Decay Rates of the Full Compressible Hall-MHD Equations for Quantum Plasmas

In this paper, we are concerned with the Cauchy problem of the full compressible Hall-magnetohydrodynamic equations in three-dimensional whole space. By the energy method, global existence of a unique strong solution is established. If further that the L ¹-norm of the perturbation is bounded, we prove the decay rates in time of the solution and its first-order derivatives in L ²-norm via some L ^p- L ^q estimates by the linearized operator.

• Full Compressible Hall-Magnetohydrodynamic Equations
• Global Existence
• Decay Rates

1. Introduction

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 [1] first derived the quantum correction to the energy density for thermodynamic equilibrium, and the quantum correction term goes hand in hand with Bohm potential [2] [3]. The -term was advanced in [4]. People might refer to Haas [5] for more physical interpretations of the model. Pu et al. [6] 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 [7]. 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 [8] [9] ), formation and evolution of stars and neutron stars [10] [11] [12] and geo-dynamo [13] (see [14] 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. [15], where the small initial disturbance belongs to. More precisely, optimal time decay rate was also established. Later, the result from [15] was improved by Gao and Yao [16]. 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. [17] 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 [18]. Obviously, system (1) becomes incompressible Hall-MHD system when and there are many interesting global results, see [19] - [24] 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 [25] - [30] ). For the corresponding full compressible MHD model, we can refer to [31] [32] [33] [34] [35] and references therein. Hu and Wang [31] 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 [33]. He et al. [35] 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 L^p-L^q estimates by the linearized operator, we prove the decay rates.

2. Reformations

In this subsection, we first reformulate the problem as follows. Set

then takes the form

where

To obtain a symmetric system, we denote

and

(11) can be rewritten in the perturbation form as follows

where the source terms are

and

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

and

for any. Then

Proof.

The proof can be done by using the standard iteration arguments. Refer, for instance, to [36] [37] [38].

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

Similarly,

Let, similarly

In a similar way,

We have

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

We get

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,

where

in a similar way

For,

For the term, we know

For the first term of, after integrating by parts, we infer from (20) that

In addition,

where

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

Then

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

where

Similarly, we get

where

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. [39].

Lemma 4.1. Let be integers. satisfies the inequalities with the initial data,

where.

Lemma 4.2. Let, then it holds that

for an arbitrarily small.

To use the L^p-L^q estimates of the linear problem for the nonlinear system (12) as, then (12) becomes

where. Such that

and then

Lemma 4.3. We assume is a smooth solution

where.

Proof. We can know

It is easy to know,

And the nonlinear source terms can be estimated as follows:

and

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

and

In a similar way, we get

and

Let, we have by Hölder’s inequality and Lemma A.1 that

and

For, in a similar way, we have

where.

Summing up, we obtain

The terms and can be bounded by

and

For, in a similar way, we have

Similarly, it holds that

and

Then using the similar way, we arrive at

or equivalently,

In a similar way, we have

Summing these terms, we get

That is, we obtain

where.

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

assume that

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

which implies

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.

5. Conclusion

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.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

Appendix

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

and

where and

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