In this paper, the problem of continuous data assimilation of three dimensional primitive equations with magnetic field in thin domain is studied. We establish the well-posedness of the assimilation system and prove that the -strong solution of the assimilation system converges exponentially to the reference solution in the sense of as . We also study the sensitivity analysis of the assimilation system and prove that a sequence of solutions of the difference quotient equation converge to the unique solution of the formal sensitivity equation.
Continuous data assimilation is one of the classical data assimilation methods, which is to apply the time continuous and space discrete observation data to the assimilation process. In 2014, based on the background of 2D Navier–Stokes equations, Azouani, Olson and Titi [3,4] gave a new algorithm of continuous data assimilation inspired by ideas from control theory. In this new algorithm, they introduced a feedback control term into the original system. [29–31,38] also studied different systems separately by using finite parameter feedback control algorithms. Let us consider the reference system
Solutions of the system are called the reference solutions. A main difficulty in obtaining the reference solution is that we largely do not know the initial data . Under the assumption that there are no observation errors and the observation grid resolution is h, a linear interpolation operator and a relaxation (nudging) parameter β were introduced into the system in [3,4]. Thus, we obtain the corresponding assimilation system
where is an arbitrary initial data. The solution of assimilation system (1.3)–(1.4) is called the assimilation solution. The assimilation system (1.3)–(1.4) obviously eliminates the need for complete initial data and incorporates incoming data into simulations as well. [3,4] showed that the assimilation solution converges exponentially to the reference solution as time t approaches infinity, under certain conditions on the spatial scale h and the relaxation parameter β. The continuous data assimilation method introduced in [3,4] is also called the data assimilation for simplicity. After the work in [3,4], the data assimilation method was also used to study the data assimilation problem for the different systems [1,2,5,7,9,13,22,24,25,28,39,40].
Both the primitive equations and the Navier–Stokes equations are widely studied in various aspects ([15,33,34]). Cao and Titi [15] achieved an important result on the global well-posedness of strong solutions for the three-dimensional primitive equations in a general cylindrical domain. Several works addressed the continuous data assimilation problem for the 2D Navier–Stokes equations [6,8,10,12,23,26,27,35]. In 2021, Carlson and Larios [16,17] performed parameter recovery for the 2D Navier–Stokes equations and further examined the sensitivity of their recovery algorithm by proving that a sequence of difference quotients converges to a unique solution of the sensitivity equations. Li and Titi [36] established the small aspect ratio limit of the Navier–Stokes equations to the primitive equations in 2019. In [32,41], Korn and Pei investigated data assimilation for the primitive equations of ocean dynamics, respectively.
The magnetohydrodynamic (MHD) equations, which combine the Navier–Stokes equations of fluid dynamics and Maxwell’s equations of electromagnetism, have much richer structure than the Navier–Stokes equations. Duvaut and Lions [21] established the existence and uniqueness results for weak and strong solutions of the 2D MHD equations. Biswas, et al in [11] used Elsässer variables to investigate the exponential convergence of continuous data assimilation for the 2D MHD equations in 2021. In [18], Du and Li analyzed the 3D incompressible MHD equations by using the scaling technique to derive the primitive equations with magnetic field (PEM) on thin 3D domains, and proved the global existence and uniqueness of strong solutions for this PEM without any small assumption on the initial data (see also [20] for the model with only horizontal viscosity). In [19], they further proved that the global strong solutions of the scaled MHD equations strong converge to the global strong solutions of the PEM.
In this paper, we continue the work of Du and Li [18,19] to further investigate the continuous data assimilation for three-dimensional PEM on the 3D thin domains. First, we obtain some uniform estimates to ensure the well-posedness of both the reference system and the assimilation system. Here, to obtain -uniform estimates, we use the governing equation for Elsässer variables (see [11]) to ensure that the nonlinear terms on the right-hand side of the energy inequality are absorbed by the left-hand side. For the assimilation system, we further estimate the assimilation terms using the properties of the linear interpolation operator and the incompressibility conditions. Then, we apply the uniformly bounded estimates obtained previously to show that the solution of the assimilation system exponentially converges to that of the reference system. Lastly, we study the sensitivity analysis problem for the assimilation system and establish the well-posedness of the formal sensitivity equations by proving that a sequence of solutions of the difference quotients equations converge to a unique solution of the formal sensitivity equations.
