Decay Rates of the Full Compressible Hall-MHD Equations for Quantum Plasmas ()
1. Introduction
We consider with the Cauchy problem of the full compressible Hall-magnetohydrodynamic (in short, Hall-MHD) equations in the whole space:
(1)
for
with the initial data:
(2)
here
,
,
and
represent the density, velocity, absolute temperature and magnetic field, respectively.
is the deformation tensor given by
(3)
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
(4)
then the Cauchy problem (1)-(2) admits a unique global solution
satisfying
(5)
Moreover, if
, then we have
(6)
(7)
(8)
(9)
(10)
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.
2. Reformations
In this subsection, we first reformulate the problem as follows. Set
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then
takes the form
(11)
where
![]()
![]()
![]()
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To obtain a symmetric system, we denote
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and
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(11) can be rewritten in the perturbation form as follows
(12)
where the source terms
are
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![]()
![]()
![]()
and
![]()
(13)
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 ![]()
(14)
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
(15)
(16)
(17)
(18)
(19)
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).
(20)
By Lemma A.2, which yields directly
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Hence, we immediately have
![]()
![]()
(21)
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
(22)
Proof. Multiplying (12)1, (12)2, (12)3 and (12)4 by n, v, z and B respectively, the integration over
gives
(23)
The five terms on the right-hand side of the above equation can be estimated as follows.
Firstly, we get
![]()
(24)
Secondly, we obtain
(25)
Next, we have
(26)
it follows from (20) and (21) that
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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
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Similarly to the proof of
, we have
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We similarly obtain
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In light of the estimates
, we see that
(27)
For the fourth term, we have
(28)
It follows from Hölder’s inequality, Lemma A.1 and (20) that
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In the same way as above, we know
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Similarly,
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Let
, similarly
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In a similar way,
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We have
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A similar argument shows that
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Summing up
, we can get
(29)
Finally, we have
(30)
In a similar way, we know
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By a direct computation, we have
(31)
We get
(32)
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
(33)
Proof. Applying
with
to (12) and then taking
-inner product with
, we obtain
![]()
![]()
![]()
(34)
First of all,
is written as
(35)
The first term
can be rewritten as
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where the first and two terms can be estimated as
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The third term in
is written as
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Moreover,
and
can be estimated similarly
![]()
For the term
, we see that
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For the second term of
,
![]()
where
![]()
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in a similar way
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For
,
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For the term
, we know
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For the first term of
, after integrating by parts, we infer from (20) that
![]()
In addition,
![]()
where
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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
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Then
![]()
can be rewritten as
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The first term
can be bounded by
(36)
A similar argument shows that
(37)
By the estimates (36) and (37), we get
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Similarly, we see that
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![]()
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Let
. For the first term
, we exploit the (20), Lemma A.1 and Hölder’s inequality to obtain
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For
, we see that
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Let
. We integrate by parts and use Hölder’s inequality to obatin
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The same estimate holds for
. Therefore
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As in
, we have
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In the same manner, it is easy to deduce
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Similar to the estimation, we obtain
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where
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Similarly, we get
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where
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Similarly, for the terms
and
, recalling from the estimate of
, we have
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That is to say
(38)
it follows from (38) that
(39)
For (39), we have
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Similar to (39), we see
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Consequently, in light of
, we get
(40)
Since
is small, (33) is given.
In the following, we consider the energy estimates on the entropy n.
Lemma 3.3. It holds that
(41)
Proof. When
to
, applying
to the second Equation in (12) and testing by
, we obtain
![]()
(42)
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
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Similar to the estimation on
, we have
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![]()
![]()
Let
. For the terms
, integrating by parts and Hölder’s inequality yields,
![]()
It is easy to say
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Finally, Combing with
and
, we get
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Plugging these estimates into (42), we obtain
(43)
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
(44)
In view of Lemma 3.3, we have
(45)
Multiplying (45) by
and adding the result to (44)
![]()
where
is sufficiently small. Consequently, using the fact
![]()
We show that
(46)
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).
(47)
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
, ![]()
![]()
![]()
![]()
(48)
where
.
Lemma 4.2. Let
, then it holds that
(49)
for an arbitrarily small
.
To use the Lp-Lq estimates of the linear problem for the nonlinear system (12) as
, then (12) becomes
(50)
where
. Such that
![]()
and then
(51)
Lemma 4.3. We assume
is a smooth solution
(52)
where
.
Proof. We can know
![]()
It is easy to know,
(53)
(54)
(55)
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
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For
, in a similar way, we have
![]()
where
.
Summing up
, we obtain
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The terms
and
can be bounded by
![]()
and
![]()
For
, in a similar way, we have
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Similarly, it holds that
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and
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Then using the similar way, we arrive at
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or equivalently,
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In a similar way, we have
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![]()
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Summing these terms, we get
(56)
(57)
(58)
That is, we obtain
(59)
where
.
The inequality reads that
(60)
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
(61)
assume that
(62)
Notice that
is non-decreasing
![]()
Then it follows from Lemma 4.3 that
(63)
In fact, applying Gronwall’s inequality to the Lyapunov-type inequality (61) and using (62), we find that
(64)
In view of (62), we have
![]()
which implies
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Since
is sufficiently small. Consequently,
(65)
Using (65), we thus get
![]()
![]()
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which also implies from Lemma A.1 that
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Therefore (16), (17) and (18) are obtained. Then
![]()
Meanwhile via Lemma 4.1 and (48), we have
![]()
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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.
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
(66)
and
(67)
where
and
(68)