Abstract
We have investigated an initial-boundary problem for the perturbation equations of rotating, incompressible, and viscous magnetohydrodynamic (MHD) fluids with zero resistivity in a horizontally periodic domain. The velocity of the fluid in the domain is non-slip on both upper and lower flat boundaries. We switch the analysis of the initial-boundary problem from Euler coordinates to Lagrangian coordinates under proper initial data, and get a so-called transformed MHD problem. Then, we exploit the two-tiers energy method. We deduce the time-decay estimates for the transformed MHD problem which, together with a local well-posedness result, implies that there exists a unique time-decay solution to the transformed MHD problem. By an inverse transformation of coordinates, we also obtain the existence of a unique time-decay solution to the original initial-boundary problem with proper initial data.
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1 Introduction
The three-dimensional (3D) rotating, incompressible and viscous magnetohydrodynamic (MHD) equations with zero resistivity in a domain \(\Omega\subset\mathbb{R}^{3}\) read as follows:
Here the unknowns \({v}= {v}(x,t)\), \(M:= {M}(x,t)\) and \(p=p(x,t)\) denote the velocity, the magnetic field, and the pressure of the incompressible MHD fluid respectively; \(\mu>0\), ρ and \(\lambda_{0}\) stand for the coefficients of the shear viscosity, the density constant, and the permeability of vacuum, respectively. \(2\rho(\vec{\omega}\times v)\) represents the Coriolis force, and \(\vec{\omega}=(0,0,\omega)\) denotes the constant angular velocity in the vertical direction. In system (1.1), equation (1.1)1 describes the balance law of momentum, while (1.1)2 is called the induction equation. As for the constraint \(\operatorname{div} M=0\), it can be seen just as a restriction on the initial value of M since \((\operatorname{div} M)_{t}=0\) due to (1.1)2.
Let \(\bar{M}:=(m_{1},m_{2},m_{3})\) be a constant vector with \(m_{3}\neq0\), and \((0,\bar{M},\bar{p})\) be a rest state of the system (1.1). We denote the perturbation to an equilibrium state \((0,\bar{M})\) by
Then, \(({v},N,q)\) satisfies the perturbation equations
where we have used the relation \(\vec{\omega}\times v=\omega( v_{1}e_{2}-v_{2} e_{1})\). For system (1.2), we impose the initial and the boundary conditions:
where \(v_{0}\) and \(N_{0}\) should satisfy the compatibility conditions \(\operatorname {div}v_{0}=\operatorname{div}N_{0}=0\). We call the initial-boundary value problem (1.2)-(1.4) the MHD problem (with rotation) for simplicity. In this article, we always assume that the domain is horizontally periodic with finite height, i.e.,
where \(\mathcal{T} :=(2\pi L_{1}\mathbb{T})\times(2\pi L_{2}\mathbb{T}) \), \(\mathbb{T}=\mathbb{R}/\mathbb{Z}\), and \(2\pi L_{1}, 2\pi L_{2}>0\) are the periodicity lengths.
The effects of magnetic fields and rotation on the motion of pure fluids were widely investigated; see [1–10] and the references cited therein. In particular, Tan and Wang [11] showed that the well-posedness problem of the initial-boundary problem (1.2)-(1.4) for \(\omega=0\) (i.e., without the effect of rotation). In this article, we further consider \(\omega\neq0\), and show that there also exists a unique time-decay solution to the initial-boundary problem (1.2)-(1.4) in Lagrangian coordinates (see Theorem 2.1), which, together with the inverse transformation of coordinates, implies the existence of a time-decay solution to the original initial-boundary problem (1.2)-(1.4) with proper initial data in \(H^{7}(\Omega)\). Our result also holds for the case \(\omega=0\), thus improves Tan and Wang’s result in [11], in which the sufficiently small initial data at least belong to \(H^{16}(\Omega)\).
In the next section we introduce the form of the initial-boundary problem (1.2)-(1.4) in Lagrangian coordinates, and the details of our result.
2 Main results
2.1 Reformulation
In general, it is difficult to directly show the existence of a unique global-in-time solution to (1.2)-(1.4). Instead, we switch our analysis to Lagrangian coordinates as in [12, 13]. To this end, we assume that there is an invertible mapping \(\zeta_{0}:=\zeta_{0}(y):\Omega\to\Omega\), such that
where \(\zeta_{3}^{0}\) denotes the third component of \(\zeta_{0}\). We define the flow map ζ as the solution to
We denote the Eulerian coordinates by \((x,t)\) with \(x=\zeta(y,t)\), where \((y,t)\in\Omega\times\mathbb{R}^{+}\) stand for the Lagrangian coordinates. In order to switch back and forth from Lagrangian to Eulerian coordinates, we assume that \(\zeta(\cdot,t)\) is invertible and \(\Omega=\zeta(\Omega, t)\). In other words, the Eulerian domain of the fluid is the image of Ω under mapping ζ. In view of the non-slip boundary condition \(v|_{\partial\Omega}=0\), we have \(\partial\Omega=\zeta(\partial\Omega, t)\). In addition, since \(\det\nabla\zeta_{0}=1\), we have
due to \(\operatorname{div}v=0\); see [14], Proposition 1.4.