We organize the rest of this paper as follows. In Section 2, we give the mathematical formulation of primitive equations with magnetic data assimilation system. In Section 3, we first obtain the well-posedness of the PEM and its assimilation system, and then we get the convergence relation between the reference solution and the assimilation solution. In Section 4, we study the sensitivity analysis problem for the assimilation equations.
Mathmatical model
We first consider the incompressible three-dimensional MHD equations in , where is a small parameter, and for two positive constants and of order with respect to ε. We notice the anisotropic MHD system
where
and u, b are the horizontal velocity and magnetic field, respectively, , are vertical velocity and magnetic field. The pressure p is scalar function. Throughout this paper, we set , , and and as their three-dimensional counterparts, respectively. Without loss of generality, we assume that , , since μ, κ are of order and , are of order, respectively, so that ν, σ are of order as well. Next, we consider the above anisotropic viscosity and magnetic diffusivity scaling in the horizontal and vertical directions, so that the MHD equations can converge to PEM, as the aspect ratio ε tends to zero.
We perform the scaling transformation on the MHD equations so that the resulting system is defined on a fixed domain that does not depend on ε, as in the case considered in [18,36]. To this end, we introduce the new unknowns
for any , and for any . Then, , and satisfy the scaled MHD equations (SMHD)
defined in the fixed domain Ω. In addition, we set the SMHD system with the initial value conditions
and periodic boundary conditions
where and are given. We can notice that the periods in x and y are and , respectively, while that in z is 2. We suppose in addition that , , , , are even, odd, odd, even and odd with respect to z, respectively. And we always assume that the initial horizontal velocity and magnetic field , are periodic in x, y, z, and are even in z.
By taking the limit as in SMHD, we obtain the incompressible three-dimensional PEM system
where
We recall that the periodic initial-boundary value problem for the SMHD equations, and it is clear that we should impose the same boundary conditions and symmetry conditions on the corresponding limiting PEM system. However, we only need to impose the initial conditions on the horizontal components of the velocity field and the magnetic field. Since and are odd in z, we have . Then, , can be determined uniquely by the incompressibility conditions, namely,
Similarly, can also be determined uniquely by u, b via the incompressibility conditions as
The reference system (2.1)–(2.4) corresponding to the assimilation system is
where the initial-boundary value is the same as the reference system. We suppose that the linear interpolation operator satisfies that
For simplicity, we denote
and
A useful property of B is , which was verified in [15]. Throughout this paper, we use to express the standard norm and to express the Sobolev norm.
Recall the definition of the strong solutions as follows.
Let . We say that is the strong solution of system (2.1)–(2.4) if it satisfies system (2.1)–(2.4) in the weak sense and also we have
Note that the strong solution definition of assimilation system (2.9)–(2.12) is similar to that of reference system (2.1)–(2.4).
The proofs in this paper rely on the following lemmas.
([14, Lemma 2.1]).
The inequalitiesandhold true, for any f, g, h such that the right-hand sides make sense and are finite, where C is a positive constant depending only onand.
Let, ϕ and ψ be periodic functions with basic domain Ω. Suppose that, within Ω,, and,and. Denote bythe horizontal components of the function φ. Then, we have the estimatewhere C is a positive constant depending only onand.
Let V, H,be three Hilbert spaces such that, whereandare the dual spaces of H and V respectively. Supposeand. Then u is almost everywhere equal to a function continuous frominto H.
Well-posedness and exponential convergence
In this section, we investigate the well-posedness of the PEM and its assimilation system, and consider the convergence relationship between the assimilation solution and the reference solution. To obtain these results, we first derive some uniform estimates. For simplicity, let’s set and which can guarantee that the left-hand side of the energy inequality absorbs the nonlinear terms on the right-hand side.
To get these results, we first derive some consistent estimates. For simplicity, let’s set and . This ensures that the energy inequality on the left absorbs the nonlinear term on the right.
