Asymptotic Behaviors of Global Solutions to the Two-Dimensional Non-resistive MHD Equations with Large Initial Perturbations
aa r X i v : . [ m a t h . A P ] F e b Asymptotic Behaviors of Global Solutions to the Two-DimensionalNon-resistive MHD Equations with Large Initial Perturbations
Fei Jiang a , Song Jiang b a College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350108, China. b Institute of Applied Physics and Computational Mathematics, Huayuan Road 6, Beijing 100088, China.
Abstract
This paper is concerned with the asymptotic behaviors of global strong solutions to the incom-pressible non-resistive viscous magnetohydrodynamic (MHD) equations with large initial per-turbations in two-dimensional periodic domains in Lagrangian coordinates. First, motivated bythe odevity conditions imposed in [Arch. Ration. Mech. Anal. 227 (2018), 637–662], we provethe existence and uniqueness of strong solutions under some class of large initial perturbations,where the strength of impressive magnetic fields depends increasingly on the H -norm of theinitial perturbation values of both velocity and magnetic field. Then, we establish time-decayrates of strong solutions. Moreover, we find that H -norm of the velocity decays faster than theperturbed magnetic field. Finally, by developing some new analysis techniques, we show that thestrong solution convergence in a rate of the field strength to the solution of the correspondinglinearized problem as the strength of the impressive magnetic field goes to infinity. In addition,an extension of similar results to the corresponding inviscid case with damping is presented. Keywords:
Incompressible MHD fluids; damping; algebraic decay-in-time; exponentialdecay-in-time; viscosity vanishing limit.
1. Introduction
We investigate the asymptotic behaviors of the global (-in-time) solutions to the followingequations of incompressible magnetohydrodynamic (MHD) fluids with zero resistivity: ρv t + ρv · ∇ v + ∇ p − µ ∆ v = λM · ∇ M/ π,M t + v · ∇ M = M · ∇ v, div v = div M = 0 , (1.1)where the unknowns v := v ( x, t ), M := M ( x, t ) and p := p ( x, t ) denote the velocity, magneticfield and the sum of both magnetic and kinetic pressures of MHD fluids respectively, and thethree positive (physical) parameters ρ , µ and λ stand for the density, shear viscosity coefficientand permeability of vacuum, respectivly.The global well-posedness of the system (1.1), for which the initial data is a small perturbationaround a non-zero trivial stationary state (i.e., v = 0, and M is a non-zero constant vector ¯ M ,often called the impressive magnetic field), has been widely investigated, see [20, 36, 37] and [35]on the 2D and 3D Cauchy problems for (1.1) respectively, and see [25] and [29] on 2D and 3Dinitial-boundary value problems for (1.1) respectively. The existence of global solutions to the 2D Email addresses: [email protected] (Fei Jiang), [email protected] (Song Jiang)
Preprint submitted to Elsevier February 16, 2021 auchy problem for (1.1) with large initial perturbations was obtained by Zhang under strongimpressive magnetic fields [37]. As for the well-poseness of the 3D Cauchy and initial-boundaryvalue problems for (1.1) with large initial perturbations, to our best knowledge, all availableresults are about the local (-in-time) existence, see [5, 10, 11] for examples. We mention herethat the corresponding compressible case has been also widely studied, see [18, 19, 33] and thereferences cited therein. Since we are interested in the large-time behavior of a global solution andthe asymptotic behavior of (a family of ) solutions with respect to the strength of the impressivemagnetic field, we next briefly introduce the relevant progress on this topics, and our main resultsin this paper.