Now, we further define the Lagrangian unknowns by
Thus in Lagrangian coordinates the evolution equations for u, p̃ and B read as
with initial and boundary conditions
Moreover, \(\operatorname {div}_{\mathcal{A}}B=0\) if the initial data \(\zeta_{0}\) and \(B_{0}\) satisfy
Here \(\mathcal{A}_{0}\) denotes the initial value of \(\mathcal{A}\), the matrix \(\mathcal{A}:=(\mathcal{A}_{ij})_{3\times3}\) via \(\mathcal{A}^{T}=(\nabla\zeta)^{-1}:=(\partial_{j} \zeta _{i})^{-1}_{3\times 3}\), and the differential operators \(\nabla_{\mathcal{A}}\), \(\operatorname {div}\mathcal{A}\) and \(\Delta_{\mathcal{A}}\) are defined by \(\nabla_{\mathcal {A}}f:= (\mathcal {A}_{1k}\partial_{k}f, \mathcal{A}_{2k}\partial_{k}f,\mathcal{A}_{3k}\partial_{k}f)^{T}\), \(\operatorname{div}_{\mathcal {A}}(X_{1},X_{2},X_{3})^{T}:=\mathcal{A}_{lk}\partial_{k} X_{l}\), and \(\Delta _{\mathcal{A}}f:=\operatorname{div}_{\mathcal{A}}\nabla_{\mathcal{A}}f\) for appropriate f and X. It should be noted that we have used the Einstein convention of summation over repeated indices, and \(\partial _{k}=\partial_{y_{k}}\). In addition, in view of the definition of \(\mathcal{A}\) and (2.3), we can see that \(\mathcal{A}=(A^{*}_{ij})_{3\times3}\), where \(A^{*}_{ij}\) is the algebraic complement minor of the \((i,j)\)th entry \(\partial_{j} \zeta_{i}\). Since \(\partial_{k}A^{*}_{ik}=0\), we can get an important relation
which will be used in the derivation of temporal derivative estimates.
Our next goal is to eliminate B by expressing it in terms of ζ. This can be achieved in the same manner as in [12, 13]. For the reader’s convenience, we give the derivation here. In view of the definition of \(\mathcal{A}\), one has
where \(\delta_{ij}=0\) for \(i\neq j\), and \(\delta_{ij}=1\) for \(i=j\). Thus, applying \(\mathcal{A}_{jl}\) to (2.5)4, we obtain
which implies that \(\partial_{t}(\mathcal{A}_{jl}B_{j})=0\) (i.e., \((\mathcal{A}^{T} B)_{t}=0\)). Hence,
which yields \(B_{i}=\partial_{l} \zeta_{i}\mathcal{A}_{jl}^{0}B_{j}^{0}\), i.e.,
Here and in what follows, the notation \(f^{0}\) also denotes the initial data of the function f. To obtain the asymptotic stability in time, we naturally expect
Thus (2.9) formally implies
Putting the above expression of \(B_{0}\) into (2.9), we get
Moreover, in view of (2.8), (2.11) and (2.12), the Lorentz force term can be represented by
Summing up the above analyses, we can see that, under the initial conditions (2.1) and (2.11), one can use the relation (2.13) to change (2.5) into the following Navier-Stokes system:
and B is defined by (2.12). Now, we introduce the shift functions
Then the evolution equations for the shift functions \((\eta,q)\) and u read as
where \(\mathcal{A}=(I+\nabla\eta)^{-1}\), \(I=(\delta_{ij})_{3\times3}\). The associated initial and boundary conditions read as follows:
It should be noted that the shift function q is the sum of the perturbed fluid and the magnetic pressures in Lagrangian coordinates. Hence we still call q the perturbation pressure for the sake of simplicity. In this article, we call the initial-boundary value problem (2.15)-(2.16) the transformed MHD problem.
2.2 Main results
Before stating our first main result on the transformed MHD problem in detail, we introduce some simplified notations that shall be used throughout this paper:
where, and in what follows, the letter c denotes a generic constant which may depend on the domain Ω and some physical parameters, such as \(\lambda_{0}\), M̄, g, μ and ρ in the MHD equations (2.15). It should be noted that a product space \((X)^{n}\) of vector functions is still denoted by X, for example, a vector function \({u}\in(H^{2})^{3}\) is denoted by \({u}\in H^{2}\) with norm \(\| {u}\|_{H^{2}}:=(\sum_{k=1}^{3}\|u_{k}\| _{H^{2}}^{2})^{1/2}\). Finally, we define some functionals:
Next, we introduce our main result.
