Small-time global approximate controllability for incompressible MHD with coupled Navier slip boundary conditions
Abstract
We study the small-time global approximate controllability for incompressible magnetohydrodynamic (MHD) flows in smoothly bounded two- or three-dimensional domains. The controls act on arbitrary nonempty open portions of each connected boundary component, while linearly coupled Navier slip-with-friction conditions are imposed along the uncontrolled parts of the boundary. Some choices for the friction coefficients give rise to interacting velocity and magnetic field boundary layers. We obtain sufficient dissipation properties of these layers by a detailed analysis of the corresponding asymptotic expansions. For certain friction coefficients, or if the obtained controls are not compatible with the induction equation, an additional pressure-like term appears. We show that such a term does not exist for problems defined in planar simply-connected domains and various choices of Navier slip-with-friction boundary conditions.
Keywords and phrases: magnetohydrodynamics; global approximate controllability; Navier slip-with-friction boundary conditions; boundary layers
AMS Mathematical Subject classification (2020): 93B05; 93C20; 35Q35; 76D55
1 Introduction
Let be a bounded domain of dimension with being smooth and denoting the outward unit normal vector to along . The goal of this article is to steer incompressible magnetohydrodynamic (MHD) flows from a prescribed initial state approximately towards a desired terminal state, without placing restrictions on the distance between these states or on the control time. This is accomplished by acting on the system via boundary controls supported in a possibly small subset , the interior of which is assumed to intersect non-trivially with all connected components of .
When it comes to nonlinear evolution equations with controls localized in an arbitrary open subset of the boundary or an interior sub-domain, establishing the global approximate controllability usually constitutes a challenging task and few methods for tackling such questions are available. In the context of fluid dynamics, but not limited to, one successful approach is known as the return method, which has first been introduced by Coron in [Coron1992] for the stabilization of certain mechanical systems and shall also be employed here. A comprehensive introduction to the return method and its applications to nonlinear partial differential equations may be found in [Coron2007, Part 2, Chapter 6]. In contrast to the Navier–Stokes equations, for which the global approximate controllability has been actively investigated in the past, nothing seems to be known regarding the global approximate controllability of viscous MHD in the presence of physical boundaries. This article therefore aims to initiate further research in this direction. A broader motivation of this topic is provided by a well-known open problem due to J.-L. Lions, asking for the global approximate controllability of the Navier–Stokes equations with the no-slip condition (cf. [LionsJL1991] and also [CoronMarbachSueur2020, CoronMarbachSueurZhang2019] for recent progress).
In this article, we focus on incompressible flows of viscosity and resistivity , for which the velocity , the magnetic field , and the total pressure are described, until a given terminal time , as a solution to the initial boundary value problem
| (1.1) |
while the underlying state space, for both the velocity and the magnetic field, is taken as
In (1.1), the vector fields and represent the given initial data. The parameter quantifies the magnetic permeability. As explained in 1.7 below, the controls are sought in an implicit form and, therefore, no boundary conditions are prescribed along the controlled boundary . Before introducing the general boundary operators and acting at , it is emphasized that they particularly include all boundary conditions of the form
| (1.2) |
with symmetric and denoting the tangential part.
Remark 1.1.
The common three-dimensional notations for the cross product and curl operator are employed whenever is possible. If considering the case , one has to replace by , while the curl of a scalar function refers to . Moreover, the cross product in two-dimensions becomes . Consequentially, for planar configurations, some objects denoted here as vectors might be scalars and means .
The Navier slip-with-friction boundary conditions.
Let represent any smoothly bounded domain with outward unit normal field . The tangent space of at a point is denoted by . Given any , we introduce the Weingarten map (or shape operator)
Then, for , and friction coefficient matrices
| (1.3) |
the linearly coupled Navier slip-with-friction operators in (1.1) are defined as
| (1.4) |
where the tangential part and the symmetrized gradient are respectively denoted by
The Weingarten map is smooth, cf. [CoronMarbachSueur2020, Lemma 1] and [ClopeauMikelicRobert1998, GieKelliher2012], and when is tangential to one has the relation
| (1.5) |
As a consequence of (1.5), the boundary conditions prescribed in (1.1) along are equivalent to
| (1.6) | |||||
Our main motivation is to treat simply-connected domains and the boundary conditions (1.2). This already constitutes a general setup, which has not been studied in terms of controllability but recently attracted increased attention in view of inviscid limit problems. However, the constructions of the controls naturally extend, at least in parts, also to the case of more general domains and allow the prescription of boundary conditions of the form (1.6). Attention shall be paid to situations where or holds, since such configurations lead to interesting challenges that are not fully resolved here.
The Navier slip-with-friction boundary conditions, as already proposed by Navier [Navier1823] two centuries ago, are relevant to a range of applications, thus have been studied in the context of the Navier–Stokes equations from various points of view. For instance, in the absence of magnetic fields, inviscid limit problems are treated in [IftimieSueur2011, ClopeauMikelicRobert1998, Kelliher2006, XiaoXin2013], regularity questions are investigated in [AmrouchePenelSeloula2013, AlBabaAmroucheEscobedo2017, Shibata2007, Shimada2007, AlBaba2019] and controllability problems are tackled in [CoronMarbachSueur2020, Guerrero2006, LionsZuazua1998, Coron1996]. Concerning the situation of incompressible viscous MHD, several singular limit problems involving uncoupled Navier slip-with-friction boundary conditions are addressed in [GuoWang2016, XiaoXinWu2009, MengWang2016]; comparing with these references, the here employed boundary conditions are more general in that the shear stresses of the velocity and the magnetic field at the boundary are linearly coupled with tangential velocity and magnetic field contributions. While (1.6) includes the classical Navier slip condition for the velocity, it can capture also more complex interactions in the presence of magnetic fields.
From the global approximate controllability point of view, several difficulties appear however when the magnetic shear stress is coupled with the tangential velocity: a magnetic field boundary layer potentially enters the analysis of Section 3. This in turn challenges the construction of magnetic field boundary controls without generating a pressure gradient term or additional control forces in the induction equation.
1.1 Main results
The statements of the main theorems anticipate the notion of weak controlled trajectories, as introduced later in Section 2.4. Briefly speaking, a weak controlled trajectory will be defined as the restriction to of a Leray–Hopf weak solution to a version of the problem (1.1), posed in an enlarged domain and driven by interior controls.
Theorem 1.2.
Assume that is simply-connected, that , and that is connected. Then, for arbitrarily fixed , , and , there exists at least one weak controlled trajectory
to the MHD equations (1.1) which obeys the terminal condition
| (1.7) |
Remark 1.3.
The next theorem holds for , but it involves a pressure-like unknown and a control , which however satisfies when . When , we need additional assumptions, since, to our knowledge, there is currently no literature providing and strong solutions for MHD under general (non-symmetric) Navier slip-with-friction conditions; this prevents us to prove 4.1 in the general case. Thus, when , we introduce the following class of the initial data.
The class .
When , are symmetric, , and the domain is simply-connected, then . Otherwise, the class consists of all which are restrictions of divergence-free vector fields that are tangential to , where is a smoothly bounded domain extension for of the type introduced in Section 2.1.
Example 1.4.
All states which vanish at and have vanishing normal derivatives up to the second order at belong to .
Theorem 1.5.
For any given time , fixed initial states , belonging to the class when , target states , and , there exists a smooth function , with when , such that the MHD system
| (1.8) |
admits at least one weak controlled trajectory
obeying the terminal condition
| (1.9) |
Remark 1.6.
When , the control may enter (1.8) if the magnetic field boundary layer described in Section 3.4.1 is not divergence-free. In order to illustrate that this statement is not sharp, we consider, as in Figure 2(b), a cylinder , for and a smoothly bounded connected open set , with controlled part . In this case, 1.5 is valid for all with . This will be illustrated by means of 3.4 combined with the discussion in Section 3.4.2.
Remark 1.7.
