Global well-posedness of the full compressible Hall-MHD equations

In this paper, we study the three-dimensional full compressible Hall-magnetohydrodynamic (for short, HallMHD) system, which is governed by the following equations (see, e.g. [4, 11]):  ρt + div(ρu) = 0, (ρu)t + div(ρu⊗ u) − μ∆u − (μ + λ)∇divu +∇P = (curlB) × B, Cv ( (ρθ)t + div(ρuθ) ) − κ∆θ + θ∂θPdivu = 2μ|D(u)|2 + λ|divu|2 + v|curlB|2, Bt − v∆B + εcurl ( curlB×B ρ ) = curl(u × B), divB = 0, (1.1)


Introduction
In this paper, we study the three-dimensional full compressible Hall-magnetohydrodynamic (for short, Hall-MHD) system, which is governed by the following equations (see, e.g. [4,11]): with t ≥ and x ∈ R . Here ρ, u, P, θ and B represent the uid density, velocity, pressure, absolute temperature and magnetic eld, respectively. Deformation tensor D(u) := [∇u + (∇u) tr ]. The constant viscosity coe cients µ and λ satisfy the physical restrictions: Here, we investigate the ideal polytropic uids so that the pressure P and the physical constant Cv satisfy where γ > is the adiabatic constant, and for simplicity, we assume Cv = R = . κ and v are positive constants. ϵ > is the Hall coe cient. The Hall-MHD system can be derived from uid mechanics with appropriate modi cations to account for electrical forces and Hall e ects. This compressible system(1.1) describes the dynamics of plasma ows with strong shear of magnetic elds such as in the solar ares, neutron stars and geo-dynamo, we refer to [1, 3,10,12,23,26,30] for the physical background of this system. The Hall term curl (curlB)×B) ρ in ( . ) has been put forward by Ghosh et al. [10] to restores the in uence of the electric current in the Lorentz force occurring in Ohms law. And the Hall coe cient ϵ is de ned by the quotient of Alfven frequency of the lowest wave number ω A and the ion cyclotron frequency Ω i , it means ϵ := ω A Ω i . When the Hall e ect term is neglected (ϵ = ), the equations (1.1) is reduced to the well-known compressible full MHD system, whose applications cover a broad range of physical elds from liquid metals to cosmic plasmas. The mathematical results on this compressible heat conducting MHD system can refer for example to [2, 6-8, 15, 16, 20, 25]. Fan and Yu [7] gave the local strong solution and Huang and Li established the blowup criterion. The global weak solutions was proved in [2,6,15,16]. The long time behavior was discussed in [8,25]. While, Jiang and his collaborators solved the low Mach number limit problem in [19,20].
The compressible Hall-MHD equations are also mathematically signi cant. The solvability and stability of the equations has attracted considerable attention recently. For the isentropic case, Fan et al. [5] studied the global existence of strong solution and established the optimal time decay rates under the small initial perturbation condition. Gao and Yao [9] improved their work and established optimal decay rates for higherorder spatial derivatives of classical solutions. In [29], Tao, Yang and Yao established the global existence, uniqueness and exponential stability of strong solutions with large initial data for the one-dimensional case. Xiang [31] established the uniform estimates and optimal decay rates to global solution with respect to the Hall coe cient ϵ under the condition that H -norm of initial data is small enough. If the temperature is taken into account, Fan et al. [4] rst proved the local well-posedness for the full compressible Hall-MHD equations, and obtained a blow-up criterion of strong solution. The boundedness and time decay of the higher-order spatial derivatives of the smooth solution under the condition that H k -norm (k ≥ ) of initial data is small and bounded inḢ s ( < s < ) are established by He, Samet and Zhou in [11]. Recently, Lai, Xu and Zhang [21] generalized Xiang' results [31] into the non-isentropic case. For other works on the compressible Hall-MHD system, we refer to [27,28] and references therein. However, to our knowledge, all the results on the global smooth solutions for the three dimensional compressible Hall-MHD equation need the initial data has at least H small norm.
In this paper, we consider an initial value problem of the Hall-MHD compressible ows ( . ) supplied with initial data (ρ, u, θ, B)(x, ) = (ρ , u , θ , B )(x), for x ∈ R (1.2) and the far eld behavior Motivated by the works for compressible Navier-Stokes equation [14,18] and the compressible MHD equation [13,22], we will rst established global existence and uniqueness of solution with smooth initial data which is of small energy. It is worth mentioning that H -norm of the initial data are not necessarily small. Then, the large time behavior of the solution will be given as well. Compared with the works in [13,14,18,22], we also have to deal with the essential di culties caused by Hall term in the present paper. This term includes the strong coupling between the density and the magnetic eld, which together with the second-order derivative structure make the derivations of estimates more di cult. Therefore, the methods used in the MHD equation to show the bounds for the magnetic eld are no longer applicable here. In order to overcome the di culties from the Hall term, we introduce two kinds of estimates for derivatives of the density, which play an important role to establish the time-independent lower-order estimates. In addition, the temperature is considered in this paper, which brings us more nonlinear term, for instance, |curlB| in temperature equation ( . ) and makes the system more complex. Throughout this paper, we use H s (R )(s ∈ N) to denote the usual Sobolev spaces with norm · H s and L p (R )( ≤ p ≤ ∞) to denote the L p spaces with norm · L p . For given initial data (ρ , u , θ , B ), we de ne the initial energy E , Now, the main result in this paper is stated as follows.
for any q ∈ ( , ∞]. (1.9) Remark 1.1. From ( . ) and the small initial energy, we can nd that the initial data in Theorem 1.1 have small H -norm for (ρ , u , B ), which is weaker than that in [21,31]. Indeed, by Gagliardo-Nirenberg inequality, the smallness of H -norm of the initial data is required in [21,31]. Moreover, the absolute temperature is considered in this paper and the initial temperature allows large oscillations.
Remark 1.2. Since the Hall term involves second-order derivative of magnetic eld and rst-order derivative of density, in order to establish the global existence of the solution, we need to get the bound of ∇ρ H , which leads ( . ) including H -norm of initial data.
The rest of this paper is organized as follow. In section 2, we derive the time-independent lower-order estimates and the higher-order estimates depending on time of the solutions. In section 3, the proof of Theorem 1.1 will be showed.

