Abstract
In this paper, we consider the Cauchy problem of a full compressible Hall-MHD system. \(\displaystyle \dot{H}^{-s}\ \ \left( 0<s\le \frac{3}{2}\right) \) Sobolev norms are shown to be preserved along time evolution. Boundedness and time decay of the higher-order spatial derivatives of the smooth solutions are given.
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1 Introduction
Magnetohydrodynamics (MHD) studies the electromagnetic fields and conducting fluids. In this paper, we consider the time decay of the following 3D full compressible Hall-MHD system [16]:
Here, \(\rho , u, \theta \), and B denote the density, velocity, temperature, and the magnetic field, respectively. The pressure \(p:=R\rho \theta \) with physical constant \(R>0\). \(\mu \) and \(\lambda \) are the shear viscosity and bulk viscosity of the flow and satisfy \(\mu >0\) and \(\displaystyle \lambda +\frac{2}{3}\mu \ge 0\). The positive constant \(C_V\) is the specific heat at constant volume. For simplicity, we will take \(R=C_V=1\). \(\nabla u^t\) is the transpose of the \(\nabla u\).
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.
Continuity equation (1.1) allows the simplification of Eqs. (1.2) and (1.3) in their first two terms as \(\rho \partial _tu+\rho u\cdot \nabla u\) and \(C_V\rho \partial _t\theta +C_V\rho u\cdot \nabla \theta \), respectively.
When the Hall effect term \(\displaystyle {\mathrm {rot}\,\left( \frac{\mathrm {rot}\,B\times B}{\rho }\right) }\) is neglected, the system (1.1)–(1.4) reduces to the well-known full compressible MHD system, which has received many studies [2, 10, 11, 13, 14, 19]. The local strong solution was proved by Fan and Yu [10]. Fan and Yu [11], Ducomet-Feireisl [2], and Hu and Wang [13, 14] established the global weak solutions. Wei et al. [19] showed the time decay of the smooth solutions.
Very recently, in [4, 5, 8, 9], local well-posedness and low Mach number limit and blow-up criteria of smooth solutions to the problem (1.1)–(1.6) were established.
The aim of this paper is to use the method in [19] to prove the time decay rate of the Cauchy problem (1.1)–(1.6). We will prove
Theorem 1.1
Let \(N\ge 3\) and \(\rho _0-1, u_0, \theta _0-1, B_0\in H^N\). Then, there exists a positive constant \(\delta _0\) such that if
then the Cauchy problem (1.1)–(1.6) has a unique global solution \((\rho , u, \theta , B)\), satisfying that for all \(t\ge 0\),
If further, \((\rho _0-1, u_0, \theta _0-1, B_0)\in \dot{H}^{-s}\) for some \(\displaystyle {s\in \left( 0,\frac{3}{2}\right) }\), then for all \(t\ge 0\),
and
where \(\Lambda :=(-\Delta )^\frac{1}{2}\). Here, \(C_0\) is a positive constant depending on \(N,s,\lambda ,\mu ,R,C_V\), and the initial data.
Remark 1.1
Wei et al. [19] showed the same time decay rate (1.8) and (1.10) when the Hall effect term is neglected. The new input of this paper is that it includes the Hall effect term.
Remark 1.2
For the corresponding incompressible model (incompressible Hall-MHD), we refer to [1, 3, 6, 7, 12, 17, 18] and references therein.
Notation. Throughout this paper, \(D^k\) with an integer k stands for the usual any spatial derivatives of order k. We define the operator \(\Lambda ^s, s\in \mathbb {R}\) by \(\displaystyle \Lambda ^sf:=\int |\xi |^s{\hat{f}}(\xi )e^{2\pi ix\cdot \xi }\hbox {d}\xi \), where \({\hat{f}}\) is the Fourier transform of f. We define the homogeneous Sobolev space \(\dot{H}^s\) of all f for which \(\Vert f\Vert _{\dot{H}^s}\) is finite, where \(\Vert f\Vert _{\dot{H}^s}:=\Vert \Lambda ^sf\Vert _{L^2}=\Vert |\xi |^s{\hat{f}}\Vert _{L^2}\), and we use \(H^s\) to denote the usual Sobolev spaces with norm \(\Vert \cdot \Vert _{H^s}\). We will employ the notation \(A\lesssim B\) to mean that \(A\le CB\) for a universal constant \(C>0\).