Theorem 2.1
Let Ω be a horizontally periodic domain with finite height, ω be an arbitrary real number, and \(m_{3}\neq0\). Then there is a sufficiently small constant \(\delta>0\), such that for any \((\eta_{0}, u_{0})\in H^{7}\times H^{6}\) satisfying the following conditions:
-
(1)
\(\| \eta_{0} \|_{7}^{2}+\|u_{0}\|_{6}^{2} \leq\delta\);
-
(2)
\(\zeta_{0}:=y+\eta_{0}\) satisfies (2.1);
-
(3)
\((\eta_{0}, u_{0})\) satisfies necessary compatibility conditions (i.e., \(\partial_{t}^{j} u(x,0)|_{\partial\Omega }=0\) for \(j=1\) and 2),
there exists a unique global solution \((\eta,u)\in C^{0}(\mathbb {R}^{+}_{0},H^{7}\times H^{6})\) to the transformed MHD problem (2.15)-(2.16) with an associated perturbation pressure q. Moreover, \((\eta,u,q)\) enjoys the following stability estimate:
Here the positive constants δ and c depend on the domain Ω and other known physical parameters \(\lambda_{0}\), M̄, μ and ρ.
Remark 2.1
Exploiting the inverse transformation of ζ, we can easily deduce from Theorem 2.1 the global well-posedness of the original MHD problem (1.2)-(1.4). More precisely, there is a sufficiently small constant \(\delta_{0}> 0\), such that, for any \(( v_{0},N_{0})\in H^{6}\) satisfying the following conditions:
-
(1)
there exists an invertible mapping \(\zeta_{0}:=\zeta _{0}(x):\Omega\to\Omega\), such that (2.1) holds, where \(\mathcal{A}^{T}_{0}=(\nabla\zeta_{0})^{-1}\);
-
(2)
\((\bar{M}+N_{0})(\zeta_{0})=\bar{M}\cdot\nabla\zeta_{0} \);
-
(3)
\(\| \zeta_{0} -x\|_{7}^{2}+\|v_{0}\|_{6}^{2} \leq\delta_{0}\);
-
(4)
the initial data \(\varrho_{0}\), \(v_{0}\), \(N_{0}\) satisfy necessary compatibility conditions (i.e., \(\partial _{t}^{j}v(x,0)|_{\partial\Omega}=0\) for \(j=1\) and 2),
there exists a unique global solution \(( v, N)\in C^{0}(\mathbb{R}^{+}_{0},H^{6})\) to the original MHD problem (1.2)-(1.4) with an associated perturbation pressure q̃. Moreover, \((v,N,\tilde {q})\) enjoys the following stability estimate:
Now we briefly describe the basic idea in the proof of Theorem 2.1. By the standard energy method, there are two functionals \(\tilde {\mathcal{E}}^{L}\) and Q of \((\eta,u)\) satisfying the lower-order energy inequality (see Proposition 3.1)
where the functional \(\tilde{\mathcal{E}}^{L}\) is equivalent to \(\mathcal{E}^{L}\). Unfortunately, we can not close the energy estimates only based on (2.19), since \(\mathcal{Q}\) can not be controlled by \(\tilde{\mathcal{E}}^{L}\). However, we observe that the structure of the energy inequality above is very similar to that of the surface wave problem [15], for which Guo and Tice developed a two-tier energy method to overcome this difficulty. In the spirit of the two-tier energy method, we look after a higher-order energy inequality to match the lower-order energy inequality (2.19). Since \(\tilde{\mathcal{E}}^{L}\) contains \(\|\eta\|_{3}\), we find that the higher-order energy at least includes \(\|\eta\|_{6}\). Thus, similar to (2.19), we establish the higher-order energy inequality (see Proposition 4.1)
where the functional \(\tilde{\mathcal{E}}^{H}\) is equivalent to \(\mathcal{E}^{H}\). Moreover, the highest-order norm \(\|(\eta,u)\|_{7}^{2}\) enjoys the highest-order energy inequality
where the norm \(\|\eta\|_{7,*} \) is equivalent to \(\|\eta\|_{7} \). In the derivation of the a priori estimates, we have \(\mathcal {Q}\lesssim\mathcal{E}^{H}\), and thus (2.19) implies (see Proposition 3.1)
Consequently, by the two-tier energy method, we can deduce the global-in-time stability estimate (2.17) based on (2.20)-(2.22).
The rest of the sections are mainly devoted to the proof of Theorem 2.1. In Section 3, we first derive the lower-order energy inequality (2.22) for the transformed MHD problem. Then in Section 4 we derive the higher-order energy inequality (2.20) and the highest-order energy inequality (2.21). Based on these three energy inequalities, we prove Theorem 2.1 by adapting the two-tier energy method in Section 5.