Global existence
In this section, a known inequality and some facts are rst collected, and then we will establish some suitable a priori estimates by the energy method.

. Preliminaries
For the convenience of the proof below, let us rewrite the system (1.1) as follows, here we use the equalities In addition, the initial data satis es 3) The following Gagliardo-Nirenberg inequality are well-known (see for example [24]).
where ≤ θ ≤ and α satisfy Now, we are ready to de ne some functions which will be frequently used later. First of all, let σ = σ(t) = min{ , t} and σ ′ = d dt σ(t). Then, we set: In what follows, we denote the generic constant and suitably small constant by C > and δ > depending only on some known constants µ, λ, k, v and ϵ but independent of time t, respectively. Particularly, we will use C(M) to emphasize that C may depend on M = max ( + C − )M , ( + C )M , where the given constants C and C are de ned in Lemma 2.8 and 2.10.

. Time-independent lower-order estimates.
The aim of this subsection is to derive the lower-order estimates on the solutions which are independent of time. Now, let (ρ, u, θ, B) be a solution to system ( . )-( . ) on R × ( , T) for some positive time T > and without loss of generality, let E ≤ .

5)
provided E ≤ δ, where δ is a positive constant depending on µ, λ, κ, v, ϵ and M but independent of T.
The proof of Proposition 2.1 consists of Lemma . -. and is to be completed by the end of this subsection.

Lemma 2.2. Under all the assumption of Proposition 2.1, it holds that
Proof: By ( . ) and Sobolev inequality, we obtain Proof: It follows from [18] and maximum principle, we have θ > for all (x, t) ∈ R × ( , T). Multiplying ( . ) -( . ) by u, − θ − and B, respectively, then adding them up and integrating by parts over R , using ( . ) and the equality curl curlB×B We thus derive ( . ) directly by integrating the above equality over ( , T) and nish the proof of Lemma . .

Lemma 2.4. Under the assumptions of Proposition 2.1, it holds that
The proof of Lemma 2.4 is the same as Lemma 3.1 in [18], so we omit it for brevity.
The following lemma is given to estimate A (T).

Lemma 2.5. Under the assumptions of Proposition 2.1, it holds that
}.