2 Proof of Theorem 1.1
This section is devoted to the proof of Theorem 1.1. We only need to establish a priori estimates.
First, we rewrite (1.1)–(1.3) as
In the following proofs, we will use the following bilinear product estimate due to Kato and Ponce [15]:
with \(s>0\) and \(\frac{1}{p}=\frac{1}{p_1}+\frac{1}{q_1}=\frac{1}{p_2}+\frac{1}{q_2}\).
We assume a priori that for sufficiently small \(\delta >0\),
By (2.5) and Sobolev’s inequality, we see that
Lemma 2.1
Let (2.5) hold true, we have
for \(k=0,1,\ldots ,N-1\).
Proof
Applying \(D^k\) to (2.1), (2.2), (2.3), and (1.4), testing by \(D^k(\rho -1), D^ku, D^k(\theta -1)\), and \(D^kB\), respectively, and summing up them, we find that
It has been proved in [19] that
We use (2.4) and (2.5) to bound \(I_2\) as follows:
Inserting the above estimates into (2.8) gives the lemma. \(\square \)
Lemma 2.2
Let (2.5) hold true, we have
for \(k=0,1,\ldots ,N-1\).
Proof
Applying \(D^{k+1}\) to (2.1), (2.2), (2.3), and (1.4), testing by \(D^{k+1}(\rho -1), D^{k+1}u, D^{k+1}(\theta -1)\), and \(D^{k+1}B\), respectively, and summing up them, we obtain
It has been proved in [19] that
Similarly to \(I_2\), we use (2.4) and (2.5) to bound \(I_4\) as
Inserting the above estimates into (2.10) gives the lemma. \(\square \)
In [19], it is proved that
for \(k=0,1,\ldots ,N-1\).
Lemma 2.3
For \(s\in \left( 0,\frac{1}{2}\right] \), we have
and for \(\displaystyle {s\in \left( \frac{1}{2},\frac{3}{2}\right) }\), we have
Proof
Applying \(\Lambda ^{-s}\) to (2.1), (2.2), (2.3), and (1.4), testing by \(\Lambda ^{-s}(\rho -1), \Lambda ^{-s}u, \Lambda ^{-s}(\theta -1)\), and \(\Lambda ^{-s}B\), respectively, and summing up them, we derive
It has been proved in [19] that
and
for \(\displaystyle {s\in \left( \frac{1}{2},\frac{3}{2}\right) }\).
In the following proofs, we will use the following inequality [19]:
and the Gagliardo–Nirenberg inequalities [19]:
Using (2.15), (2.16), and (2.17), we bound \(I_6\) as follows:
When \(\displaystyle 0<s\le \frac{1}{2}\), we have
When \(\displaystyle \frac{1}{2}<s<\frac{3}{2}\), using the Gagliardo–Nirenberg inequality
we bound \(I_6\) as
Inserting the above estimates (2.18) and (2.19) into (2.14) leads to the lemma. \(\square \)
Now, by the same calculations as that in [19], we can finish the proof of Theorem 1.1.