The systems 1.1 and 1.8 are under-determined since no boundary condition is prescribed along . Once a weak controlled trajectory is found via Theorem 1.2 or 1.5, one obtains explicit boundary controls by taking traces along , see also [Coron1996, CoronMarbachSueur2020, Fernandez-CaraSantosSouza2016, Glass2000].
Remark 1.8.
Since the proofs for Theorems 1.2 and 1.5 will be carried out in a certain extended domain, one can allow the interior of to be part of a Lipschitz continuous Jordan curve. Moreover, the controlled boundary is allowed to meet in a non-smooth way, as long as one can define domain extensions in the sense of Section 2.1.
1.2 Related literature and organization of the article
The global approximate controllability for viscous- and resistive MHD in non-periodic domains has to our knowledge not been studied, neither for incompressible- nor for compressible models. Therefore, the present work constitutes a first step in this direction. As a possible continuation, it would be interesting to generalize 1.2 for arbitrary and without additional interior control. Also, the question of global exact controllability to zero or towards trajectories remains open.
Concerning local exact controllability for MHD, where the initial state lies in the vicinity of the target trajectory, there have been some interesting works when the velocity satisfies the no-slip boundary condition. For incompressible viscous MHD, Badra obtained in [Badra2014] the local exact controllability to trajectories, while maintaining truly localized and solenoidal interior controls. However, since the boundary conditions are different from those employed here, one cannot deduce the small-time global exact controllability towards trajectories by combining the approaches given in [Badra2014] with our global approximate results. A variety of previous local exact controllability results may also be found in [BarbuHavarneanuPopaSritharan2005] by Barbu et al. and in [HavarneanuPopaSritharan2006, HavarneanuPopaSritharan2007] by Havârneanu et al., while approximate interior controllability for certain toroidal configurations without boundary has been investigated by Galan in [Galan2013]. Moreover, Anh and Toi studied in [AnhToi2017] the local exact controllability to trajectories for magneto-micropolar fluids, while Tao considered the local exact controllability for planar compressible MHD in the recent work [Tao2018]. Recently, we have studied the global exact controllability for the ideal incompressible MHD in [RisselWang2021], in which the small-time global exact controllability in rectangular channels is obtained in the presence of a harmonic unknown as in (1.8). Subsequently, Kukavica et al. demonstrated in [KukavicaNovackVicol2022], likewise restricted to a rectangular domain controlled at two opposing walls, how to find boundary controls such that either vanishes or is explicitly characterized.
Aside of various MHD specific constructions, this article combines the return method and the well-prepared dissipation method as described by Coron et al. in [CoronMarbachSueur2020], where the small-time global exact controllability to trajectories has been studied for incompressible Navier–Stokes equations in two- and three-dimensional domains with Navier slip-with-friction conditions. Meanwhile, we shall also extend certain asymptotic expansions, obtained by Iftimie and Sueur in [IftimieSueur2011] for the incompressible Navier–Stokes equations, to the present MHD system. Due to the structure of the induction equation, the return method has to be carefully implemented in order to avoid generating pressure-like and additional forcing terms in the induction equation. To this end, under the assumptions of 1.2, we modify the return method trajectory from [CoronMarbachSueur2020] to be everywhere divergence-free, but allow a nonzero curl in the control region; this approach seems new and might be useful for further studies on the controllability of the ideal MHD equations.
Let us also mention other recent works on global controllability problems for fluids that employ the return- and well-prepared dissipation methods. For instance, an incompressible Boussinesq system with Navier slip-with-friction boundary conditions for the velocity is considered by Chaves-Silva et al. in [ChavesSilva2020SmalltimeGE]. Moreover, the question of smooth controllability for the Navier–Stokes equations with Navier slip-with-friction boundary conditions is investigated in [LiaoSueurZhang2022]. Further, Coron et al. obtain in [CoronMarbachSueurZhang2019] global exact controllability results for the Navier–Stokes equations under the no-slip condition in a rectangular domain.
Organization of this article.
Section 2 collects several preliminaries and defines notions of weak controlled trajectories. In Section 3, the global approximate controllability from sufficiently regular initial data towards smooth states is shown. The main theorems are concluded in Section 4. In Appendices A and B, boundary layer estimates and a proof of 4.1 are provided.
2 Preliminaries
A domain extension for is introduced in Section 2.1, several function spaces and norms are defined in Section 2.2, initial data extensions to are discussed in Section 2.3, notions of weak controlled trajectories for (1.1) and (1.8) are discussed in Section 2.4. Finally, Section 2.5 briefly outlines the strategy of the paper. Throughout, if not indicated otherwise, constants of the form are generic and can change from line to line during the estimates.
2.1 Domain extensions
In what follows, the sets denote the connected components of and stand for the respective intersections , hence
Let be a smoothly bounded domain, which is an extension of as shown in Figure 3, satisfying
Such an extension exists by the requirements on . Throughout, the outward unit normal at is denoted by , or simply by if no confusion can arise. We also make the following assumptions:
-
•
the extension is selected such that and are tangential at ;
-
•
for the sake of simplifying the notations, to each connected component of at most one connected component of is attached.
Moreover, given , we denote
When is a multiply-connected domain, there is a number of smooth -dimensional and mutually disjoint cuts , which meet transversely, such that one obtains a simply-connected set via ; see, e.g., [Temam2001, Appendix I]. Next, for each , a unit normal field to is denoted by . When is simply-connected, we set .
The following Korn and Poincaré type inequalities for possibly multiply-connected domains are well-known.
Lemma 2.1.
There exists a constant such that for any , one has the estimate
| (2.1) | ||||
Proof.
It is known (e.g., see [AmroucheSeloula2013, Corollary 3.4]), that all with along obey
| (2.2) |
Moreover, as demonstrated in [BoyerFabrie2013, Theorem III.4.3], there exists a function which solves the Neumann problem
and satisfies
| (2.3) |
By employing trace estimates and the properties of , the potential field is seen to satisfy
Consequently, by means of the estimates (2.2) and (2.3), the first inequality in (2.1) follows with from
Concerning the second inequality in (2.1), let the multi-valued functions be chosen such that are smooth and form a basis for the space of curl-free and divergence-free vector fields tangential at . As in [Temam2001, Appendix I], one can select this basis such that , where denotes the jump of across and is the usual Kronecker symbol. Therefore, one has
which allows to conclude the proof. ∎
Let be sufficiently small so that represents a thin tubular neighborhood in of the boundary . Further, let satisfy for all and
This implies that for all , assuming without loss of generality that is sufficiently thin. Now, a smooth extension of to is provided by
| (2.4) |
In this sense, the tangential part of is then defined everywhere in . Moreover, the Weingarten map and the general friction matrices are smoothly continued to such that
while also extending the assumptions (such as or ) that might have been made in Theorems 1.2, 1.5, and 2.5.
For describing boundary layers in the vicinity of , when a parameter is assumed small, some functions will depend on a slow variable , the time and a fast variable
In this case, for a map we denote
By convention, differential operators are always taken with respect to only, if not indicated otherwise by the notation. Therefore, as also remarked in [CoronMarbachSueur2020, IftimieSueur2011], one has the commutation formulas
| (2.5) | ||||||
and consequently
2.2 Function spaces
The Hilbert spaces and of divergence-free and tangential vector fields are defined by means of
and
where denotes the closure in . For any , the weakly continuous functions from to are denoted by . The space for weak MHD solutions is
Moreover, for , we employ the weighted Sobolev spaces
where denotes for functions the seminorm
2.3 Initial data extensions
We extend the original initial data to . Whether divergence-free extensions are possible depends on the normal traces of the initial states fixed in Theorems 1.2 or 1.5. More specifically, we either choose extensions of the type
or we select continuations with defined normal trace at and which obey
These extensions will be made precise below in 2.2, which is a modification of [ChavesSilva2020SmalltimeGE, Proposition 2.1]; hereto, given any , the following notations are fixed beforehand.