3 Lower-order energy inequality
In this section, we start to derive the lower-order energy inequality in the a priori estimates for the transformed MHD problem. To this end, let \((\eta,u)\) be a solution of the transformed MHD problem with perturbed pressure q, such that
where δ is sufficiently small. It should be noted that the smallness depends on the known physical parameters in (1.4), and will be repeatedly used in what follows. Moreover, we assume that the solution \((\eta,{u},q)\) possesses proper regularity, so that the formal calculation makes sense. We remind the reader that in the calculations, we shall repeatedly use Cauchy-Schwarz’s inequality, Hölder’s inequality, the embedding inequalities (see [16], 4.12 Theorem)
and the interpolation inequalities (see [16], 5.2 Theorem)
for any \(0\leq j< i\) and any constant \(\epsilon>0\), where the constant \(C_{\epsilon}\) depends on the domain Ω and ϵ. In addition, we shall also frequently use the following two estimates:
and
where \(\kappa(j)=j\) for \(j\geq2\) and \(\kappa(j)=2\) for \(j\leq1\). We also introduce the following inequality, see (3.4) in [17]:
Before deriving the lower-order energy inequality defined on \((0,T]\), we first give some preliminary estimates, temporal derivative estimates, horizontal spatial estimates and Stokes estimates in sequence.
3.1 Preliminary estimates
In this subsection, we derive some preliminary estimates for \(\mathcal {A}\). To begin with, we give an expression of \(\mathcal{A}\). Using (2.15)1, we have
where \(A^{*}_{ij}\) is the algebraic complement minor of the \((i,j)\)th entry in the matrix \(I+\nabla\eta\). Recalling the definition of \(\mathcal{A}\), we see that
Inserting this relation into (3.7), we get
which, together with initial condition \(\det(I+\nabla\eta_{0})=1\), implies
Thus we obtain
Now, exploiting (2.15)1, (3.1), (3.4) and (3.8), we easily see that
Similarly, we further deduce that
Letting \(\tilde{\mathcal{A}}:=\mathcal{A}-I\), we next bound \(\tilde {\mathcal{A}}\). To this end, we assume that δ is so small that the following expansion holds:
whence
Using (3.1), (3.4) and (3.9), we find that
3.2 Temporal derivative estimates
In this subsection, we try to control temporal derivatives. For this purpose, we apply \(\partial_{t}^{j}\) to (2.15) to get
where
and we have used relation (2.7) in (3.11). Then from (3.10) we show the following estimates:
Lemma 3.1
It holds that for \(j=0\) and 1,
Proof
(1) We only prove (3.12) for \(j=1\), since the derivation of the case \(j=0\) is similar. Multiplying (3.10)2 with \(j=1\) by \(u_{t}\), integrating (by parts) the resulting equality over Ω, and using (3.10)1, we get
The last three integrals \(I_{2}^{L},\ldots,I^{L}_{4}\) can be estimated as follows:
where we have used (3.10)3 in (3.15). Consequently, the desired estimate (3.12) follows from (3.15)-(3.17).
(2) Now we turn to the proof of (3.13). Multiplying (3.10)2 with \(j=1\) by \(u_{tt}\), integrating (by parts) the resulting equality over Ω, and using (3.10)1, we conclude
On the other hand, the five integrals \(J^{H}_{1},\ldots,J^{H}_{5}\) can be bounded as follows:
Thus, substituting (3.19)-(3.23) into (3.18) and using Cauchy-Schwarz’s inequality, we immediately get (3.13). □
3.3 Horizontal spatial estimates
In this subsection, we establish the estimates of horizontal spatial derivatives. For this purpose, we rewrite (2.15) as the following non-homogeneous linear form:
where
Then we have the following estimate on horizontal spatial derivatives of η.
Lemma 3.2
It holds that
Proof
We only show the case \(j=3\); the remaining three cases can be verified similarly. Applying \(\partial_{h}^{3}\) to (3.24)2, multiplying the resulting equation by \(\partial^{3}_{h}\eta\), and then using (3.24)1, we get
If we integrate (by parts) the above identity over Ω, we obtain
where the first three integrals on the right-hand side of the first equality in (3.25) are denoted by \(K^{L}_{1}\), \(K^{L}_{2}\) and \(K^{L}_{3}\), respectively.
We have the boundedness
Noting that
we find that
Next we estimate the third integral \(K_{3}^{L}\). To start with, we analyze the property of divη. Since
we have by Sarrus’ rule
Multiplying the above identity by a smooth test function ϕ, and then integrating (by parts) the resulting identity over Ω, we derive that
where
This means that
Thus, it follows immediately that
Now, substituting (3.26), (3.27) and (3.28) into (3.25), we immediately obtain the desired estimate for the case \(j=3\). □
Similarly, we also establish the following estimates of horizontal spatial derivatives of u:
Lemma 3.3
We have
Proof
We only prove the case \(j=3\); the remaining two cases can be shown similarly. Applying \(\partial_{h}^{3}\) to (3.24)2, and multiplying the resulting equality by \(\partial^{3}_{h} u\), we make use of (3.24)1 to get
Integrating (by parts) the above identity over Ω, we have
On the other hand, similarly to (3.27) and (3.28), the two integrals \(M^{L}_{1}\) and \(M^{L}_{2}\) can be estimated as follows:
where we have used (3.24)3 in (3.31). Consequently, putting the above two estimates into (3.29), we obtain Lemma 3.3 for the case \(j=3\). □
3.4 Stokes problem and stability condition
In this subsection, we use the regularity theory of the Stokes problem to derive more estimates of \((\eta,u)\). To this end, we rewrite (3.24)2 and (3.24)3 as the following Stokes problem:
coupled with boundary condition
where \(\omega=\lambda_{0} m^{2}_{3} \eta+\mu u\).
