We consider with the Cauchy problem of the full compressible Hall-magnetohydrodynamic (in short, Hall-MHD) equations in the whole space:
for with the initial data:
here, , and represent the density, velocity, absolute temperature and magnetic field, respectively. is the deformation tensor given by
The smooth function is the pressure satisfying and and for all,. is the coefficient of heat conduction. is the specific heat at constant volume. For and are the first and second viscosity coefficients satisfying the usual physical condition
And is the Planck constant. The represents the divergence and the symbol denotes the kronecker multiplication such that
In the last couple of decades, the magneto-hydrodynamic equations and associated models with quantum effects are widely studied. According to the quantum correction, Wigner  first derived the quantum correction to the energy density for thermodynamic equilibrium, and the quantum correction term goes hand in hand with Bohm potential  . The -term was advanced in . People might refer to Haas  for more physical interpretations of the model. Pu et al.  recently got global existence of classical solutions for the full compressible quantum Navier-Stokes. Global existence and decay rate of smooth solutions to the constant profile is considered by Pu and Xu . The system (1) is itself interesting because the energy equation also includes the quantum effects through the energy density, which gives the system new features. This makes it differ in the previous given results.
If there is no quantum corrections (i.e.), this system reduces to the usual compressible Hall-MHD equations. Hall-MHD is needed in many current physics problems. The Hall-MHD is indeed necessary to solve problems, for example, magnetic reconnection in space plasma (see   ), formation and evolution of stars and neutron stars    and geo-dynamo  (see  for a detailed description of these physical processes). In contrast to the general MHD equations, the Hall-MHD equations have the Hall term, which plays a significant role in magnetic reconnection. However, as far as we know, few achievements have been made in the study of the dynamics of global solutions to the 3D compressible Hall-MHD system, especially on the temporal decay of solutions. Very recently, global existence of smooth solutions to the 3D compressible Hall-MHD equations was first proved by Fan et al. , where the small initial disturbance belongs to. More precisely, optimal time decay rate was also established. Later, the result from  was improved by Gao and Yao . They obtained the global existence of strong solutions with the initial data are obtained in the lower regular spaces and proved optimal decay rates for the constructed global strong solutions in -norm. Xu et al.  took a pure energy method to prove the fast time decay rates for the higher-order spatial derivative of solutions when the initial data are close to a stable equilibrium state in
for some. Recently, for the case of initial data
are close to a stable equilibrium state in critical Besov spaces, the unique global solvability of strong solutions to the system was established by them . Obviously, system (1) becomes incompressible Hall-MHD system when and there are many interesting global results, see  -  to list only a few.
When the Hall effect term is ignored, the system (1) is reverted to the well-known MHD system. The MHD systems have been studied by many authors (see  -  ). For the corresponding full compressible MHD model, we can refer to      and references therein. Hu and Wang  constructed the solution of the initial-boundary value problem and established the global weak solutions. The global smooth solutions and their decay were given by Pu and Guo in . He et al.  considered boundedness and time decay of the higher-order spatial derivatives of the smooth solutions for a full compressible Hall-MHD system.
Although important, there are few results on the large-time behaviors of the Cauchy problem to the best of our knowledge. Much more complicate nonlinear terms, quantum effect term and the Hall effect term in the system (1) lead to new difficulties in decay analysis. The main novelty is to introduce (20) to cooperate with the special structure of (1). Fortunately, we can finally establish an optimal decay results for (1) under this norm, that is to say, the unknowns near the constant steady solution of (1) are more convenient to show.
For the main results of this paper, we have the following:
Theorem 1.1 Assume that , there exists a constant such that if
then the Cauchy problem (1)-(2) admits a unique global solution satisfying
Moreover, if, then we have
for some positive constant.
Notation. Throughout this paper, the norms in the Sobolev Spaces and are denoted respectively by and for and. In particular, when, we will simply use and. Moreover,
and for any integer, denotes all derivatives of order of the function f. In addition, denotes the inner product in, i.e., for f and,
First of all, we rewrite the Cauchy problem (1)-(2) into a more suitable form. Secondlly, we do a priori estimate and establish the global existence of solutions. Then, based on some Lp-Lq estimates by the linearized operator, we prove the decay rates.