-
•
The sets enumerate the connected components of the -th controlled boundary part .
-
•
The set is the extension attached to at , namely the maximal union of connected components of with .
-
•
For each , the set is the connected component of attached to . If is attached to and with , then .
Lemma 2.2.
There exists a constant such that for each there are functions with and satisfying
When the vector field additionally obeys at the conditions
| (2.6) |
then one can choose .
Proof.
Let denote the outward unit normal to at , while the outward unit normal to at is written as . It is known (cf. [BoyerFabrie2013, Chapter IV, Section 3.2]) that there exists a continuous normal trace operator
Then, for each , a smooth function is fixed such that
This guarantees that one can solve for each a weak formulation of the respective elliptic problem
| (2.7) |
In particular, if (2.6) holds for , then can be chosen. Accordingly, the proof is concluded by taking in and in . The continuity of the extension operator follows from (2.7) and the divergence-free condition encoded in . ∎
2.4 Weak controlled trajectories
To define notions of weak controlled trajectories for the problems (1.1) and (1.8), we follow the idea from [CoronMarbachSueur2020] and first introduce Leray–Hopf weak solutions for interior controlled MHD problems posed in the respectively enlarged domain . Then, by restricting such solutions to , one obtains a notion of boundary controlled weak solutions to (1.1) and (1.8). The plan is as follows.
-
•
Section 2.4.1 defines weak controlled trajectories for initial data in .
-
•
Section 2.4.2 discusses the appearance of .
-
•
Section 2.4.3 formulates weak controlled trajectories in Elasser variables.
-
•
Section 2.4.4 defines weak controlled trajectories for more general initial data.
Given any time , let us recall the notation of the space-time cylinder and its mantle
In this subsection, if forces and appear in the right-hand sides of MHD problems, then it is assumed that
for some . In particular, will coincide with that in 1.5.
2.4.1 The case of data (e.g., 1.2)
We focus now on the situation of 1.2; however, if the initial data can be extended as -functions, the following definitions make also sense for the setting of 1.5. To streamline the presentation, the three-dimensional cross product and curl notations are employed.
Definition of weak controlled trajectories.
When the initial data admit extensions to , as emphasized in 2.3 for 1.2, a weak controlled trajectory for (1.1) is defined as any pair of vector fields that are of the form
where denotes a Leray–Hopf weak solution to the viscous and resistive incompressible MHD system
| (2.8) |
A pair is called a Leray–Hopf weak solution to (2.8), if it satisfies for all and for almost all , the variational formulation
| (2.9) |
together with the following energy inequality for almost all :
| (2.10) |
The weak formulation (2.9) and energy inequality (2.10) are derived by utilizing the identity for any sufficiently regular vector field with , while also using the integration by parts and vector calculus formulas
where stands for the surface measure on .
Existence of weak solutions.
By analysis similar to the Navier–Stokes equations, for instance via the Galerkin method explained in [Temam2001, Chapter 3], one can obtain the existence of Leray–Hopf weak solutions satisfying (2.9) and (2.10). Regarding the energy inequality (2.10), we refer to [IftimieSueur2011, Section 3] for a strategy that carries over to the present MHD model. In particular, the boundary integrals in 2.9 and (2.10) are not causing additional difficulties in comparison with the references mentioned above. Indeed, when and , trace inequalities and interpolation imply
| (2.11) | ||||
for any and a generic constant . The estimates (2.1) and (2.11) facilitate a Galerkin method of the type described in [Temam2001, Chapter 3]. In this way, one obtains approximate solutions to (2.9) that are bounded in , satisfy a discrete version of (2.10), and converge in to a Leray–Hopf weak solution as . For passing the limit in the discrete version of the energy inequality (2.10), one needs to show
To this end, if denotes either or , and represents either or , then, by means of trace theorems and interpolation, one has
| (2.12) | ||||
which implies for and that
2.4.2 A pressure-like gradient in the induction equation
It is important to verify that sufficiently regular functions which satisfy the variational formulation (2.9) are classical solutions to the original problem (2.8). On the one hand, if the pair possesses the necessary regularity and satisfies (2.9), then also classically obeys a version of (2.8) where the induction equation is replaced by
| (2.13) |
On the other hand, because in (2.9) is divergence-free and tangential at , one cannot generally conclude that (2.8) is satisfied. However, let us now suppose that , and that the control in (2.8) additionally obeys
| (2.14) |
In this situation, by acting with the divergence operator on (2.13), while also taking the normal traces of (2.13) at , one finds that solves the Neumann problem
| (2.15) |
hence holds in .
Remark 2.4.
If or , even for the uncontrolled model with in , we shall use the modified induction equation (2.13). Indeed, assuming, e.g., for , that there would exist a classical solution to (2.8) on a short time interval, then necessarily on . Since is supported in , it is unclear whether this could contradict
For instance, if is the identity matrix and , then it would follow for that is a constant of time at ; thus, would satisfy a Navier-slip-with-friction and a Dirichlet boundary condition at the same time.
Despite that possibly when or , it still remains an interesting question whether magnetic field interior controls with (2.14) can be constructed, as this would provide more insights on the nature of the term in a control theoretic context. In order to obtain a partial result in that direction (cf. 2.5), we assume for now that and let be the linear operator that assigns to the gradient of the (unique up to a constant) solution to the Neumann problem
Then, concerning the general case where or , the problem (2.8) with induction equation replaced by (2.13) can be reformulated as
| (2.16) |
By analysis similar to the D incompressible Navier–Stokes system, weak solutions to (2.16) are unique and initial data give rise to strong solutions when (2.14) is satisfied. In this article we shall prove in parallel to Theorems 1.2 and 1.5 the following controllability result for (2.16) via distributed controls and satisfying (2.14).
Theorem 2.5.
Let be an annulus and an open simply-connected control region such that is simply-connected and contains an annulus sector. The friction coefficient matrices are arbitrarily fixed with
Then, for any control time , accuracy parameter , and
-
•
initial states with and vanishing on ,
-
•
target states with for a stream function that vanishes on ,
there exist controls , supported in and obeying (2.14), such that the solution to (2.16) satisfies the terminal condition
2.4.3 Change of unknowns
A few exceptions aside, the subsequent analysis can be streamlined by introducing the symmetrized notations
| (2.17) |
as well as
and
where
By utilizing the inner product structure of , one can verify that the energy
satisfies
| (2.18) | ||||
Therefore, if is a Leray–Hopf weak solution to (2.8), by inserting (2.18) to (2.10) and using the transformations from (2.17), it follows for almost all the inequality
| (2.19) |
2.4.4 The case of 1.5
Given the assumptions of 1.5, an application of 2.2 provides initial data extensions with at . Moreover, the scalar functions and belong to and are supported in . A weak controlled trajectory for (1.8) is then defined as any pair
with
where solve in the below specified Leray–Hopf weak sense the Elsasser system111Systems of the form (2.20), but with different boundary conditions, have been considered by Elsasser in [Elsasser1950].
| (2.20) |
where . In (2.20), the boundary operators are as defined in Section 2.4.3. Moreover, the functions are assumed to be smooth, supported in , and of zero average in , that is . Since and are likewise smooth, a notion for weak solutions to (2.20) can now be introduced similarly to the Navier–Stokes case in [CoronMarbachSueur2020]. In order to lift the nonzero divergence constraints, let solve the linear Elsasser system
| (2.21) |
For defining a weak formulation for (2.21), it is of advantage to first eliminate the inhomogeneous divergence data . Hereto, one may decompose , where are for each smooth solutions to the elliptic Neumann problems
| (2.22) |
while obey the inhomogeneous system
The regularity of weak solutions to this linear problem can be investigated with the help of estimates that are known for the Navier–Stokes system (cf. B.1). As a result, one finds that are smooth for . Finally, the ansatz for the solutions of (2.20) provides a description of by means of the perturbed Elsasser equations
A Leray–Hopf weak solution to the above system is any pair which satisfies for all and almost all the variational formulation
| (2.23) |
and for almost all the energy inequality
| (2.24) |
Since the profiles are smooth and , the existence of satisfying (2.23) and (2.24) can be obtained through a Galerkin method as explained in Section 2.4.1.