Now, applying \(\partial^{k}_{h}\) to (3.32) and (3.33), we get
Then we apply the classical regularity theory to the Stokes problem as in [18], Proposition 2.3, to deduce that
where
In addition, applying \(\partial_{t}^{k}\) to (3.24)2-(3.24)3, we see that
Hence, we apply again the classical regularity theory to the Stokes problem to get
where \(S_{k,i}^{u}: =\|\partial_{t}^{k} ( N^{h}_{u}, N^{h}_{q})\|_{i-2k}^{2}+\| \partial_{t}^{k} D^{h}_{u}\|_{i-2k+1}^{2}\). As a result of (3.34) and (3.35), one has the following estimates.
Lemma 3.4
We have
where \(E^{L}:=\mathcal{E}^{L}-\|\nabla\eta\|_{3,0}^{2}\) and \(\|\eta\| _{3,*}\) is equivalent to \(\|\eta\|_{3}\).
Proof
Noting that, by virtue of (2.15)1,
we deduce from (3.34) that
In particular, we take \((i,k)=(1,0)\) and \((i,k)=(1,1)\) to get
and
On the other hand, it is easy to show that
Thus we immediately obtain (3.36) from (3.40)-(3.42).
Now we turn to the derivation of (3.37). In view of (3.35) with \((i,k)=(1,0)\), we have
On the other hand, we can use (3.42) to infer that
Hence, (3.37) follows from the above two estimates.
Finally, to show (3.38), we take \((i,k)=(2,1)\) in (3.35) to deduce that
Keeping in mind that
we get (3.38) from the above two estimates. □
3.5 Lower-order energy inequality
Now, we are able to build the lower-order energy inequality. In what follows, the letters \(c_{i}^{L}\) and \(i=1,\ldots,7\) will denote generic positive constants which may depend on the domain Ω and some physical parameters in the transformed MHD equations (2.15).
Proposition 3.1
Under the assumption (3.1), if δ is sufficiently small, then there is an energy functional \(\tilde{\mathcal{E}}^{L}\) which is equivalent to \(\mathcal{E}^{L}\), such that
Proof
We choose δ so small that
Then, thanks to (3.5), (3.6) and (3.44), we deduce from (3.12) and Lemmas 3.2-3.3 that there are constants \(c_{1}^{L}\), \(c_{2}^{L}\) and \(\sigma_{0}\geq1\), such that
where \(\sigma_{0}\) depends on the domain and the known physical parameters, and
Utilizing (3.13), (3.36), the interpolation inequality, and the estimate
we find that
Now, multiplying (3.47) by \(c_{2}^{L}/(2c_{5}^{L})\) and adding the resulting inequality to (3.45), we obtain
where \(\tilde{{\mathcal{E}}}^{L}_{2}\) and \(\tilde{\mathcal{D}}^{L} \) are defined by
On the other hand, from (3.38) we get \(\mathcal{D}^{L}\lesssim\tilde{\mathcal{D}}^{L}+\sqrt{\mathcal {E}^{H}}\mathcal {D}^{L}\), which implies \(\mathcal{D}^{L}\lesssim\tilde{\mathcal{D}}^{L}\) for sufficiently small δ. Therefore, (3.48) can be rewritten as follows:
Next we show that \(\mathcal{E}^{L}\) can be controlled by \(\tilde {{\mathcal{E}}}^{L}_{2}\). By virtue of Cauchy-Schwarz’s inequality, (3.5) and (3.44), there is an appropriately large constant σ, depending on ρ, μ and Ω, such that
In view of (3.37), we further have
which implies \({\mathcal{E}}^{L}\lesssim\tilde{\mathcal{E}}^{L}_{2}\) for sufficiently small δ. In addition, obviously \(\tilde{{\mathcal{E}}}^{L}_{2}\lesssim\mathcal{E}^{L}\). Hence, the energy functional \(\tilde{{\mathcal{E}}}^{L}_{2}\) is equivalent to \(\mathcal{E}^{L}\). Finally, letting \(\delta\leq c_{7}^{L} /2 c_{6}^{L}(1+\sigma)\) and noting \(\tilde {\mathcal{E}}^{L} =2\tilde{{\mathcal{E}}}^{L}_{2}/c_{7}^{L}\), we see that (3.49) immediately implies (3.43). Obviously, the energy functional \(\tilde{\mathcal{E}}^{L} \) is still equivalent to \(\mathcal{E}^{L}\). This completes the proof. □
4 Higher-order and highest-order energy inequalities
In this section, we derive the estimates (2.20) and (2.21) for the transformed magnetic RT problem. First we shall establish the higher-order version of Lemmas 3.1-3.4 in the following:
Lemma 4.1
We have
Proof
We only prove the case \(j=3\), and the case \(j=2\) can be shown similarly. Multiplying (3.10)2 with \(j=3\) by \(\partial_{t}^{3}u\), integrating (by parts) the resulting equation over Ω, and using (3.10)1 and (3.10)3, we obtain
On the other hand, the integrals \(I^{H}_{1}\), \(I_{2}\) and \(I^{H}_{3}\) can be estimated as follows:
Inserting the above three inequalities into (4.1), we get the desired conclusion immediately. □
Lemma 4.2
We have
Proof
We only show the case \(k=6\), since the derivation of the case \(k=5\) is similar. Since the derivation involves the norm \(\|\nabla q\|_{5}\), we first estimate \(\|\nabla q\|_{5}\). It follows from (3.35) that
The last two terms on the right-hand side of the above inequality can be bounded as follows:
Hence, if δ is sufficiently small, then one gets from the above two estimates that
Now, applying \(\partial_{h}^{6}\) to (3.24)2, multiplying the resulting equality by \(\partial^{6}_{h}\eta\), we utilize (3.24)1 to have
Integrating (by parts) the above identity over Ω, we obtain
where the first four integrals on the right-hand side are denoted by \(J^{H}_{1},\ldots,J^{H}_{4}\), respectively.
