In this paper, we consider the following compressible Hall-MHD equations for quantum plasmas in three dimensional whole space :
for with the initial conditions:
Here , and denote the density, the velocity and magnetic field, respectively. The pressure is a smooth function with for , and are referred to as the shear viscosity and the bulk viscosity coefficients of the fluid, which satisfy the usual condition
where is the Planck constant. The -term is referred to as the quantum potential or Bohm potential term  , which is strongly nonlinearly degenerate and leads to the system non-diagonal and should be regarded as a consequence from dispersive properties of the quantum fluid.
The quantum terms date back to Wigner  , where quantum corrections were considered for the thermodynamic equilibrium. The quantum correction to the stress tensor was proposed in   . One may see Hass  for many physics backgrounds and mathematical derivation of many interesting models. Pu and Guo  established the global existence of strong solutions and the semiclassical limit for the full compressible quantum Navier-Stokes. Later, they  obtained the following decay rates
with . Recently, Pu and Xu  showed the decay rates for smooth solutions of the magnetohydrodynamic model for quantum plasmas as follows:
where . The interested reader can refer to   and references therein for more results of the quantum term.
Without the quantum effects, the above system (1.1) is usual compressible Hall-MHD equations, which represent the momentum conservation equation for the plasma fluid. Compared with the classical MHD equations, there exists the Hall term in (1.1)3, which makes Hall-MHD equations entirely different from MHD equations for understanding the problem of magnetic reconnection, due to the froze-field effect. Thus, we note that the Hall-MHD equations are useful in describing many phenomena such as magnetic reconnection in space plasmas, star formation, neutron stars and geo-dynamo (see    and references therein).
The compressible Hall-MHD equations have received some results in recent years. In particular, Fan et al.  proved the local existence of strong solutions with positive initial density and global small classical solutions with small initial perturbation belongs to . They also obtained optimal time decay rate for strong solutions as follows:
Motivated by Fan et al., Gao and Yao  improved the optimal time decay rates for higher order spatial derivatives of classical solutions under the condition that the initial data belongs to . For the case of initial data belonging to some negative Sobolev space, Xu et al. showed the fast time decay rates for the higher-order spatial derivatives of solutions in  . Recently, they  established the unique global solvability and the optimal time decay rates of strong solutions in Besov spaces. On the other hand, there are also many works of incompressible Hall-MHD equations, see  -  .
To our knowledge, so far there is no result on the large-time behaviors of the Cauchy problem (1.1)-(1.2). Therefore, the main purpose of this paper is to investigate global existence and decay rate in time of smooth solutions in H4-framework. The decay rate of solutions towards the steady state has been an important problem in the PDE theory, which has been investigated extensively, see for instance  -  and the references therein. Compared with the general compressible H-MHD equations    , the quantum term (higher order) appears in (1.1)2, which leads to new difficulties in decay analysis than those results. The major method is to make a hypothesis (3.1) to cooperate with the special structure of (1.1). We first construct the global existence of strong solutions by the standard energy method under the condition that the initial data are close to the equilibrium state in H4-norm. Furthermore, by assuming that the initial data in L1-norm are finite additionally, we establish the optimal time decay rates of strong solutions by the method of spectral analysis and energy estimates. More precisely, we obtain the following time decay rates
for all .
Our main results of this paper are stated as the following theorem.
Theorem 1.1 Assume that the initial condition satisfies the constraints (1.2), there exists a constant such that if
then there exists a unique global solution of the Cauchy problem (1.1)-(1.2) satisfying
Furthermore, if , the solution enjoys the following decay properties
for some positive constant .
Notation. Throughout this paper, we denote the norms in Sobolev spaces and by and for and respectively. In particular, for , we shall simply use and . Moreover, , and for any integer , denotes all derivatives of order of the function f. In addition, C denotes the generic positive constant which may vary in different places and the integration domain will be always omitted without any ambiguity. Finally, denotes the inner product in .