-Uniform estimates for u, b, ,
Taking the inner product of equation (2.1) and (2.3) with u and b respectively, we have
Thanks to , and using the Poincaré inequality, we can infer that
so we can get
where . Using the Gronwall inequality, we get
so u and b are uniformly bounded. Integrating in time 0 to t for (3.1), we can get that there exists such that
for . For convenience, is used for boundedness throughout the rest of this paper. Integrating in time t to for (3.1), we can get the uniform boundedness of the time average of and , i.e.,
Thus, we get
Similarly, taking the inner product of assimilation equation (2.9) and (2.11) with and respectively, we have
Because of (2.13) and (2.14), using the Hölder and Young inequalities, we can get
where , . Now choosing h small enough so that
and by the Poincaré inequality, we get
So by the Gronwall inequality and the estimate (3.2), we get that and are uniformly bounded. Similarly, by the estimate (3.3), we can obtain
-Uniform estimates for u, b, ,
Adding and subtracting the equations (2.1) and (2.3) respectively, we obtain
Letting , , we get
For simplicity, we assume , . Taking the inner product of equations (3.5) and (3.6) with and respectively, we obtain
Using Lemma 2.1, we infer that
From [18, Proposition 3.3], we know that
Because of the Poincaré and Young inequalities, we get
Similarly, we can get
Combining all the above estimates, we obtain
Considering the Gronwall inequality and estimate (3.2), we can get A and are uniformly bounded. Integrating in time t to and by (3.3), we can get
Similarly, adding and subtracting the equations (2.9) and (2.11) respectively, we obtain
We also take the inner product of equations above with and , respectively, and get
In order to obtain , we apply the operator to equation (3.10) and obtain
Because of , , and from (2.7)–(2.8), we get
By the Hölder inequality, we have
Similarly, we can get
So by the linear of and using the divergence free conditions, we can get the key estimates
Thus, we have
Similarly, for (3.11), we can get
So we can get
where
By the Hölder and Young inequalities, we get
Similarly, we can get
Thanks to the above estimates, using the Gronwall inequality and the estimates (3.3)–(3.4), we have
-Uniform estimates for , , ,
In order to obtain the -uniform bounds on u and b, we first need to get -uniform estimates for , . Taking the inner product of equation (2.1) and (2.3) with and respectively, we can get
Using the method in [18, Proposition 3.4], we can get
Considering the Gronwall inequality and (3.9), we get that and are uniformly bounded. Integrating in time t to , we can get the uniform boundedness of the time average of and . Thus
Similarly, in order to obtain the -uniform bounds on and , we first need to get -uniform estimates for , . Taking the inner product of equation (2.9) and (2.11) with and respectively, by Poincaré’s inequality, we get the key estimate
and
So we can get
Using the Gronwall inequality, we get that and are uniformly bounded. Integrating in time t to , we can get the uniform boundedness of the time average of and . Thus
-Uniform estimates for u, b, ,
Taking the inner product of equation (2.1) and (2.3) with and respectively, we have
where
Using Lemma 2.1, the Young and Poincaré inequalities, we infer that
Because of , and from (2.7), we get
So we can obtain
Therefore, we have
Similarly, we can get
Thus
Considering Gronwall’s inequality, we get that u and b are uniformly bounded. Integrating in time t to , we can get the uniform boundedness of the time average of and . Thus
Similarly, taking the inner product of equation (2.9) and (2.11) with and respectively, we have
We can get the key estimates
and
By the above estimates, we have
Using Gronwall’s inequality, we get that and are uniformly bounded. Integrating in time t to , we can get the uniform boundedness of the time average of and . Thus
-Uniform estimates for u, b, ,
Taking the inner product of equation (2.1) and (2.3) with and respectively, we have
where
Using Lemma 2.1, the Poincaré and Young inequalities, we infer that
Therefore, we have
Similarly, we can get
Thus
Using Gronwall’s inequality, we get that u and b are uniformly bounded. Integrating in time t to , we can get the uniform boundedness of the time average of and . Thus
Similarly, taking the inner product of equation (2.9) and (2.11) with and respectively, we have
We can get the key estimates
and
By the above estimates, we have
Using the uniformly bounded estimates obtained previously, we follow similar arguments as in [18, Proposition 3.6] to show that and are also uniformly bounded. Hence, following the argument in [43], we can get the existence of a global attractor for both the PEM and its assimilation system. Moreover, we derive the following two propositions.
Let. Then system (
2.1
)–(
2.4
) has a unique strong solution satisfying thatMoreover, the semigroup generated by problem (
2.1
)–(
2.4
) has a global attractor.