On the other hand, the four integrals \(J^{H}_{1},\ldots,J^{H}_{4}\) can be bounded as follows:
where the interpolation inequality (3.3) has been employed in (4.6) and (4.7). Consequently, putting the above four estimates into (4.3), we obtain Lemma 4.2. □
Lemma 4.3
We have
Proof
We only show the case \(k=6\), since the derivation of the case \(k=5\) is similar. Applying \(\partial_{h}^{6}\) to (3.24)2, multiplying the resulting equality by \(\partial^{6}_{h} u\), we make use of (3.24)1 to have
Integrating (by parts) the above identity over Ω, we get
Analogously to (4.4)-(4.7), the three integrals \(M^{H}_{1}\), \(M^{H}_{2}\) and \(M^{H}_{3}\) can be bounded as follows:
Substituting the above three inequalities into (4.8), we obtain Lemma 4.3. □
Lemma 4.4
The following estimates hold:
where the norm \(\|\eta\|_{6,*}^{2}\) is equivalent to \(\|\eta\|_{6}^{2}\) and \({{E}^{H}}:=\mathcal{E}^{H}-\|\nabla\eta\|_{6,0}^{2}\).
Proof
(1) We begin with the derivation of (4.9). Taking \((i,k)=(4,0)\) in the Stokes estimate (3.35), we have
Noting that
we get
Using the recursion formula (3.35) for \(i=4\) from \(k=1\) to 2, we obtain
On the other hand, \(S_{k,4}^{u}\) can be bounded from above by
Therefore,
which, together with (4.12), yields
In view of the interpolation inequality (3.3), we have
Thus, inserting (4.15) into (4.14), one gets (4.9).
(2) We proceed to prove the estimate (4.10). Exploiting the recursion formula (3.39) for \(i=4\) from \(k=0\) to 4, we see that there are positive constants \(c_{k}\), \(k=0,\ldots,4\), such that
which combined with (4.13) gives
where we have used the fact that \(S_{k ,4}^{\omega}\leq S_{0,4}^{\omega}\) for \(0\leq k\leq4\). Since
and \(E^{H}\leq\mathcal{D}^{H}\), we further infer that
Using the interpolation inequality (3.3), we get (4.10) from (4.16), where \(\|\eta\|_{6,*}^{2}\) equals to \(\sum_{k=0}^{4}c_{k}\|\eta\|_{{k,6-k}}^{2}\) multiplied by some positive constant.
(3) Finally, we derive the estimate (4.11) for higher-order dissipation estimates. We use the recursion formula (3.35) for \(i=5\) from \(k=1\) to 2 to deduce that
Noting that
we obtain (4.11) from (4.17). □
Now we are in a position to build the higher-order and highest-order energy inequalities. In what follows, the letters \(c_{i}^{H}\) and \(i=1,\ldots,6\) will denote generic constants which may depend on the domain Ω and some physical parameters in the transformed MHD equations (2.15).