The rest of this paper is organized as follows. In Section 2 we reformulate the system (1.1)-(1.2) into a more convenient form. In Section 3, we make some crucial energy estimates for the solution that will play an essential role for us to construct the global existence of strong solutions. In Section 4, we use the energy estimates derived in Section 3 to build the global existence of the solution, which combine with the linear decay estimates imply Theorem 1.1. In Appendix, we list some useful inequalities.
To make it more convenient to prove Theorem 1.1, in this section, we will reformulate the problem (1.1) and (1.2). More precisely, we set
then the system (1.1) and (1.2) can be rewritten as
where and the source terms are
We defined the two nonlinear function of n by
In the following, we will establish the global existence and time decay rates of the solution to the stead state . We first define the solution space of the initial value problem (2.1) by
for any . By the standard continuity argument, the global existence of solutions to (2.1) will be obtained by combining the local existence result together with a priori estimates.
Proposition 2.1 (Local existence). Assume that and
Then there exists a positive constant depending on such that the initial value problem (2.1) has a unique solution satisfying and
Proposition 2.2 (A priori estimate). Let . Suppose that the initial value problem (2.1) has a solution for some . Then there exist a small constant and a constant , which are independent of T, such that if
then for any , it holds that
Furthermore, there is a constant such that for any , the global solution has the decay properties
The proof of Theorem 1.1 is followed from Proposition 2.1 and Proposition 2.2 by the standard iteration arguments. The proof of Proposition 2.1 is standard and thus omitted. Proposition 2.2 will be proved in Section 3 and Section 4.
3. Energy Estimates
In this section we will drive some a priori energy estimates for the solutions to the system (2.1). We assume a priori that for sufficiently small ,
By (2.1) and Sobolev’s inequality, we then obtain
Therefore, for , we have
In the first place, we will obtain the dissipation estimate for v.
Lemma 3.1 Let be a smooth solution to (2.1), then it holds that
Proof. Multiplying (2.1)1, (2.1)2 and (2.1)3 by n, v and B respectively, and then integrating them over , we have
We will estimate the three terms on the right-hand side.
Firstly, for the first term, by the continuity equation and integration by parts twice, we have
Secondly, for the second term, it follows from Lemma 5.1, the assumption (3.1), the Hölder inequality and the Young inequality that
Next, for the third term, we have
For the term and , using (3.1), (3.2), Hölder’s inequality, Young’s inequality and Lemma 5.1, we obtain
For the term , we have by Hölder’s inequality, Lemma 5.1 and (3.1) that
Let . For the term , by (3.1), (3.2), the Hölder inequality and integration by parts, we have
In a similar way, we have
For the term , we similarly obtain
In light of the estimates , we can get
Finally, for the last term, we have
Similarly, we bound the first and second terms on the right hand side of (3.9) by
For the last term on the right hand side of (3.9), by integration by part, we have
Combined with (3.10) and (3.11), we get
Substituting (3.5), (3.6), (3.8) and (3.12) yields into (3.4), by the smallness of , we get (3.3). □
In the following lemma, we derive the higher-order dissipative estimates.
Lemma 3.2 Let be a smooth solution to (2.1), then
Proof. For , applying to (2.1)1-(2.1)3 and then taking L2-inner product with , we have
We will estimate each term on the right-hand side. At first, we split as
By the continuity equation and integration by parts, the first term can be rewritten as
where the first two terms can be estimated as
Note that the last term in is much more complicated, so we can further decompose it into
The first two terms and can be bounded by
For the term , by the continuity equation and the Hölder inequality, we have
For the second term of , separating the case of and from the order cases, we bound the summation by
Similarly, we bound the first and the last term in by
Collecting these terms, we get
For the term , we have
For the first term of , we have by integration by parts and (3.1) that
For the second term of , similarly, we separate the case of and from the order cases and bound the summation by
Collecting these term, we get
For the second term of (3.15), we have by the assumption (3.1), Hölder’s inequality, Lemma 5.1, (3.2) and integration by parts that
Summing up and , we have
For the term , we can rewrite it as
The first term can be bounded by
For the second term , similarly, separating the case of from the order cases, we bound the summation by
In light of (3.16) and (3.17), we obtain
Recalling from the estimates of , we have
Let . For the first term , we have by (3.1), Lemma 5.1, Hölder’s inequality and integration by parts that
The same estimates hold for and . Combining all the estimates for , we get
Let . We have by integration by parts and Hölder’s inequality that
The same estimate holds for . Combining all the estimates for , we obtain
For the term , we have
Similarly, for the terms and , recalling from the estimate of , we have
Indeed, computing directly, it is easy to deduce
then for the term , we have by integration by parts and (3.18) that
To estimate the first factor on the right-hand side of (3.19), using Lemma 5.1, 5.2 and Hölder’s inequality, we obtain
The similar estimate holds for the second factor on the right-hand side of (3.19). Thus, for the term , we have
Consequently, summing up , by the smallness of , we have
Summing up above estimates for from to , by the smallness of , we get (3.13). □
Next, we derive the dissipation estimate for n.