Remark 2.6.
The weak formulation from Section 2.4.1 for (2.8) can be regarded as a special case of (2.23) and (2.24). The two formulations are presented separately in order to make a stronger distinction between the case without and non-physical situations, where even is possible in .
2.5 Brief description of the strategy
To prove Theorems 1.2, 1.5, and 2.5, we develop the approach from [CoronMarbachSueur2020]. Essentially, the time interval will be divided into two sub-intervals and which correspond to the two main stages of the control strategy.
Stage 1 (Section 4).
Stage 2 (Section 3).
During , controls and are applied in . More precisely, we determine and such that each Leray–Hopf weak solution to (2.8) or (2.20), starting from at , approaches the final state in at . As discussed in Section 2.4, for proving Theorems 1.2 and 2.5, we have to ensure that is supported in and obeys in and at .
3 Approximate controllability between regular states
Let , , and . Then, assuming that sufficiently regular , and are given, the different configurations in Theorems 1.2, 1.5, and 2.5 are treated simultaneously as follows.
-
•
To show Theorems 1.2 and 2.5, the Leray–Hopf weak solution to (2.8) with the data is fixed and, by means of Section 2.4.3, rewritten in the symmetrized variables .
- •
It will be shown that, if the a priori selected functions are of a certain form, then satisfy
| (3.1) |
More generally, given arbitrary states , it will be demonstrated that for suitable choices one has
| (3.2) |
which implies
where
3.1 Asymptotic expansions
The systems (2.8) and (2.20) are reformulated as a small-dissipation perturbation of an ideal MHD type system in the variables . Hereto, for any small , the following scaling is performed
| (3.3) |
and for the controls
| (3.4) |
As a result, the profiles are seen to satisfy a weak formulation and strong energy inequality for the following problem
| (3.5) |
In order to achieve the desired estimate (3.2), it shall be verified that, for and being of specific forms, all solutions to (3.5) obey
| (3.6) |
Hence, after choosing sufficiently small, the asymptotic behavior (3.6) implies 3.2. To prove (3.6) with , see Section 3.6 for the general case, the selected solution to (3.5) is expanded according to the ansatz
| (3.7) | ||||
and for the controls
| (3.8) |
For all , we fix
and
On the time interval , the profiles
are chosen in the following way:
- •
- •
- •
The profiles are, later on, defined on via 3.8 as the solutions to (3.14), together with associated pressure terms , and interior controls . The vector field fails in general to obey the boundary condition at , giving rise to weak amplitude boundary layers in the zero dissipation limit . These boundary layers are of a similar nature as those studied in [IftimieSueur2011, CoronMarbachSueur2020]. In the particular case , there appears not only a velocity boundary layer, but also one for the magnetic field. The profiles
which are related to such boundary layers, will be described in Section 3.4. In the presence of magnetic field boundary layers, the controls shall be defined in Section 3.4.2. The boundary layer dissipation controls will be obtained in Section 3.4.3. Subsequently, in Section 3.5, the remainders are estimated. Then, concerning approximate null controllability, the asymptotic behavior (3.6) with is shown in 3.28. The approximate controllability towards arbitrary smooth states is concluded in Section 3.6.
3.2 A return method trajectory
The zero order profiles , , , and are chosen for as a special solution to the controlled Euler system
| (3.9) |
Given a smooth vector field , let denote for the unique flow which solves the ordinary differential equation
| (3.10) |
Remark 3.1.
Under the assumptions of 1.5, the profiles , , , and are chosen by means of 3.2 below, which is taken from [CoronMarbachSueur2020, Lemma 2] and has been proved in [Coron1993, Coron1996EulerEq, Glass1997, Glass2000].
Lemma 3.2 ([CoronMarbachSueur2020, Lemma 2]).
Remark 3.3.
As shown in [Coron1993], for two-dimensional simply-connected domains one can replace (3.11) by a uniform flushing property. That is to say, there exists a smoothly bounded open set such that and for all one has .
Example 3.4.
In order to illustrate 3.3 by means of a very specific example, and to provide more details regarding 1.6, let us consider, as in Figure 5, for a smoothly bounded connected open set the cylindrical setup
Let be the planar simply-connected extension of with . One can extend through , in the sense of Section 2.1, to a smoothly bounded domain with
Now, let with for and extend by zero to . Moreover, for some large number , take such that when . Then, for and choose
Since does not depend on the spatial variables in , the above profiles solve the controllability problem (3.9) with the support of and being located away from . Also, for large enough, one has a uniform flushing property as mentioned in 3.3, see for instance the proof of [RisselWang2021, Lemma 3.1] which carries over to a three-dimensional pipe.
Lemma 3.5.
Proof.
At first, let be the profiles obtained from 3.2; thus, one might have at some points. However, as sketched in Figure 6, there exist an open set and a number satisfying
such that
It is not restrictive to assume that is simply-connected; hence, there is a scalar potential with
Indeed, if denotes a rotation by , then
Let be any extension of to and take a smooth cutoff function with
Since is tangential at , one can observe along the vanishing tangential derivatives
Therefore, there is a time-dependent constant such that at for each . Finally, we define
which implies in and at . Since can only differ from if , one has , and the flushing property (3.11) remains valid for . The proof is then concluded by renaming as , emphasizing that in and renaming as , and by modifying inside such that (3.9) holds. ∎
Remark 3.6.
In the proof of 3.5, one can explicitly choose the size and location of , as long as is open and .
Remark 3.7.
When is not connected, the proof of 3.5 can still be applied to situation where is simply-connected; for instance, by selecting such that all its integral curves cross the same connected component of . To avoid creating a gradient term during the regularization stage described in Section 4, the initial data should then obey relations of the type (2.6).
The special case of an annulus.
Concerning 2.5, where is an annulus (cf. Figure 7), we introduce an explicit return method trajectory , which is curl-free, divergence-free, and tangential at . This is possible because annuli are doubly-connected. More precisely, we define
and choose for a constant a smooth function , satisfying
for all . Then, for , we denote the vector field
which possesses in the properties
Due to the symmetry of , the extended unit normal field can be chosen everywhere orthogonal to . Now, for sufficiently large, we define
| (3.12) | ||||||||
where satisfies whenever , for some independent of . In particular, assuming that is fixed sufficiently large, a flushing property of the type (3.11) holds. Indeed, the profile never vanishes in and the associated flow propagates information along circular trajectories around the annulus.
Finally, we take and set , while choosing general friction coefficient matrices . Then, satisfies the relations
| (3.13) | ||||||
because
3.3 Flushing the initial data
Due to the scaling in (3.3) and (3.4), the contributions of to are at . In order to avoid that impact the remainder estimates in Section 3.5 below, the goal is to cancel their influence for by using the controls , which are supported only in . After inserting (3.7) and (3.8) into (3.5), motivated by [CoronMarbachSueur2020], one observes that a good strategy consists of defining as the solutions to the linear problem
| (3.14) |
We shall determine the controls such that the corresponding solution to (3.14) satisfies . This is achieved by combining [CoronMarbachSueur2020, Lemma 3] with new ideas for the cases of Theorems 1.2 and 2.5, where we have to maintain the properties
| (3.15) |
Lemma 3.8.
Proof.
When is determined via 3.2, the proof is a direct application of the arguments from [CoronMarbachSueur2020, Lemma 3] to the uncoupled systems solved by . Hence, we consider here only the two-dimensional situations of Theorems 1.2 and 2.5, where is obtained either via 3.5 or by (3.12). Our strategy is close that from [CoronMarbachSueur2020, Lemma 3], but compared to [CoronMarbachSueur2020] there are two new challenges:
Step 1. Preliminaries.