Proposition 4.1
Under the assumption (3.1), if δ is sufficiently small, then there are two norms \(\tilde{\mathcal{E}}^{H}\) and \(\|\eta\|_{7,*}^{2}\) which are equivalent to \(\mathcal{E}^{H}\) and \(\|\eta\|_{7}^{2}\) respectively, such that
Proof
(1) We first prove (4.18). Similarly to (3.48), we make use of (3.5), (3.6), (3.44) and (3.46) to deduce from Lemmas 4.1-4.3 and (4.10) that there are constants \(c_{1}^{H}\), \(c_{2}^{H}\), \(c_{3}^{H}\) and \(\alpha_{0}\geq1\), such that
where \(\alpha_{0}\) depends on the domain and the known physical parameters, and
Moreover, by (4.20) and Proposition 3.1 we find that
where \(\tilde{\mathcal{E}}^{H}_{2}:=\tilde{\mathcal{E}}^{H}_{1}+ c_{4}^{H}\tilde{\mathcal{E}}^{L}\) and \(\tilde{\mathcal{D}}^{H} := \tilde{\mathcal{D}}^{H}_{1} + c_{5}^{H} {\mathcal{D}}^{L}\). By (4.11), we have \(\mathcal{D}^{H}\lesssim\tilde {\mathcal{D}}^{H}+{\mathcal{E}^{H}}\mathcal{D}^{H}\). So, (4.21) can be rewritten as
Next, we show that \(\mathcal{E}^{H}\) can be controlled by \(\tilde {\mathcal{E}}^{H}_{2}\). Keeping in mind that
we use Cauchy-Schwarz’s inequality, (3.5) and (4.23) to infer that there is an appropriately large constant α, depending on ρ, μ and Ω, such that
Recalling (4.9) and the fact that \(E^{H}\leq\mathcal {E}^{H}\), we have
which implies \({\mathcal{E}}^{H} \lesssim\tilde{\mathcal{E}}^{H}_{2}\) for sufficiently small δ.
On the other hand, \(\tilde{\mathcal{E}}^{H}_{2}\lesssim\mathcal{E}^{H}\) obviously. Hence, the energy functional \(\tilde{\mathcal{E}}^{H}_{2}\) is equivalent to \(\mathcal{E}^{H}\). Finally, letting \(\delta\leq c_{6}^{H} /2 c_{3}^{H}(1+\sigma)\) and denoting \(\tilde{\mathcal{E}}^{H}=2\tilde{\mathcal{E}}_{2}^{H}/c_{6}^{H}\), we see that (4.22) implies (4.18).
(2) We proceed to derive the highest-order estimate (4.19). We start with employing the recursion formula (3.39) with \(i=5\) from \(k=0\) to 4 to find that there are positive constants \(d_{k}\) (\(k=0,1,2\)), such that
On the other hand, by virtue of (4.2),
Thus, we conclude
Hence, (4.19) holds by defining \(\|\eta\|_{7,*}^{2}: =2\sum _{k=0}^{5}d_{k}\|\eta\|_{k,7-k}^{2}\), provided δ is sufficiently small. □
The following lemma will be needed in the next section.
Lemma 4.5
Under the assumption (3.1), if δ is sufficiently small, then
Proof
In view of (4.9), we have
which results in
for sufficiently small δ. Below we show that the \(L^{2}\)-norm of \(u_{t}\) and \(\partial_{t}^{3}u\) can be controlled by \(\sqrt{\mathcal{E}}\).
(1) First, to bound \(u_{t}\), we multiply (3.10)2 with \(j=0\) by \(u_{t}\) and integrate the resulting equation over Ω to obtain
where the last term on the right-hand side can be bounded from above by \(\mathcal{E}\). Hence,
(2) Then we control the term \(u_{tt}\). Multiplying (3.10)2 with \(j=1\) by \(u_{tt}\) and integrating the resulting equation over Ω, we infer that
which yields
Noting that
and
we see that
(3) Finally, we estimate the term \(\partial_{t}^{3}u\). If we multiply (3.10)2 with \(j=2\) by \(\partial_{t}^{3}u\) in \(L^{2}(\Omega)\), we obtain
whence
On the other hand, it is easy to show that
and
Therefore,
Now, we control the term \(\|u_{tt}\|_{2}\) in (4.28). By (3.10)2 with \(j=1\), we deduce that
where the last term on the right-hand side can be bounded by \(\mathcal {E}\). Consequently,
To estimate \(\nabla q_{t}\), we rewrite (3.24)2 as follows:
with boundary condition
where
In view of the standard elliptic regularity estimates [19], we find that
Putting the above inequality into (4.29), we obtain
which, together with the interpolation inequality (3.3), yields
Consequently, from (4.28) and (4.30) it follows that
Next, we control \(\|u_{t}\|_{4}\) in (4.31). Using (2.15)2, we derive that
where, by virtue of (3.24)2, ∇q satisfies
with boundary condition
and
Thus, we can use the elliptic regularity estimate to get
Hence, from (4.32) we get
which, together with the interpolation inequality (3.3), implies that
Inserting (4.33) into (4.31), we conclude
(4) Now we are able to show (4.24). Summing up the estimates (4.26), (4.27), (4.33) and (4.34), we arrive at
5 Proof of Theorem 2.1
This section is devoted to the proof of Theorem 2.1. Roughly speaking, Theorem 2.1 is shown by combining the a priori stability estimate (2.17) and the local well-posedness of the transformed MHD problem. Before we derive the a priori stability estimate (2.17), we begin with estimating the terms \(\mathcal{G}_{1},\ldots,\mathcal{G}_{4}\).