We first close the energy estimates at each \(l\hbox {th}\) level in our weaker sense. Let \(N\ge 3\) and \(0\le l\le m-1\) with \(1\le m\le N\). Summing up the estimates (2.7) of Lemma 2.1 for from \(k=l\) to \(m-1\), we obtain
Let \(k=m-1\) in the estimates (2.9) of Lemma 2.2, we have
Adding the inequality (2.20) with (2.21), we get
Summing up the estimates (2.11) for from \(k=l\) to \(m-1\), we have
Multiplying (2.23) by \(\displaystyle {\frac{2C_2\delta }{C_3}}\) and adding it with (2.22), since \(\delta >0\) is small, we deduce that there exists a constant \(C_5>0\) such that for \(0\le l\le m-1\)
Next, we define \(\mathcal {E}_l^m(t)\) to be \(C_5^{-1}\) times the expression under the time derivative in (2.24). Observe that since \(\delta \) is small, \(\mathcal {E}_l^m(t)\) is equivalent to \(\Vert D^l(\rho -1,u,\theta -1,B)(t)\Vert _{H^{m-l}}^2\), that is, there exists a constant \(C_6>0\) such that for \(0\le l\le m-1\)
Then, we may write (2.24) as that for \(0\le l\le m-1\)
Proof of (1.8). Taking \(l=0\) and \(m=3\) in (2.26) and then integrating directly in time, we get
By a standard continuity argument, this closes the a priori estimates if at the initial time we assume that \(\Vert (\rho _0-1,u_0,\theta _0-1,B_0)\Vert _{H^3}^2\le \delta _0\) is sufficiently small. This in turn allows us to take \(l=0\) and \(m=N\) in (2.26) and then integrate it directly in time to obtain
This proved (1.8).
Next, we turn to prove (1.10). However, we are not able to prove them for all \(\displaystyle s\in \left[ 0,\frac{3}{2}\right) \), and at this moment, we shall first prove them for \(\displaystyle s\in \left[ 0,\frac{1}{2}\right] \).
Define \(\mathcal {E}_{-s}(t):=\Vert \Lambda ^{-s}(\rho -1,u,\theta -1,B)(t)\Vert _{L^2}^2\). Then, integrating in time (2.12) of Lemma 2.3, by the bound (1.8), we obtain that for \(\displaystyle s\in \left( 0,\frac{1}{2}\right] \),
This implies (1.10) for \(\displaystyle s\in \left[ 0,\frac{1}{2}\right] \), that is,
If \(l=1,\ldots ,N-1\), we may use the Gagliardo–Nirenberg inequality [19]:
By this fact and (2.29), we may find
This together with (1.8) implies in particular that for \(l=1,\ldots ,N-1\),
Thus, we deduce from (2.26) with \(m=N\) the following time differential inequality
Solving this inequality directly gives
This implies that for \(s\in \left[ 0,\frac{1}{2}\right] \),
Hence, by (2.29), (2.32), and the interpolation, we get (1.10) for \(\displaystyle s\in \left[ 0,\frac{1}{2}\right] \).
For \(\displaystyle s\in \left( \frac{1}{2},\frac{3}{2}\right) \), notice that the arguments for the case \(\displaystyle s\in \left[ 0,\frac{1}{2}\right] \) cannot be applied to this case. However, observing that we have \(\rho _0-1,u_0,\theta _0-1,B_0\in \dot{H}^\frac{1}{2}\) since \(\dot{H}^{-s}\cap L^2\subset \dot{H}^{-s'}\) for any \(\displaystyle s'\in \left[ 0,\frac{1}{2}\right] \), we then deduce from what we have proved for (1.10) with \(s=\frac{1}{2}\) that the following decay result holds:
Hence, by (2.33), we deduce from (2.13) that for \(\displaystyle s\in \left( \frac{1}{2},\frac{3}{2}\right) \),
This implies (1.10) for \(\displaystyle s\in \left( \frac{1}{2},\frac{3}{2}\right) \), that is,
Now that we have proved (2.35), we may repeat the arguments leading to (1.10) for \(\displaystyle s\in \left[ 0,\frac{1}{2}\right] \) to prove that they hold also for \(\displaystyle s\in \left( \frac{1}{2},\frac{3}{2}\right) \). This completes the whole proof of Theorem 1.1. \(\square \)
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Acknowledgements
The authors are indebted to the referees for nice suggestions. This work is partially supported by NSFC (Grant No. 71772153). The second author extends his appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group No RGP-237 (Saudi Arabia).
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Communicated by Ahmad Izani Md.
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He, F., Samet, B. & Zhou, Y. Boundedness and Time Decay of Solutions to a Full Compressible Hall-MHD System. Bull. Malays. Math. Sci. Soc. 41, 2151–2162 (2018). https://doi.org/10.1007/s40840-018-0640-y
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DOI: https://doi.org/10.1007/s40840-018-0640-y