Step 2. A partition of unity.
Due to the regularity of and the flushing property (3.11), as provided by 3.2, 3.5, or by the definition for in (3.12), there exists a small number such that
Hence, one can select a smoothly bounded closed set with and
Moreover, for some , we fix a finite covering of which consists of interior and boundary squares. The boundary squares are centered in points of , fully included inside , and one side lies in the interior of . The interior squares are centered in points of and belong to . Consequently, there exists and a number of balls which cover such that for each index one has
| (3.17) |
With respect to the balls , let be any fixed smooth partition of unity in the sense that
| (3.18) |
Step 3. Flushing the initial magnetic field.
Since holds in for the presently case, the initial magnetic field can be flushed without pressure term. To this end, we rely on the existence of a stream function with
Indeed, under the assumptions of 1.2 this follows from being simply-connected, while for 2.5 it is part of the hypotheses. Then, by the decomposition in (3.16), the vector field satisfies the linear problem
| (3.19) |
with . By employing the partition of unity given in (3.18), we first solve for the homogeneous problems
| (3.20) |
Concerning the pressure gradient, after multiplying in (3.20) with and integrating by parts, one finds
Finally, we take supported in , satisfying (3.15), and such that the corresponding solution to (3.19) obeys . For instance, one can choose
| (3.21) |
where is a smooth cut-off with
| (3.22) |
for from (3.17).
Step 4. Flushing the initial velocity: the idea.
The vector field obeys with the linear problem
| (3.23) |
The pressure gradient is eliminated by taking the curl in (3.23), leading to a transport equation for with non-local terms. The goal is to determine , spatially supported in , such that the corresponding solution to (3.23) satisfies
| (3.24) |
Regarding 1.2, where is simply-connected, (3.24) implies in (cf. (2.1)). Concerning 2.5, since might in that case be multiply-connected, one can from (3.24) only conclude that , where and spans the one-dimensional space of divergence-free, curl-free, and tangential vector fields on the annulus . In fact, one can take . Therefore, we need to be able to steer any state of the form to zero. To this end, note that . Moreover, since is simply-connected, for with for and when , there exists , with for all , such that
Thus, the function solves together with a smooth pressure and a smooth control , which is spatially supported in , the controllability problem
| (3.25) |
Indeed, one can take and in . In , one may choose any smooth extension of and fix . Summarized, first one employs for steering to , followed by utilizing the controls to connect with .
Step 5. Flushing the initial velocity: showing (3.24).
To determine such that (3.24) holds, we rewrite (3.23) in vorticity form, which describes . That is, given the partition of unity from (3.18), we make an ansatz of the form
| (3.26) |
where each triple is sought to satisfy
| (3.27) |
Since is supported in , the transport problem decouples in from the div-curl system. To see this, let for each be the solution to
| (3.28) |
Then, we take and define via
| (3.29) |
where is the function from (3.22). Owing to (3.17) and (3.22), one finds for all and the relations
Therefore, if it would be possible to choose for each such that
| (3.30) |
then would satisfy (3.27). To construct such , the following observations are remarked.
-
•
The right-hand side of (3.30) is supported in .
- •
Since should be supported in and obey (3.30), the average of the right-hand side in (3.30) must vanish on each interior cube. This indeed happens because solves the homogeneous transport equation (3.28); in fact, for each with , the average of on vanishes at and is transported by . Finally, by undoing the curl in (3.30) (cf. [CoronMarbachSueur2020, Section A.2] for explicit formulas), one obtains the desired controls . Thanks to the assumption , one has and are bounded in . ∎
3.4 Boundary layers and technical profiles
In this subsection, the boundary layers and related technical profiles, appearing in (3.7)–(3.8), will be described. In addition to to the neighborhood , as defined in Section 2.1, another tubular region is denoted by
where is a small number to be fixed in 3.19 below. Moreover, given a function with
a smooth cutoff is defined by
| (3.31) |
where for as described in Section 2.1. By construction, one observes that in the vicinity of and that . Furthermore, in view of (2.4), the gradient of in can be calculated as
| (3.32) |
Remark 3.10.
3.4.1 Boundary layer equations
Our definition of the boundary layer correctors is motivated by [CoronMarbachSueur2020, IftimieSueur2011]. After plugging the relations (3.7) and (3.8) into (3.5), there appears a term at order that is not absorbed by (3.9). However, resorting to the idea from [IftimieSueur2011] of writing
this contribution is seen to behave as . In order to also offset the mismatching boundary values , the boundary layer profiles in (3.7) are introduced in as the solution to the coupled linear problem
| (3.33) |
with boundary and initial conditions
| (3.34) |
Above, the functions and are for all given by
| (3.35) |
Since on and in , the function is smooth (cf. [IftimieSueur2011, Lemma 4]). Like and , the functions are smooth as well.
Remark 3.11.
Now, several properties of the solutions to 3.33 and (3.34) are summarized; recall that is the weighted space defined in Section 2.2 via
Lemma 3.12.
Proof.
The well-posedness of the linear problem (3.33), (3.34) is analogous to that of the Navier slip-with-friction boundary layers for the Navier–Stokes equations (cf. [IftimieSueur2011, CoronMarbachSueur2020]). Since are assumed smooth, the regularity stated in (3.36) is obtained from A.1. The relation (3.37) follows by multiplying in (3.33) with , which leads to a priori estimates for similar to that given in [IftimieSueur2011, Section 5]. ∎
3.4.2 Technical profiles
For the sake of having in , the normal contributions of , which appear at when inserting (3.7) into (3.5), have been omitted in (3.33). This and the commutation formula
motivate introducing the profiles in (3.7) as the solutions to
| (3.38) |
Next, let us define the second boundary layer correctors . The normal parts of will compensate for the non-vanishing divergence of , while their tangential parts constitute a lifting for and later on enable sufficient remainder estimates. Namely,
| (3.39) | ||||||
noting that satisfy under the assumption the relations
| (3.40) | ||||||
Remark 3.13.
In order to balance the nonzero divergence contributions and normal fluxes generated by , the correctors are introduced as solutions to
| (3.41) |
For each , the corresponding Neumann problem in (3.41) is well-posed; see 3.24 below.
It remains to specify the profiles and the forces . Inserting the ansatz (3.7) into (3.5) gives at the order rise to the terms
| (3.42) |
which are not behaving well regarding the remainder estimates in Section 3.5. In particular, the norms of the boundary data in 3.41 are of order . For this reason, the pressure correctors are defined by
such that one has the representations
| (3.43) |
at all points where . Consequentially, in view of Lemmas 3.2 and 3.5, or by 3.4, the relations in (3.43) are always true in . However, in this might not be the case; but can be utilized to improve the remainder estimates. More precisely, motivated by the desired estimate (3.83) in Section 3.5.1 below, we shall define either via (3.44) or by means of (3.45), as explained next.
The first case of the definition for .
When is satisfied at , 3.24 will provide good estimates for , allowing us to choose
| (3.44) |
There are at least two situations with at .
- •
- •
Given the assumptions of 1.2 or 2.5, one can conclude also the additional properties
Indeed, either implies that , hence , or, if in the case of 2.5, one has by means of (3.12), which even provides .
The second case of the definition for .
When in and at some points of , then we define
| (3.45) |
This applies to the general situation of 1.5, where is obtained from 3.2.
Remark 3.14.
When , magnetic field boundary layers cannot arise, as emphasized by the following lemma.
Lemma 3.15.
and imply .
3.4.3 Boundary layer dissipation via vanishing moment conditions
The boundary layer controls appearing in (3.33) are now determined. For large times , in Section 3.1 we already fixed
For all , the controls will be chosen such that admit improved decay rates as . To this end, we implement the well-prepared dissipation method, described in [CoronMarbachSueur2020] for the Navier–Stokes equations and previously in [Marbach2014] for a viscous Burgers’ equation. Here, two different constructions for will be given: the first one, namely 3.17, follows closely the known results and is suitable for showing 1.5 with nonzero ; the second one, namely 3.18, allows concluding Theorems 1.2, 2.5, and the assertion for of 1.5. To begin with, we state a direct modification of [CoronMarbachSueur2020, Lemma 6], which involves the space (cf. Section 2.2)
Lemma 3.16.