Lemma 3.3 Let be a smooth solution to (2.1), then we have
Proof. For , applying to (2.1)2, multiplying them by and then integrating them over , we have
Next, we will estimate each term on the right-hand side. First, for the term , by integration by parts twice, (3.1) and the continuity equation, we have
For the terms and , similarly as the estimate of , we obtain
Similarly for the terms and , we recall from the estimate of to have
Let . For the terms , we have by integration by parts and Hölder’s inequality that
The same estimates hold for the other three terms of . Combing all the estimates for , we have
Finally, Combing with and , we get
In light of , we have
Summing up above estimates for from to , by the smallness of , we conclude Lemma 3.3. □
4. Convergence Rates
In this section, we will combine all the energy estimates that we have derived in the previous section to prove Proposition 2.2.
The linearized equations corresponding to (2.2)1-(2.2)3 read
Thus, at the level of the linearization, B is decoupled with . If we set
then the solution to (4.1)1-(4.1)2 can be written as
where is a matrix-valued differential operator given by
The solution semigroup has the following property on the decay in time, cf.  .
Lemma 4.1 Let be an integer. Assume that is the solution of the linearized system for the first two equations in (2.1) with the initial data , , then
We need the following elementary inequality  :
Lemma 4.2 Let , then it holds that
for an arbitrarily small .
If we denote the nonlinear terms for the first two equations in (2.1) as , then (2.1) becomes
where . Note that for , we have
and then there exists a constant C such that
for any and .
Lemma 4.3 Let be a smooth solution to (2.1), then
Proof. From Duhamel’s principle, it holds that
Thus from Lemma 3.1 and (4.4), we have
By (3.1), Hölder’s inequality and Lemma 5.1, the nonlinear source terms can be estimated as follows:
Put these estimates into (4.7) and (4.8), we have
Let , and in (4.5), we obtain
Putting (4.12) and (4.13) together, then we complete the proof of Lemma 4.3. □
Now we are in a position to prove Proposition 2.2.
Since is sufficiently small, from Lemma 3.1 and 3.2, we obtain
In view of Lemma 3.3, we have
Multiplying (4.14) by , adding it with (4.13) since is small, then we deduce
We have by Gronwall’s inequality that
then (4.16) gives (2.3).
We define the temporal energy functional
where it is noticed that
that is, there exists a constant such that
From Lemma 3.2 and 3.3, we have
Adding to both sides of the inequality above gives
where is a positive constant independent of . We define
From Lemma 4.2 and Lemma 4.3, we have
By Gronwall’s inequality, we have from (4.16) that
Since is non-decreasing, we have from (4.20) that
which implies that if is small enough, then
This in turn gives
From (4.21), we have
which also implies from Lemma 5.1 that
Hence (2.5) and (2.6) are proved. By Sobolev’s inequality, we have
Next, by (4.2) and (4.5), it follows from the Duhamel’s principle that
Hence, for any , we have by the interpolation that
where , this proves (2.4). On the other hand, using the estimates above (2.1), we have
Then, for any we get (2.7). Therefore, the proof of Proposition 2.2 is complete. □
In this appendix, we state some useful inequalities in the Sobolev space.
Lemma 5.1 Let . Then
Lemma 5.2 Let be an integer, then we have
Proof. Please refer for instance to  . □
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