Letand. Then the assimilation system (
2.9
)–(
2.12
) has a unique strong solution satisfying thatMoreover, the semigroup generated by problem (
2.9
)–(
2.12
) has a global attractor.
Next, we consider the convergence relation between the assimilation solution and the reference solution.
Letandbe the unique solution of the PEM system (
2.1
)–(
2.4
) and its assimilation system (
2.9
)–(
2.12
), respectively. Suppose that,satisfyandthenconverges toexponentially inas. Here,,are constants defined in (
3.23
)–(
3.25
) below.
Set
Subtracting the assimilation equations from the PEM equations, we have
Taking the inner product of equation (3.19) and (3.21) with and respectively, we have
Using Lemma 2.1, we infer that
Using the Young inequality, we infer that
Thus
Using Lemma 2.1 and the Young inequality, we infer that
Using the Young inequality, we infer that
Thus
Using Lemma 2.1 and the Young inequality, we infer that
Using the Young inequality, we infer that
Thus
Using Lemma 2.1 and the Young inequality, we infer that
Using the Young inequality, we infer that
Thus
Using Lemma 2.1 and the Young inequality, we infer that
Using the Young inequality, we infer that
Thus
Using Lemma 2.1, we infer that
Using the Young inequality, we infer that
Thus
We can also get
Thus
where
So if , satisfy
and
then the Gronwall inequality implies that converges to exponentially in . □
Sensitivity analysis
In this section, we consider the sensitivity analysis problem for the assimilation system (2.9)–(2.12), by proving that a sequence of solutions of difference quotients equations converges to a unique solution of the formal sensitivity equations. We only analyze the horizontal viscosity and magnetic diffusivity, and the vertical direction is similar. To simplify the notation, we set , . We first present the sensitivity equations and the difference quotients equations, respectively.
By taking the derivative of the assimilation system (2.9)–(2.12) with respect to μ, we can get the sensitivity equations with respect to the horizontal direction:
where , , , , .
The difference quotients equations is obtained as the following by taking the solutions and of the assimilation system and respectively:
where , , .
Next, we consider the well-posedness of the sensitivity equations and the difference quotients equations, respectively.
Suppose thatandare two strong solutions of the assimilation system (
2.9
)–(
2.12
) with viscositiesand, respectively. Then, the difference quotients system (
4.5
)–(
4.8
) has a unique solution.
It is estimated from uniform boundedness that the difference quotients equation has a strong solution. Next, we prove the unique.
Let and be two solutions of the difference quotients system, and , . We have
Taking the inner product of equation with and respectively, we can get
Using the Hölder and Young inequality, we infer that
Using Lemma 2.3 and Young’s inequality, we infer that
and
Combining all the above estimates, we obtain
then by Gronwall’s inequality, the difference quotients system has a unique solution. □
Letandbesolutions of the assimilation system (
2.9
)–(
2.12
) in μ and, andbestrong solutions of the difference quotients equation (
4.5
)–(
4.8
) with viscosity. If,, then there is a subsequence ofinthat converges to the unique strong solutionof the sensitive system (
4.1
)–(
4.4
).
Let be given. Let be large enough, which implies .
Considering these estimates for strong solutions , we can easy to verify that
where is a decreasing function respect to . So we can get that is uniformly bounded in respect to n. Then there exists a subsequence such that . Similarly available is also uniformly bounded in respect to n. Therefore, there exists a subsequence such that
So is a strong solution to assilation systems. Due to the uniqueness of solutions, we know that .
For difference quotients equation, we take , . is the corresponding strong solution to the difference quotients equation with viscosity .
Similar to the estimate of Proposition 4.1, we can get that is uniformly bounded in respect to n. Then there exists a subsequence such that
Similarly, is also uniformly bounded in respect to n. Combining the Aubin Lemma, we deduce that in . By Lemma 2.4, we get . □
Footnotes
Acknowledgements
This work was supported by the National Natural Science Foundation of China (No. 12271261), the Qing Lan Project of Jiangsu Province and Postgraduate Research and Practice Innovation Program of Jiangsu Province (Grant No. KYCX22_1125).
Conflict of interest
This work does not have any conflicts of interest.
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