Let and suppose that satisfy for all integers the vanishing moment conditions
| (3.48) |
Furthermore, assume that solve the coupled parabolic system
Then, for all one has the decay estimate
| (3.49) |
where is a constant independent of and the initial data .
Proof.
The following result, which is a consequence of 3.16 and [CoronMarbachSueur2020, Lemma 7], provides the controls on the time interval . It will be applied in the general situation of 1.5.
Lemma 3.17.
For any , there exist satisfying
such that obey the decay rate
| (3.51) |
for all and , with a constant not depending on the time .
Proof.
Consider the even extensions of to plus lifted boundary data, defined via
| (3.52) |
For , one has by construction
Thus, are governed by the parabolic system
| (3.53) |
where is a parameter. Therefore, in view of 3.16 and (3.52), the decay estimate (3.51) follows if enough vanishing moment conditions of the type (3.48) are satisfied. To see this, the proof of [CoronMarbachSueur2020, Lemma 7] can be applied individually to the equations satisfied by and . This provides controls such that for each the Fourier transformed functions
obey the relations
| (3.54) |
Since (3.54) implies sufficient vanishing moment conditions and 3.12 provides uniform bounds for on the time interval , the proof is complete. ∎
The next lemma provides magnetic field boundary layer controls that are not only tangential at , but also divergence-free in ; this is required for showing 2.5. Also regarding 1.2, and 1.5 with , the proof below will explain how can be selected with . The approach is based on ideas from [CoronMarbachSueur2020, Lemma 7].
Lemma 3.18.
Given any , the controls with the properties from 3.17 can under additional assumptions be chosen as follows.
-
1)
When is valid for all , then can be fixed with
-
2)
When the profile selected in Section 3.1 satisfies
(3.55) then after fixing the number from the definition of sufficiently small, one can construct with
(3.56) - 3)
Proof.
If in , one may take . Indeed, the magnetic field boundary layer solves in that case a well-posed linear problem with zero data, which yields in as shown by 3.15. In any case, we apply the arguments from the proof of 3.17 for determining such that
Thus, to show 2) and 3), it remains to identify a suitable control (having the desired properties), which acts in the equation satisfied by in a way that
Step 1. Preliminaries.
In view of (3.52), the vector field can with be written as
| (3.57) |
Under the assumption , noting that is true by construction, it follows from (3.37) that holds as well. Also, 3.32 and 3.55 imply by means of
| (3.58) | ||||
Even more, being a gradient, hence , ensures for two vector fields and defined in that
Therefore, one can infer at all points where holds the relation
Next, while from (3.55) it only follows that in , we temporarily guarantee that in all of by means of the artificially strong assumption, which will be removed in the last step by choosing small222In this article, (3.59) is always satisfied when we employ 3.18 with the hypotheses in (3.55); see (3.13).:
| (3.59) |
As a result of the foregoing considerations, and by understanding the yet unspecified controls as extended evenly to all , the function satisfies the following problem:
| (3.60) |
where
Consequently, the partial Fourier transform
satisfies the problem
| (3.61) |
Therefore, during the time interval , for each the evolution of the evaluated derivatives
is governed by the transport equation
| (3.62) |
which contains the source term
| (3.63) |
Step 2. Determining .
Let denote the integer part of . Since is symmetric about the axis, it remains to steer for the even moments to zero. Hereto, we make for the ansatz
| (3.64) |
where the even functions are chosen such that
obey the relations
For instance, for any even smooth cutoff with in a neighborhood of the origin one may take as the inverse Fourier transforms of the functions
Subsequently, inserting the ansatz (3.64) for each even choice into (3.62) provides the cascade system of transport equations
| (3.65) |
with zero initial conditions
Let us begin with determining the control in (3.65). Hereto, the auxiliary function is taken as the unique solution to
| (3.66) |
Since holds in by (3.58), and is true in as well, for all one may observe that
Furthermore, the profile is without loss of generality assumed to satisfy a flushing property of the type (3.11). Indeed, when is defined via (3.12), one notices that the flow associated with simply moves particles around the annulus in the opposite direction. More generally, the flushing property of can be ensured by constructing with replaced by , followed by gluing the resulting profile at the time to a time-reversed version. Now, we take as the unique solution to
| (3.67) |
where the control is chosen such that
In order to find , we proceed analogously to the constructions of controlled solutions to (3.19) in the proof of 3.8, noting that the problems (3.19) and (3.67) are of the same type. As a result, we can define in the vector field
which solves the first equation in (3.65) with and obeys the desired initial and terminal conditions
To determine , the same arguments as for finding can be repeated, but now with the known source term . Due to the cascade structure of (3.65), all controls are obtained in this way. Finally, is constructed via (3.64) and obeys (3.56).
Step 3. Removing the assumption (3.59).
Without assuming (3.59), the equation (3.60) for might not be correct in , since (3.33) contains the terms . However, for small it will be shown below that the control obtained in the previous step already ensures
| (3.68) |
for the solution to
| (3.69) |
Then, because the assumptions in (3.55) together with (3.68) imply
in all of , it follows that satisfies a version of (3.69) where the first line is replaced by
which corresponds to the equation for derived from (3.33). Thus, after verifying (3.68) for some , it follows retrospectively that the equation (3.60) for is correct even without assuming (3.59).
To show (3.68), we apply the ideas from [CoronMarbachSueur2020, Section 3.4]. Hereto, for any , consider the tube and define its maximal distance of influence during the time interval under the flow via
The controls in (3.65) only act where pollution, caused by , , or , arrives via the flow associated with . Thus, is supported in and its action travels at most into . Consequently, the effects of are propagated into . Since the -support of is contained in by the definition of , and in view of (3.57), maintaining the -support of within is achieved by adjusting the support of . Indeed, is continuous and one has because is tangential to , which yields the existence of with . Then, every choice is suitable. ∎
The last part of the proof of 3.18 shows why one can take in the definition of such that in the -support of . A similar argument ensures such a property for and is valid for the general situation of 3.17 (cf. [CoronMarbachSueur2020, Section 3.4]).
Lemma 3.19.
3.4.4 Properties of the boundary layers and technical profiles
Due to the fast variable scaling for the boundary layer profiles , , and , several estimates will profit from a gain of order as stated below.
Lemma 3.21 ([IftimieSueur2011, Lemma 3]).
There exists a constant such that, for all and functions in with , it holds
Lemma 3.22.
The functions from (3.38) satisfy for all . In addition, there exists a constant independent of such that
| (3.70) |
Proof.
Lemma 3.23.
For all , the profiles determined in (3.39) satisfy
| (3.71) | ||||
| (3.72) | ||||
| (3.73) |
Proof.
One can show (3.71) by separately estimating the tangential and normal parts
For instance, integration by parts yields
Hence, for arbitrary one has
Thus, by choosing small and a new constant , one obtains
The other aspects of (3.71) are along the same lines, noting that estimating costs one regularity level in due the application of a trace theorem. The estimate (3.72) follows from a combination of 3.21, the above idea for showing (3.71) and the identity
Regarding (3.73), the starting point is to derive from (3.39) the representation
into which one can subsequently insert the equation (3.33) and proceed as before. ∎
Lemma 3.24.
The Neumann problems (3.41) are well-posed, with uniqueness of solutions up to a constant, and all solutions obey for the estimates
| (3.74) |
If for all , it additionally holds
| (3.75) |
Furthermore, for all , one has
| (3.76) |
Proof.