In this subsection, we first reformulate the problem as follows. Set
then takes the form
To obtain a symmetric system, we denote
(11) can be rewritten in the perturbation form as follows
where the source terms are
We will obtain a global solution by a combination of the local existence result and a priori estimates.
Proposition 2.1 (Local existence). Let be such that
There exists a positive constant depending on and satisfies
for any. Then
The proof can be done by using the standard iteration arguments. Refer, for instance, to   .
Proposition 2.2. (A priori estimate). Let. is a solution of the initial value problem (12) on the time interval. For any fixed, then we have
the following a priori estimate holds for
where and are independent of T.
Remark 2.1. The global existence and uniqueness of the solution stated in Theorem 1.1 follows from Proposition 2.1 and 2.2.
Proposition 2.3. (Decay rates). Under the assumptions of Proposition 2.2, if, for any, there exists a constant such that
3. Energy Estimates
This section is devoted to the proof of Proposition 2.2. We deduce energy estimates that play an important role for establishing the global existence of solutions under the problem (12).
By Lemma A.2, which yields directly
Hence, we immediately have
Before proving Proposition 2.2, we need Lemmas 3.1, 3.2 and 3.3.
Lemma 3.1 Let be defined in (12), then it holds that
Proof. Multiplying (12)1, (12)2, (12)3 and (12)4 by n, v, z and B respectively, the integration over gives
The five terms on the right-hand side of the above equation can be estimated as follows.
Firstly, we get
Secondly, we obtain
Next, we have
it follows from (20) and (21) that
To deal with the term, we arrive at
Let. For, we have by (20), (21), the Hölder inequality, Young inequality, Lemma A.1 and integration by parts that
Similarly to the proof of, we have
We similarly obtain
In light of the estimates, we see that
For the fourth term, we have
It follows from Hölder’s inequality, Lemma A.1 and (20) that
In the same way as above, we know
In a similar way,
A similar argument shows that
Summing up, we can get
Finally, we have
In a similar way, we know
By a direct computation, we have
Plugging these estimates into (23), we deduce (22).
Then, we give a energy estimate of the higher-order for.
Lemma 3.2. Let be defined in (12), then we have
Proof. Applying with to (12) and then taking -inner product with, we obtain
First of all, is written as
The first term can be rewritten as
where the first and two terms can be estimated as
The third term in is written as
Moreover, and can be estimated similarly
For the term, we see that
For the second term of,
in a similar way
For the term, we know
For the first term of, after integrating by parts, we infer from (20) that
Collecting these terms, we get
For the second term of (35), In view of (20), (21), Hölder’s inequality and Lemma A.1, there holds
can be rewritten as
The first term can be bounded by
A similar argument shows that
By the estimates (36) and (37), we get
Similarly, we see that
Let. For the first term, we exploit the (20), Lemma A.1 and Hölder’s inequality to obtain
For, we see that
Let. We integrate by parts and use Hölder’s inequality to obatin
The same estimate holds for. Therefore
As in, we have
In the same manner, it is easy to deduce
Similar to the estimation, we obtain
Similarly, we get
Similarly, for the terms and, recalling from the estimate of, we have
That is to say
it follows from (38) that
For (39), we have
Similar to (39), we see
Consequently, in light of, we get
Since is small, (33) is given.
In the following, we consider the energy estimates on the entropy n.
Lemma 3.3. It holds that
Proof. When to, applying to the second Equation in (12) and testing by, we obtain
To estimate each term on the right-hand side, we integrate by parts twice, (20) and the continuity equation to deduce
Similarly, as the estimate of, we obtain
Similar to the estimation on, we have
Let. For the terms, integrating by parts and Hölder’s inequality yields,
It is easy to say
Finally, Combing with and, we get
Plugging these estimates into (42), we obtain
Integrating (43) with respect to t and taking sufficiently small, we conclude Lemma 3.3.
