s)‖2L∞ ds ≤ ∫ t1 t0 [‖ρẇ(·, s)‖2L2 + ‖w(·, s)‖2L2 ] ds + ∫ t1 t0 [‖ρẇ(·, s)‖2L4 + ‖w(·, s)‖2L4 ] ds ≤ M̃Cτ0 + ∫ t1 t0 [‖ẇ(·, s)‖1/2L2 ‖∇ẇ(·, s)‖3/2L2 + ‖w(·, s)‖1/2L2 ‖∇w(·, s)‖3/2L2 ] ds ≤ M̃Cτ0 + (∫ t1 1 ∫ R3 |ẇ|2 dxds )1/4(∫ t1 1 ∫ R3 |∇ẇ|2 dxds )3/4 + (∫ t1 1 ∫ R3 |w|2 dxds )1/4(∫ t1 1 ∫ R3 |∇w|2 dx ds )3/4 ≤ M̃Cτ0 . The last inequality is derived from the Proposition 3.1. This completes the proof. 5. Proof of Theorem 1.2 The proof is done by constructing the weak solution as the limits of the smooth solutions. More specifically, provided that the assumptions of Proposition 3.1 be valid, and the initial data (ρ0, u0, w0, H0) of the solution satisfy the assumptions (1.10)-(1.13) and (1.20). By convolving (ρ0, u0, w0, H0) with a standard mollify- ing kernel of width ξ > 0, we can obtain the smooth approximate initial data (ρξ0, u ξ 0, w ξ 0, H ξ 0 ). Then using the local existence result of Theorem 2.3, (1.1)-(1.2) has a unique local solution (ρξ, uξ, wξ, Hξ) on R3× [0, T ]. Applying a conventional energy estimate similar to [26, Theorem 4.1], the local solution (ρξ, uξ, wξ, Hξ) can be extended to any T > 0. We establish the global-in-time existence of smooth solutions with the initial data (1.2) satisfying low energy condition (1.20). Then sup 0≤s≤T ‖(ρ− ρ̃, u, w,H)(·, s)‖H3 + ∫ T 0 ‖(u,w,H)(·, s)‖2H4 ds ≤M ′(T ). (5.1) By Theorem 2.3, for every ξ we have a global solution that satisfies A(t) ≤MCτ0 , and ρ ≤ ρξ(x, t) ≤ ρ̄. (5.2) Then (ρ, u, w,H) of A(t) can be replaced by (ρξ, uξ, wξ, Hξ) in (3.1). These esti- mates will offer the compactness needed to extract the required solution (ρ, u, w,H) in the limit as ξ → 0. Next, we establish the uniform Hölder continuity away from t = 0. Lemma 5.1. Taking ε > 0, there exists a constant C = C(ε) for all ξ > 0 such that 〈u(·, t)〉1/2,1/8R3×[ε,∞), 〈w(·, t)〉1/2,1/8R3×[ε,∞), 〈H(·, t)〉1/2,1/8R3×[ε,∞) ≤ C(ε)Cτ0 . (5.3) Proof. First, notice that we proved the Hölder- 1 2 continuity of uξ in (3.2). To obtain the Hölder continuity in time, we need to fix x and t2 ≥ t1 ≥ ε, |uξ(x, t2)− uξ(x, t1)| ≤ 1 |BR̆(x)| ∫ BR̆(x) |uξ(z, t2)− uξ(z, t1)|dz + C(ε)Cτ0 R̆ 1/2 EJDE-2023/86 GLOBAL LOW-ENERGY WEAK SOLUTIONS 31 ≤ R̆− 3 2 |t2 − t1|1/2 sup t≥ε ∫ |ut(z, t)ξ|2 dx+ C(ε)Cτ0 R̆ 1/2 ≤ C(ε)Cτ0 [R̆− 3 2 |t2 − t1|1/2 + R̆1/2], by the estimates in (5.2). Given R̆ = |t2 − t1|1/4, we obtain the bound of uξ in (5.3). The proofs for wξ and Hξ are similar. � Compactness of the approximate solutions (ρξ, uξ, wξ, Hξ) now follows. From Lemma 5.1 and Ascoli-Arzela Theorem, we obtain uξσ , wξσ , Hξσ → u,w,H uniformly on compact sets in R3 × (0,∞); (5.4) for a sequence ξσ → 0. Then, according to the same sequence from (5.1) and based on the elementary consideration that the weak-L2 derivative and the distribution derivatives are