Proof:
The lemma can be established by a similar way in [14]. For completeness, we prove it here. Multiplying ( . ) by u, integrating the result over R and using integration by parts, we have By Hölder inequality and using (2.8) and (2.9), the right-hand side of (2.11) is estimated as follows, (2.12) Taking (2.12) into (2.11), we have On the order hand, multiplying ( . ) by θ − , integrating the resulting inequality over R , we obtain (2.14) By ( . ), ( . ) and ( . ), using Hölder, Sobolev and Young inequalities, we can bound E as follows, Similarly, we have following estimate for E , Inserting the estimates of E and E into ( . ), we get  Then, the combination of ( . ) with ( . ) yields }. Thus, we complete the proof of Lemma 2.5.
The estimates of ∇B L and ∇ B L will be given by following Lemma.

Lemma 2.6. Under the assumptions of Proposition 2.1, it holds that
Proof: Applying ∇ to ( . ) and multiplying by ∇B, then integrating it over R , we obtain :=I + I . (2.24) Using the integration by parts and with the help of Hölder, Sobolev and Young inequalities, I and I can be bounded as Putting the bounds of I and I into ( . ), and then applying ( . ) and ( . ), it leads to Applying ∇ to ( . ) and multiplying it by ∇ B, then integrating it over R , we get :=II + II . (2.28) Using the integration by parts and with the help of Hölder, Sobolev and Young inequalities, by (2.4) and (2.10), II and II can be bounded as Inserting the estimates of II and II into ( . ), and then applying ( . ) and ( . ), it implies For ∇ρ L , we want to establish two di erent estimates in the following lemma.

Lemma 2.7. Under the assumptions of Proposition 2.1, it holds that
Proof: Multiplying ( . ) by ∇ρ ρ and integrating the resulting equality over R , one has (2.34) By ( . ) and ( . ), with the help of Hölder, Young and Sobolev inequalities, we obtain Similarly, we give the estimates for J as follows, Particularly, we established the di erent estimates for J + J , Substituting the estimates of J i (i = , · · · , ) into ( . ), it is clear that Integrating (2.35) and (2.36) from to T, and then applying ( . ) and ( . ), we obtain ( . ) and ( . ), respectively.
The following lemma is established to estimate A (T).

Lemma 2.8. Under the assumptions of Proposition 2.1, it holds that
where C is a positive constant depending on µ and λ.
Proof: Using ∇ to ( . ) and ( . ) , multiplying the resulting equations by ∇ρ and ∇u, respectively, then summing up and integrating it over R , we obtain By ( . ) and ( . ), using Hölder, Young and Sobolev inequalities, let us show the estimates of K i (i = , , ), respectively. Next, multiplying ( . ) by σ and integrating the resulting inequality from to T, applying ( . ) and ( . ), we can obtain (2.38) as follows, This completes the proof of this lemma.

Lemma 2.9. Under the assumptions of Proposition 2.1, it holds that
where C is a positive constant depending on κ. Proof: It follows from ( . ) , ( . ), Hölder and Sobolev inequalities that By above inequality and Young inequality, we obtain (2.47) Integrating ( . ) from to T, applying ( . ) and ( . ), (2.44) holds. Multiplying ( . ) by σ and integrating the resulting inequality from to T, applying ( . ) and ( . ), we obtain (2.45) as follows, Thus, we complete the proof of this lemma.
By the bound of ∇θ L + T ∇ θ L dt, we give the following lemma to estimate A (T).

Lemma 2.10. Under the assumptions of Proposition 2.1, it holds that
where C and C are positive constants depending on some known constants µ, λ, k, v.
Integrating ( Next, the following lemma is needed to bound A (T). (2.58) Similarly, it follows from ( . ) , ( . ), ( . ), Hölder, Sobolev and Young inequalities that (2.59) Summing up ( . ) and ( . ), we have  Applying ∇ to ( . ) , squaring both sides of resulting equation, then integrating it over R , we have Finally, similar to the proof of ( . ), we can bound we omit the proof of it for brevity. Thus, we complete the proof of this lemma by ( . ) and ( . ).

Proof of Theorem .
Applying Lemma . -. , . , . and . -. , we obtain the following estimate, (3.1) With the help of the existence and uniqueness of local solutions which has been proved in [4] and all the a priori estimates above, using the standard continuum arguments, we extend the local solution to the global one. Next, we investigate the large time behavior of solution. It follows from ( . ) and ( . ) that ∞ ∇ρ L + ∇u L + ∇θ L + ∇B L dt ≤ C(M).  and using Sobolev inequality, we arrive at ( . ). Thus, the proof of Theorem . is completed.