Finally, we obtain the global existence.
Proof of Proposition 2.2. Put Lemma 3.1 into Lemma 3.2 and taking is small, we see that
In view of Lemma 3.3, we have
Multiplying (45) by and adding the result to (44)
where is sufficiently small. Consequently, using the fact
We show that
then (46) gives (14).
This completes the whole proof of Propositions 2.2.
4. Convergence Rates
In this section, we shall prove the decay rates of the solution stated in Propositions 2.3. To do this, the strategy is to combine all the energy estimated.
We focus on the following homogenous linearized system of (12).
Let us denote the matrix-valued differential operator associated with (47) by
Hence, we separation of B from. Assume
by taking the Fourier transform with respect to the x-variable, we have
is the solution semigroup defined by, cf. .
Lemma 4.1. Let be integers. satisfies the inequalities with the initial data,
Lemma 4.2. Let, then it holds that
for an arbitrarily small.
To use the Lp-Lq estimates of the linear problem for the nonlinear system (12) as, then (12) becomes
where. Such that
Lemma 4.3. We assume is a smooth solution
Proof. We can know
It is easy to know,
And the nonlinear source terms can be estimated as follows:
and by Hölder’s inequality and Lemma A.1
The second term is much more complicated, which can be further decomposed into
The first term can be easily bounded by
In a similar way, we get
Let, we have by Hölder’s inequality and Lemma A.1 that
For, in a similar way, we have
Summing up, we obtain
The terms and can be bounded by
For, in a similar way, we have
Similarly, it holds that
Then using the similar way, we arrive at
In a similar way, we have
Summing these terms, we get
That is, we obtain
The inequality reads that
Combine (59) and (60), and hence this completes the proof of Lemma 4.3.
Now we are in a position to prove Propositions 2.3.
Proof of Proposition 2.3. We do it by two steps.
Step 1. First, for formula
we can assume
then, taking into
The linear combination of (44) and (45) leads to
Adding to both sides of the inequality above gives
Notice that is non-decreasing
Then it follows from Lemma 4.3 that
In fact, applying Gronwall’s inequality to the Lyapunov-type inequality (61) and using (62), we find that
In view of (62), we have
Since is sufficiently small. Consequently,
Using (65), we thus get
which also implies from Lemma A.1 that
Therefore (16), (17) and (18) are obtained. Then
Meanwhile via Lemma 4.1 and (48), we have
Hence, by interpolation, it is easy to see that for any,
where, this proves (15).
Step 2. On the other hand, where, we get it by using the estimates above (12), (16) and Lemma A.1.
Hence, (19) is proved and we complete the proof of Proposition 2.3.
Proposition 2.1 gets the local existence, Proposition 2.2 proves a priori estimate, Proposition 2.3 obtains the decay rates of solutions and then Theorem 1.1 is obtained by Propositions 2.1, 2.2 and 2.3.
In this appendix, we state some useful inequalities in the Sobolev space.
Lemma A.1. Let. Then
Lemma A.2. Let. Then we get
Lemma A.3. Let be an integer, then we have
 Pu, X. and Guo, B. (2015) Global Existence and Semiclassical Limit for Quantum Hydrodynamic Equations with Viscosity and Heat Conduction. Kinetic & Related Models, 9, 165-191.
 Polygiannakis, J. and Moussas, X. (2001) A Review of Magneto-Vorticity Induction in Hall-MHD Plasmas. Plasma Physics and Controlled Fusion, 43, 195-221.
 Fan, J., Alsaedi, A., Hayat, T., Nakamura, G. and Zhou, Y. (2015) On Strong Solutions to the Compressible Hall-Magnetohydrodynamic System. Nonlinear Analysis: Real World Applications, 22, 423-434.
 Gao, J. and Yao, Z. (2017) Global Existence and Optimal Decay Rates of Solutions for Compressible Hall-MHD Equations. Discrete & Continuous Dynamical Systems, 36, 3077-3106.