In (3.41), there is no coupling between superscribed functions. Thus, the well-posedness of (3.41) together with (3.74) and (3.76) can be established by analysis similar to [CoronMarbachSueur2020, Equations (4.29), (4.31)–(4.33) and (4.58)]. In particular, by employing (2.5), (3.37), (3.39), and (3.40), one can verify the necessary compatibility conditions for (3.41) via
It remains to show (3.75) when at . By elliptic regularity for weak solutions to the Laplace equation (3.41), 3.21, and (3.71) one obtains
∎
Finally, several properties of the remainder terms in the ansatz (3.7) are summarized.
Lemma 3.25.
The remainder terms given in (3.7) satisfy the conditions
| (3.77) | ||||||
where
Moreover, for a constant independent of , the boundary data can be estimated by
| (3.78) |
3.5 Remainder estimates
The goal is now to show that as . As various estimates are similar to those for the Navier–Stokes problem in [CoronMarbachSueur2020], we place emphasis on the arguments that are specific for the current MHD problem.
3.5.1 Equations satisfied by the remainders
In order to derive the equations satisfied by the remainders , the definitions of given in (3.38) are employed for rewriting (3.33) in the form
| (3.79) |
Then, (3.7)–(3.8) are inserted into (3.5) while using (2.5), (3.79) and 3.25. Since the terms and vanish, the remainders satisfy the following problem
| (3.80) |
where the amplification terms are given by
and the remaining terms
contain
and
Before deriving energy estimates for , several asymptotic properties of the right-hand side terms in 3.80 are summarized.
Lemma 3.26.
For all , one has
| (3.81) |
with a constant independent of , and . Furthermore, as , it holds
| (3.82) |
Proof.
Throughout, it will be used that have been fixed in Section 3.4.3 either by 3.17 (or 3.18) applied with , , and . This provides bounds for in which are uniform in , since for and one has the convergence of the integral
In order to show (3.81), let denote the functions
From 3.8 one knows that are bounded in as long as the initial data satisfy . Hence, combining Sobolev embeddings with 3.24 allows to infer
Moreover, by 3.17 or 3.18 one finds a constant independent of with
which eventually implies (3.81).
The bounds for in (3.82) follow from (3.78) and 3.17 (or 3.17) such that it remains to establish the estimates for in (3.82). We begin with the terms
According to the definition of in Section 3.4.2, the terms vanish when in and at some points of . When is satisfied at , then
Concerning the latter case, the estimate (3.75) implies
and an invocation of 3.17 or 3.18 yields
| (3.83) |
In order to treat the terms which appear with a factor in , we resort to a trick similar to [IftimieSueur2011, Equation (69)]. Indeed, the definitions for in (3.39) provide
which due to being bounded leads to
where
Since , 3.21 can be applied to and, by similar analysis for , there exists a constant independent of such that
As a result, 3.17 (respectively 3.18) allows to infer
The remaining terms contained in and behave as in the Navier–Stokes case treated in [CoronMarbachSueur2020, Section 4.4]. Carrying out these details involves the estimates (3.70), (3.72), (3.73), and (3.76). In particular, for estimating , several norms of enter through (3.73) with coefficients. Since are supported in and smooth, there are bounds for these terms. ∎
3.5.2 Energy estimates
The desired asymptotic behavior in (3.6) is now obtained as a consequence of the next proposition.
Proposition 3.27.
Proof.
The idea is to multiply the equations in the first line of (3.80) by respectively, followed by integrating over for , which however is not justified. Indeed, the regularity of does not guarantee the convergence of the integrals
Since (3.7)–(3.8) imply that the term , where is bounded in and the temporal derivatives obey , the above mentioned convergence issue can be avoided by using the strong energy inequality as explained in [IftimieSueur2011, Page 167-168].
Step 1. Employing the energy inequality.
Let us sketch the aforementioned approach of utilizing the energy inequality. For the sake of simplicity, we carry out the steps for the case from Section 2.4.1 where . First, from (3.80) and (3.5), one observes that satisfy a weak formulation for the problem
| (3.85) |
where
Multiplying the equations (3.85) with , integrating over , and summing up the results, leads to
| (3.86) | |||
In (3.86), the following cancellations, which are justified via integration by parts and by using the regularity of , have been taken into account:
Second, taking the test function in the weak formulation for , which is justified thanks to the regularity of , yields
| (3.87) | |||
Third, multiplying the energy inequality (2.19) with , followed by evaluation at and for , while performing the change of variables , one has
| (3.88) | |||
Step 2. Conclusion.
By subtracting from (3.88) the equations 3.86 and 3.87, while considering for the general case the identity
where , one arrives at the inequality
with
Since in and on , the inequality (2.1) provides
which in turn yields
| (3.89) |
where depends on and the fixed quantity . The boundary integrals containing are treated by applying for and the estimate
Thus, for and arbitrary one has
Consequently, by selecting small, employing 3.26, and applying Grönwall’s inequality in (3.89), one arrives at (3.84). ∎
Corollary 3.28.
The functions fixed in the beginning of Section 3 satisfy
3.6 Controlling towards arbitrary smooth states
Let be arbitrarily fixed. The previous arguments for approximate null controllability can be modified for the target . The idea is similar to that described in [CoronMarbachSueur2020, Section 5] for a Navier–Stokes problem. First, the ansatz (3.7) is modified such that for on the time interval one chooses an expansion of the form
| (3.90) |
while on it is assumed that
| (3.91) |
The profiles in (3.90) belong to and, by adopting the constructions from 3.8’s proof, there exist with such that solve the controllability problem
| (3.92) |
In particular, all bounds for and are independent of , as this parameter does not appear in (3.92). Because are smooth and independent of time, by analysis similar to Section 3.5, one can infer . As a result, the rescaled functions satisfy
4 Conclusion of the main results
In order to relax the assumption employed in Section 3, we connect initial data from by a weak controlled trajectory to a state which belongs to . This is done via 4.1 below, and a proof of this argument, which is a modification of [CoronMarbachSueur2020, Lemma 9], will be outlined in Appendix B.
Lemma 4.1.
When , assume that and are symmetric, , and that is simply-connected. For any given and , there exists a smooth function with such that a Leray–Hopf weak solution to (2.8) obeys for some the estimate
Let the control time , the states , and any be arbitrarily fixed. Then, the proof of 1.2 is completed by means of the ensuing steps.
-
1)
The physical domain is extended to , as explained in Section 2, and the weak formulation given in Section 2.4.1 is chosen.
- 2)
-
3)
By a density argument, one can select states with
- 4)
- 5)
The previous arguments, while skipping the initial data extension and regularization steps, also yield 2.5. In order to conclude 1.5, we proceed as follows.
-
1)
The physical domain is extended to as described in Section 2, but now the weak formulation in Section 2.4.4 is considered. When 4.1 cannot be applied, the extended initial data are chosen with . Otherwise, in order to reach a divergence-free state, one defines , with obeying and for all . A corresponding weak solution to (2.20) on with zero forces is denoted by and it follows that .
- 2)
-
3)
As before, by density, one can choose regular states with
- 4)
-
5)
By a gluing argument, one obtains a Leray–Hopf weak solution to (2.20) on , starting from the initial data and satisfying
Appendix A Boundary layer estimates
The estimates used in the proof of 3.12 are outlined; more general than there, we take now with along and for all . In 3.12, it is . We now consider in the equations (cf. (3.33))
| (A.1) |
together with the initial and boundary conditions (cf. (3.34))
| (A.2) |
While also more general data could be chosen, here we take the cutoff defined in (3.31) and consider
Due to the support of , compatibility conditions up to all orders are satisfied by the initial and boundary data in (A.2). Multiplying in (A.1) with , one may similarly to [IftimieSueur2011, Section 5] establish energy estimates which imply for all that
The goal consists now of showing the following lemma.
Lemma A.1.
Proof.
The ideas and arguments for proving A.1 are based on [IftimieSueur2011]. All constants which appear during the estimates can depend on , , , , , , , , and .
Step 1. Estimates for .
