LOW MACH NUMBER LIMIT OF THE FULL COMPRESSIBLE HALL-MHD SYSTEM

. In this paper we study the low Mach number limit of the full compressible Hall-magnetohydrodynamic (Hall-MHD) system in T 3 . We prove that, as the Mach number tends to zero, the strong solution of the full com- pressible Hall-MHD system converges to that of the incompressible Hall-MHD system.

Here ρ, u, p, e, T and b denote the density, velocity, pressure, internal energy, temperature, and magnetic field, respectively. The physical constants µ and λ are the shear viscosity and bulk viscosity of the flow and satisfy µ ą 0 and λ`2 3 µ ě 0. κ ą 0 is the heat conductivity. ą 0 is the (scaled) Mach number. ∇u t is the transpose of the ∇u. For simplicity, we shall consider the case that the fluid is a polytropic ideal gas, that is e :" C V T , p :" RρT (6) with C V ą 0 and R ą 0 being the specific heat at constant volume and the generic gas constant, respectively. The applications of the Hall-MHD system cover a very wide range of physical objects, for example, magnetic reconnection in space plasmas, star formation, neutron stars, and geo-dynamo, see [22,23,25]. Due to its physical importance and mathematical interest, there are a lot of results on the Hall-MHD system. For the incompressible Hall-MHD system, for example, the regularity criteria of the solutions were obtained in [6,11,12,15,26], and the global existence of small solution and global weak solutions were given in [3][4][5]. For the isentropic compressible Hall-MHD system, the local existence of strong solutions, global existence of small solutions were first obtained in [9] and the low Mach number limit problem was discussed in [21,27]. Very recently, the local well-posedness and a blow-up criterion of strong solutions to the 3D compressible full Hall-MHD system (1)-(5) with positive density was obtained in [10].
When the Hall effect term rot´r ot bˆb ρ¯i s neglected, the system (1)-(5) reduces to the well-known full compressible MHD system, which has received many studies [2,8,13,14,16,17,19]. The local strong solution was obtained by Fan-Yu [13]. The global weak solutions was obtained by Fan-Yu [14], Ducomet-Feireisl [8] and Hu-Wang [16] respectively. The low Mach number limit problem was studied by Jiang-Ju-Li [18] in T 3 for well-prepared initial data, Jiang-Ju-Li-Xin [19] in R 3 for ill-prepared initial data, and Cui-Ou-Ren [2] in a bounded domain for well-prepared initial data.
In this paper we study the low Mach number limit to the full Hall-MHD system (1)-(5) with well-prepared initial data in T 3 .
A local existence result for (8)- (12) in the following sense can be shown in a similar way to that in [28]. Thus we omit the details of the proof. Proposition 1.1 (Local existence). Let P p0, 1q. Suppose that the initial data pσ 0 , u 0 , θ 0 , b 0 q satisfy that 1` σ 0 ě m ą 0 for some positive constant m, and Then there exists a positive constant T ą 0 such that the problem (8)-(12) has a unique solution pσ , u , θ , b q satisfying that 1` σ ą 0 in T 3ˆp 0, T q, and for k " 0, 1, 2, To simplify the statement, we have used B t up0q to signify the quantity B t u| t"0 obtained through equation (9), and B 2 t up0q is given recursively by B t (9) in the same manner. Similarly, we can define The main result of this paper is stated as follows, which shows the uniform estimates of strong solutions to (8)- (12), and the corresponding low Mach number limit.
Then there exist positive constants T 0 and D such that pσ , u , θ , b q satisfy the uniform estimates: with D 0 , T 0 and D independent of ą 0. (7) and (14) imply ρ Ñ 1 and T Ñ 1 in certain Sobolev space as Ñ 0. Furthermore, pσ , u , θ , b q converge to pσ, u, θ, bq in certain Sobolev space as Ñ 0, and there exists a function πpx, tq such that pu, b, πq in Cpr0, T 0 ; H 2 sq solves the following problem of the incompressible Hall-MHD equations: where u 0 , b 0 are the weak limits of u 0 and b 0 , respectively, in H 2 with div u 0 " div b 0 " 0 in T 3 .
We will denote Similarly to those in [1,7,20], it suffices to show the following theorem to get the uniform estimates in (14). We will give the details on the proof of Theorem 1.3 in section 3 based on Theorem 1.2. Theorem 1.3. Let T be the maximal time of existence for the problem (8)- (12) given in Proposition 1.1. Then for any t P r0, T q, we have for some given nondecreasing continuous functions C 0 p¨q and Cp¨q.
The novelty of our paper lies in that the system has the Hall term with strong nonlinearity and therefore the difference between our paper and the references [2,7,19,20] is that we will bound some new terms I i pi " 1,¨¨¨, 6q coming from the Hall term.
However, the only different term between our system and that in [2] is the Hall term. On the other hand, we will only use the formulation of M ptq, which is same as that in [2] in our estimates. For example, we mainly use sup 2. Proof of Theorem 1.3. This section is devoted to the proof of Theorem 1.3, we only need to show the inequality (16). We shall use some ideas developed in [2,7,19,20], say, we will use ω :" rot u and J :" rot b to show a priori estimates.
Below we shall drop the super script " " of ρ , σ , u , θ , etc. for the sake of simplicity; moreover, we write M ptq and M p0q as M and M 0 , respectively. Since the physical constants κ, C V , and R do not bring any essential difficulties in our arguments, we shall take κ " C V " R " 1 for presentation simplicity.
We bound I 1 and I 2 as follows.
By the same calculations as that in [2], we have Proof. Applying B t to (21), testing by B t ω, doing as that in [2], one has Applying B t to (23), testing by B t J, we observe that :C 0 pM 0 q`I 3`I4 .
We bound I 3 and I 4 as follows.
t}B t b} L 8 p0,t;H 1 q p}B t b} L 2 p0,T ;H 2 q }u} L 8 p0,t;H 2 q }b} L 8 p0,t;H 2 q }B t u} L 2 p0,t;H 2 q q ď ? tCpM q, We point out the cancellation of the triple product like has been used in I 4 and a similar idea will also be used in I 6 below. Inserting the above estimates into (33) and using (32) lead to (31).
By taking the same calculations as that in [2], we arrive at Finally, we estimate B 2 t b in order to close the energy estimate. Lemma 2.4. For any 0 ď t ď mintT , 1u, we have Proof. Applying B 2 t to (11), testing by B 2 t b, we reach 1 2 ot bˆb ρ˙d xdsˇˇˇ" We bound I 5 and I 6 as follows.
Let t ă mintT , T 1 u. By combining the inequalities (16) and (40) with the assumption M p0q " M 0 , we have that M ptq ‰ D. Besides, we can assume without restriction that D 0 ď D, so that M p0q ď D. Since the function M ptq is continuous, we obtain M ptq ď D for t ă mintT , T 1 u and 0 ă ď 1.
Then T ą T 1 for 0 ă ď 1. Otherwise, by using the uniform estimates in (41) and applying Proposition 1.1 repeatedly, one can extend the time interval of existence to r0, T 1 s, which contradicts to the maximality of T . Therefore, M ptq ď D for any t P r0, T 1 s where T 1 is independent of 0 ă ď 1. Clearly, the conclusion is also true for T " 8 by applying the same argument. This completes the proof.