 Xu, F., Zhang, X., Wu, Y. and Liu, L. (2016) Global Existence and Temporal Decay for the 3D Compressible Hall-Magnetohydrodynamic System. Journal of Mathematical Analysis and Applications, 438, 285-310.
 Chae, D., Degond, P. and Liu, J. (2014) Well-Posedness for Hall-Magneto-hydrodynamics. Annales de l’Institut Henri Poincare (C) Non Linear Analysis, 31, 555-565.
 Chae, D. and Lee, J. (2014) On the Blow-Up Criterion and Small Data Global Existence for the Hall-Magnetohydrodynamics. Journal of Differential Equations, 256, 3835-3858.
 Fan, J., Fukumoto, Y., Nakamura, G. and Zhou, Y. (2015) Regularity Criteria for the Incompressible Hall-MHD System. Zeitschrift fur Angewandte Mathematik und Mechanik, 95, 1156-1160.
 Wan, R. and Zhou, Y. (2015) On Global Existence, Energy Decay and Blow-up Criteria for the Hall-MHD System. Journal of Differential Equations, 259, 5982-6008.
 Jia, X. and Zhou, Y. (2016) On Regularity Criteria for the 3D Incompressible MHD Equations Involving One Velocity Component. Journal of Mathematical Fluid Mechanics, 18, 187-206.
 Zhao, X. and Zhu, M. (2018) Global Well-Posedness and Asymptotic Behavior of Solutions for the Three-Dimensional MHD Equations with Hall and Ion-Slip Effects. Zeitschrift für angewandte Mathematik und Physik, 69, 22.
 Umeda, T., Kawashima, S. and Shizuta, Y. (1984) On the Decay of Solutions to the Linearized Equations of Electro-Magneto-Fluid Dynamics. Japan Journal of Applied Mathematics, 1, 435-457.
 Zhou, Y. (2006) Regularity Criteria for the 3D MHD Equations in Terms of the Pressure. International Journal of Non-Linear Mechanics, 41, 1174-1180.
 Li, F. and Yu, H. (2011) Optimal Decay Rate of Classical Solutions to the Compressible Magnetohydrodynamic Equations. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 141, 109-126.
 Tan, Z. and Wang, H. (2013) Optimal Decay Rates of the Compressible Magnetohydrodynamic Equations. Nonlinear Analysis: Real World Applications, 14, 188-201.
 Fan, J., Jia, X., Nakamura, G. and Zhou, Y. (2015) On Well-Posedness and Blowup Criteria for the Magnetohydrodynamics with the Hall and Ion-Slip Effects. Zeitschrift für angewandte Mathematik und Physik, 66, 1695-1706.
 Hu, X. and Wang, D. (2008) Global Solutions to the Three-Dimensional Full Compressible Magnetohydrodynamic Flows. Communications in Mathematical Physics, 283, 255-284.
 Pu, X. and Guo, B. (2013) Global Existence and Convergence Rates of Smooth Solutions for the Full Compressible MHD Equations. Zeitschrift für angewandte Mathematik und Physik, 64, 519-538.
 Gao, J., Tao, Q. and Yao, Z. (2016) Optimal Decay Rates of Classical Solutions for the Full Compressible MHD Equations. Zeitschrift für angewandte Mathematik und Physik, 67, 23.
 He, F., Samet, B. and Zhou, Y. (2018) Boundedness and Time Decay of Solutions to a Full Compressible Hall-MHD System. Bulletin of the Malaysian Mathematical Sciences Society, 41, 2151-2162.
 Hu, X. and Wang, D. (2010) Local Strong Solution to the Compressible Viscoelastic Flow with Large Data. Journal of Differential Equations, 249, 1179-1198.
 Zhang, Y. and Tan, Z. (2007) On the Existence of Solutions to the Navier-Stokes-Poisson Equations of a Two-Dimensional Compressible Flow. Mathematical Methods in the Applied Sciences, 30, 305-329.
 Wang, Y. and Tan, Z. (2011) Optimal Decay Rates for the Compressible Fluid Models of Korteweg Type. Journal of Mathematical Analysis and Applications, 379, 256-271.