Using (4.19), and recalling the equivalence of \(\|\eta\| _{7,*}^{2}\) and \(\|\eta\|_{7}^{2}\), we deduce that
which yields
Multiplying (4.19) by \((1+t)^{-3/2}\), we get
which implies that
An integration of (4.18) with respect to t gives
Let
From now on, we further assume \(\sqrt{\mathcal{G}_{5}(T)}\leq\delta\), which is a stronger requirement than (3.1). Thus, we make use of (5.2) to find that
which implies
Finally, we show the time decay behavior of \(\mathcal{G}_{4}(t)\), noting that \(\mathcal{E}^{L}\) can be controlled by \(\mathcal{D}^{L}\), except the term \(\|\nabla\eta\|_{3,0}\) in \(\mathcal{E}^{L}\). To deal with \(\|\nabla\eta\|_{3,0}\), we use (3.3) to get
On the other hand, we combine (5.1) with (5.3) to get
Thus,
Putting the above estimate into the lower-order energy inequality (3.43), we obtain
which yields
with \(\mathcal{I}_{0}:=c( \mathcal{E}^{H}(0)+ \|\eta_{0}\|_{7}^{2})\) for some positive constant c. Therefore,
Now we sum up the estimates (5.1)-(5.4) to conclude that
where (4.24) has been also used. Consequently, we have proved the following a priori stability estimate.
Proposition 5.1
Let \((\eta,u)\) be a solution of the transformed MHD problem with an associated perturbation pressure q. Then there is a sufficiently small \(\delta_{1}\), such that \((\eta,u,q)\) enjoys the following a priori stability estimate:
provided that \(\sqrt{\mathcal{G}_{5}(T_{1})}\leq\delta_{1}\) for some \(T_{1}>0\). Here \(C_{1}\geq1\) denotes a constant depending on the domain Ω and other physical parameters in the transformed MHD equations.
In view of the a priori stability estimate in Proposition 5.1 and the following result of local existence of a small solution to the transformed MHD problem, we immediately obtain Theorem 2.1.
Proposition 5.2
There is a sufficiently small \(\delta_{2}\), such that for any given initial data \((\eta_{0},u_{0})\in H^{7}\times H^{6}\) satisfying
and the compatibility conditions (i.e., \(\partial_{t}^{j} u(x,0)|_{\partial\Omega}=0\), \(j=1,2\)), there exist a \(T_{2}:=T_{2}(\delta_{2})>0\) which depends on \(\delta_{2}\), the domain Ω and other known physical parameters, and a unique classical solution \((\eta, u)\in C^{0}([0,T_{2}],H^{7}\times H^{6} )\) to the transformed MHD problem (2.15), (2.16) with an associated perturbation pressure q. Moreover, \(\partial_{t}^{i}u\in C^{0}([0,T_{2}],H^{6-2i})\) for \(1\leq i\leq3\), \(q\in C^{0}([0,T_{2}],H^{5})\), \(\mathcal{E}^{H}(0)\lesssim\|\eta_{0}\|_{7}^{2}+\|u_{0}\| _{6}^{2}\), and
and \(\mathcal{G}(t)\) is continuous on \([0,T_{2}]\).
Proof
The transformed MHD problem is very similar to the surface wave problem (1.4) in [20]. Moreover, the current problem is indeed simpler than the surface wave problem due to the non-slip boundary condition \(u|_{\partial\Omega}=0\). Using the standard method in [20], one can easily establish Proposition 5.2, hence we omit its proof here. In addition, the continuity, such as \((\eta, u,q)\in C^{0}([0,T],H^{7}\times H^{6}\times H^{6})\), \(\mathcal{G}(t)\) and so on, can be verified by using the regularity of \((\eta,u,q)\), the transformed MHD equations and a standard regularized method. □
6 Conclusion
We have proved the existence of a unique time-decay solution to the initial-boundary problem (1.2)-(1.4) of rotating MHD fluids in Lagrangian coordinates, which, together with the inverse transformation of coordinates, implies the existence of a time-decay solution to the original initial-boundary problem (1.2)-(1.4) with proper initial data in \(H^{7}(\Omega)\). Our result also holds for the case \(\omega=0\) (i.e., the absence of rotation), thus it improves Tan and Wang’s result in [11], in which the sufficiently small initial data at least belongs to \(H^{16}(\Omega)\). Hence our result reveals that rotation does not affect the existence of solutions of rotating MHD fluids. We mention that the phenomenon of rotating MHD fluids widely exists in nature, so our result has potential applications. In addition, based on Theorem 2.1, we will further study the Rayleigh-Taylor problem of rotating MHD fluids in the future; please refer to [21–28] for relevant results on the Rayleigh-Taylor problem.
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The current work is partially supported by NSFC under Grant Nos. 11501116 and 11671086, and the Education Department of Fujian Province under Grant No. JA15063.
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Wang, W., Zhao, Y. Time-decay solutions of the initial-boundary value problem of rotating magnetohydrodynamic fluids. Bound Value Probl 2017, 114 (2017). https://doi.org/10.1186/s13661-017-0845-2
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DOI: https://doi.org/10.1186/s13661-017-0845-2