We take in (A.1) the partial derivatives for and . As a result,
| (A.5) | ||||
Furthermore, multiplying (A.5) for arbitrary with and integrating in over yields
with the right-hand side being given by
| (A.6) | ||||
We focus now on the situations where and . For the terms , integration by parts in leads to
| (A.7) | ||||
By means of Young’s inequality and the identities , one obtains
| (A.8) | ||||
For a constant which vanishes when , one can infer
| (A.9) |
Thus, after collecting A.7, A.8, and A.9, one obtains the bound
| (A.10) |
Concerning , expanding the derivatives leads to
which implies that is less than or equal to
| (A.11) |
Hence, the interpolation inequality and (A.11) yield for arbitrary that
| (A.12) | ||||
In (A.12), we used the elementary inequality . Then, the number is less than or equal to
| (A.13) |
It remains treating and . For a vector field on , we denote by arbitrary linear combinations of components of and derivatives of such, which are taken in and are of order , while the coefficients can depend on . Given a multi-index with , the relations imply
| (A.14) |
Therefore, in view of (A.14) and the definition of the tangential part , one may write
and integration by parts implies that the second line is of the same type as the first one. Thus,
| (A.15) | ||||
Due to on , the last integral in (A.15) reads
such that (A.15) eventually implies the bound
| (A.16) |
When it comes to , the corresponding estimates are less demanding compared to those for and one finds
| (A.17) | ||||
In order to collect the previous estimates, for fixed , we sum in A.10, A.13, A.16, and A.17 over all and . Moreover, we denote
with
As a result, one obtains the estimate
| (A.18) |
On the right-hand side of (A.18), all terms containing norms of the spaces
disappear in the respective base cases when or . Thus, inductively with respect to and , and by using a Grönwall argument for (A.18) with sufficiently small, one can obtain
and the estimate (A.4) when .
Step 2. Estimates for .
From (A.2) and the regularity obtained in Step 1, one can estimate the boundary values of at in for any by a constant of the type stated in (A.3). Therefore, by acting on (A.2) with , one obtains the estimates for by analysis similar to Step 1. The boundary values of at are then again bounded via (A.2) and Step 1. After acting with on (A.2), one can derive the estimates for . By induction over , the proof of A.1 is concluded. ∎
Appendix B Proof of 4.1
To prove 4.1, we proceed essentially along the lines of [CoronMarbachSueur2020, Lemma 9] and [ChavesSilva2020SmalltimeGE, Lemma 2.1], where Navier–Stokes and Boussinesq systems have been considered. That is, in Section B.1 below, assuming and , we obtain a priori estimates that are valid for smooth solutions. To conclude 4.1 from there, it seems the existing literature does not provide a suitable theory of strong solutions under general coupled Navier slip-with-friction boundary conditions for MHD. In particular, since the boundary conditions in (2.8) are generally non-symmetric, we are unaware of eigenvector bases for respectively coupled Stokes type problems, preventing us to employ usual Galerkin method arguments, e.g., as in [Temam2001], to obtain the and strong solutions to (2.8). When the domain is simply-connected, with being positive symmetric, and , such a basis of eigenvectors is available in [GuoWang2016]. If , the argument from [XiaoXin2013] also allow taking symmetric matrices and with ; these references apparently do not provide eigenvector bases under general symmetric boundary conditions, where , are symmetric and .
In D, 4.1 follows from the estimates in Section B.1 by a different approach. First, the initial data are approximated, similarly to [Kelliher2006, Appendix], in by functions satisfying the Navier slip-with-friction boundary conditions. Then, a Leray–Hopf weak solution is constructed as the limit of a sequence of solutions. This idea relies on the assumption , which guarantees that the sequence of strong solutions is defined on a fixed time interval .
B.1 Estimates for sufficiently regular solutions
To display the estimates for and simultaneously, the symmetric notations from Section 2.4.3 are employed. If is a Leray–Hopf weak solution to (2.8), then the functions obey the energy inequality (2.19) and a corresponding weak formulation for the Elsasser system
| (B.1) |
To begin with, a priori estimates for a related stationary problem are shown based on known results for the Navier–Stokes equations.
Lemma B.1.
Let , forces , a vector tangential to , and friction operators be arbitrary. Then, every solution with to the coupled Stokes type system
| (B.2) |
obeys the estimate
| (B.3) |
Proof.
In the case of uncoupled boundary conditions, where , the functions and both obey independent Stokes problems under Navier slip-with-friction boundary conditions. Thus, from [Guerrero2006, Pages 90-94], one has for each the estimates
and
which imply (B.3) by means of the triangle inequality. For the general case, we start with and observe that every solution to (B.2) satisfies
| (B.4) |
with
Since , after applying to (B.4) the result for uncoupled boundary conditions explained above, one finds
Inductively, if (B.3) is true for a fixed , one has and the known estimates for (B.4) lead to (B.3) with being replaced by . ∎
Remark B.2.
In what follows, the operator denotes the Leray projector in onto and thus, for any selection with , B.1 provides
| (B.5) |
The proof of 4.1 is now completed by means of the following steps.
Step 1. Basic energy estimates when .
We introduce for a positive parameter the quantity
Then, the strong energy inequality (2.19) with provides
Furthermore, by (2.1) and trace estimates, together with being fixed, one has for small and the bound
Thus, for sufficiently small it follows
As a result, by employing (2.1) similarly as in Section 3.5.2 and further utilizing Grönwall’s inequality, one obtains for the energy estimate
| (B.6) |
Therefore, by a contradiction argument, there exists and a possibly small time for which
Step 2. Higher order a priori estimates when .
We apply the Leray projector in (B.1) and multiply with . Subsequently, the results are added up and integrated over . Hereto, we denote for the auxiliary function
and the nonlinear terms
Lemma B.3.
Assume that , are symmetric and . For any small , and a constant which is reciprocal to , it holds
| (B.7) |
Proof.
One has to estimate several boundary integrals of the form
with and . Thanks to the symmetry assumptions, it follows that
for arbitrary , and similarly one can treat
Therefore, the inequality (B.7) can be inferred from (2.1), the transformations provided in Section 2.4.3, and the basic energy estimate (B.6). ∎
Remark B.4.
When are arbitrary, one can resort to parallel energy estimates as used in [CoronMarbachSueur2020, Lemma 9] for the Navier–Stokes equations. Hereto, one additionally multiplies in (B.1) with , which, in combination with the estimates that arise from multiplying (B.1) with , allows to absorb norms of the form , where
Thus, similarly to [Guerrero2006, Page 995], one can use interpolation arguments to draw a conclusion as in B.3.
It remains to further bound the integrals in (B.7). Applying embedding and interpolation inequalities, one obtains for any small constant the bound
| (B.8) | ||||
Moreover, the estimate (2.1) and Young’s inequality with provide
| (B.9) |
while (B.5) allows to infer
| (B.10) |
Thus, by combining B.6, B.7, B.8, B.9, and B.10, one obtains
Therefore, for sufficiently small parameters , one arrives at
| (B.11) | ||||
In order to apply Grönwall’s lemma in (B.11), first one utilizes the elementary inequality
such that for and sufficiently small one has the estimate
Thus, for a generic constant , the function
obeys , where . Taking small enough and integrating the latter differential inequality leads for to
Consequently, for some constant and all one has the estimate
Therefore, there exists and such that
Step 3. Additional estimates for when .
By taking in (B.1), multiplying the resulting equations with respectively, and integrating over , one obtains for
the estimate
Therefore, considerations similar to the previous steps lead for some constants and , a possibly small time and all , to the bound
noting that
Hence, there exists a time and a constant such that
Step 4. Conclusion.
Acknowledgments
Funding: This research was partially supported by National Key R&D Program of China under Grant No. 2020YFA0712000, National Natural Science Foundation of China under Grant No. 12171317, Strategic Priority Research Program of Chinese Academy of Sciences under Grant No. XDA25010402, and Shanghai Municipal Education Commission under Grant No. 2021-01-07-00-02-E00087.