equal, it follows that ∇uξσ (·, t),∇wξσ (·, t),∇Hξσ (·, t),∇W ξσ 1 (·, t),∇W ξσ 2 (·, t) → ∇u(·, t),∇w(·, t),∇H(·, t),∇W1(·, t),∇W2(·, t) (5.5) weakly in L2(R3) for every t > 0; and ϑ1/2u̇ξσ , ϑ1/2ẇξσ , ϑ1/2Hξσ t , ϑ5/2∇u̇ξσ , ϑ5/2∇ẇξσ , ϑ5/2∇Hξσ t → ϑ1/2u̇, ϑ1/2ẇ, ϑ1/2Ht, ϑ 5/2∇u̇, ϑ5/2∇ẇ, ϑ5/2∇Ht (5.6) weakly in L2(R3 × [0,∞)). Using [7] and [14], we can obtain the convergence of approximate densities (5.7) as ρξσ (·, t)→ ρ(·, t) (5.7) strongly in L2 loc(R3) for all t ≥ 0. Proof of Theorem 1.2. Obviously, the definition of limiting functions (ρ, u, w,H) in (5.4)-(5.7) inherits the bounds from (5.1) and (5.2) (but please notice that there is no representations in (1.26) about u̇(·, t), ẇ(·, t) or Ht(·, t)). It is also explicit from the convergence pattern described in (1.22)-(1.27) that (ρ, u, w,H) meets the weak forms (1.16)-(1.19) of the differential equations in (1.1)-(1.2). Furthermore, the continuity statement (1.21) is easily derived from these weak forms and bounds (1.27). Therefore, (1.27) is achieved. Next we study the large-time behavior of (ρ, u, w,H) in (1.28). Taking a similar methods and proofs as for [5] and [8], we have lim t→∞ ‖ρ− ρ̃‖Ll(R3) = 0 (5.8) for all l ∈ (2,∞). Then, following the same argument as in [5], we take a sequence ul(t, x) := u(t+ l, x), for all integer l, and (x, t) ∈ R3 × [1, 2]. Then by (1.27), we can arrive at lim l→∞ ∫ 1 0 ‖∇ul‖L2(R3) = 0. Similarly, we have ‖ul‖H1(R3) ≤ C uniformly for t, l. Thus we obtain liml→∞ ‖ul‖L2(R3) = 0 uniformly for t, which indicates lim t→∞ ‖u(t)‖L2(R3) = 0. (5.9) 32 W. P. WU, Y. H. ZHANG EJDE-2023/86 For t ≥ 1, from Lemma 2.4, it is easy to deduce that ‖∇u(t)‖L6(R3) ≤ C(‖G1(t)‖L6(R3)) + ‖W1(t)‖L6(R3) + ‖(P (ρ)− P̃ )(t)‖L6(R3) ≤ C(1 + ‖∇G1(t)‖L2(R3) + ‖∇W1(t)‖L2(R3)) ≤ C(1 + ‖u̇(t)‖L2(R3) + ‖∇w(t)‖L2(R3) + ‖H · ∇H‖L2(R3) + ‖∇|H|2‖L2(R3)) ≤ C. (5.10) Taking the summation of (1.27), (5.9) and (5.10), we can infer that lim t→∞ ‖u‖W 1,r(R3) = 0, (5.11) for r ∈ (2, 6). Similarly, we have lim t→∞ ‖(w,H)−→‖W 1,r(R3) = 0, (5.12) for r ∈ (2, 6). Combining (5.8),(5.11) and (5.12), we obtain (1.28). The proof of Theorem 1.2 is complete. Acknowledgments. This work is partially supported by the National Nature Sci- ence Foundation of China( grant 12271114), by the Innovation Project of Guangxi Graduate Education (grant JGY2023061), by the Guangxi Natural Science Foun- dation (grants 2019JJG110003, 2019AC20214), by the Key Laboratory of Mathe- matical Model and Application Guangxi Normal University, Education Department of Guangxi Zhuang Autonomous Region. References [1] R. A. 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Proof of Proposition ?? 4. Point wise bounds for the density Proof of Proposition ?? 5. Proof of Theorem ?? Proof of Theorem ?? Acknowledgments References