It has been physically conjectured that in MHD fluids, the energy is dissipated at a rate thatis independent of the resistivity [4]. Hence, one can easily conclude that a non-resistive MHDfluid may still be dissipative. At present, this conclusion has been mathematically verified for theglobal small perturbation solutions of equations (1.1) around a non-zero trivial stationary state,in which the impressive magnetic field M is given by¯ M = ¯ M N e N . (1.2)Here and in what follows ¯ M N is a non-zero constant, e N a unit vector with the N -th componentbeing 1 and N the spatial dimension. In addition, we define b := M − ¯ M N e N and h t i − := (1+ t ) − .The first mathematical verification result was probably given by Ren–Wu–Xiang–Zhang forthe 2D Cauchy problem of (1.1), they established the following time-decay [26]: h t i k − ε k ∂ kx ( v, b ) k L ( R ) c I , (1.3)where ε ∈ (0 , /
2) is any given, 0 k
2, and the initial data ( v , b ) of ( v, b ) belongs to H ( R ).Here and in what follows, c I will denote a generic positive constant, which may depend on theinitial data ( v , b ). Then, Tan–Wang further obtained the almost exponential decay of solutionsto the 2D/3D initial-boundary value problems [29]: h t i n − k ( v, b ) k H n +4 (Ω) c I , (1.4)where n >
4, the initial data ( v , b ) belongs to H n (Ω), and Ω is a 2D/3D layer domain withfinite height. Later, Abidi–Zhang also got a decay rate of solutions to the 3D Cauchy problem[1]: h t i / k ( v, b ) k H ( R ) c I , (1.5)where b = 0 and v ∈ H s with s ∈ (3 / , h t ik∇ v k L ( R ) + h t i / k ( v, b ) k H ( R ) c I . (1.6)In addition, Pan–Zhou–Zhu proved the existence of a unique global solution to the initial-boundary value problem of (1.1) defined in a 3D periodic domain T , and also obtained thefollowing time-decay as a byproduct [24]: h t i (3 − σ ) / k v k H s − ( T ) + X i h t i (1 − σ +2 i ) / k ∂ ( v, b ) k H s − i ) ( T ) c I , (1.7)2here 0 < σ < v , b ) ∈ H s +1 ( T ) with s > H -norm ofthe initially perturbed values of the velocity and magnetic field, and the initially perturbed datashould satisfy some odevity conditions imposed by Pan–Zhou–Zhu in [24]. Then, we show thatthe global solution enjoys the following decay in time: u h t i / k v k H ( T ) + h t ik v k H ( T ) + h t i / k b k H ( T ) c I , (1.8)where the initial datum ( b , v ) belongs to H . We refer the reader to Theorem 2.3 for the details(or see Theorems 2.1 and 2.2 for the version in Lagrangian coordinates). We should point outhere that by virtue of (1.8), the velocity in H -norm decays faster than the perturbation magneticfield, while in (1.6), the H -norm of the velocity enjoys the same decay rate as the one of theperturbation magnetic field. It is well-known that a non-resistive MHD fluid, the motion of which is described by thesystem (1.1), exhibits elastic characteristics. In particular, a MHD fluid strains when stretchedand will quickly return to its original rest state by the magnetic tension once the stress is removed[15]. This means that the magnetic tension will have stabilizing effects in the motion of MHDfluids. Moreover, the larger the strength of an impressive magnetic field is, the stronger thisstabilizing effect will be, see the inhibition phenomenon of flow instabilities by magnetic fields[14, 16, 31]. It is worth to mention here that Bardos–Sulem–Sulem used hyperbolicity of (1.1)with µ = 0 to establish an interesting global existence result of classical solutions with smallinitial data in the H¨older space H s ( R ) [2]; also see [3, 13] for the case of Sobolev spaces [3, 13]).Remark that such a result is not known for the system (1.1) in 3D when the magnetic field isabsent (the 3D incompressible Euler equations).In [37] Zhang also found an interesting mathematical result that the solution of the nonlinearsystem (1.1) converges to the solution of some linear equations obtained from (1.1) under theuse of the stream function as ¯ M → ∞ . More precisely, let M = ( ∂ , − ∂ ) T ( ψ − ¯ M x ), then thesystem (1.1) reduces to the following system: ρv t + ρv · ∇ v + ∇ p − µ ∆ v − ¯ M λ∂ ( ∂ ψ, − ∂ ψ ) T / π = λ ( ∂ ψ, − ∂ ψ ) · ∇ ( ∂ ψ, − ∂ ψ ) T / π,ψ t + v · ∇ ψ + ¯ M v = 0 , div v = 0 , ( v, ψ ) | t =0 = ( v , ψ ) , (1.9)where the subscript T denotes the transposition, ∂ i := ∂ x i and x i is the i -th component of x ∈ R .Thus, Zhang proved that the solution ( v, ψ ) of (1.9) converges to the solution ( v L , ψ L ) of the3ollowing linear pressureless equations as ¯ M → ∞ : ρv L t − µ ∆ v L − ¯ M λ∂ ( ∂ ψ L , − ∂ ψ L ) T / π = 0 ,ψ L t + ¯ M v L2 = 0 , ( v L , ψ L ) | t =0 = ( v , ψ ) . (1.10)However, no convergence rate in the strength ¯ M is given in [37]. In this paper we shall provea similar result in Lagrangian coordinates, and further provide a convergence rate in ¯ M of theglobal solution to (1.1) by developing some new analysis techniques. More precisely, we shallshow that the difference between the solution of (1.1) in Lagrangian coordinates and the solutionof the corresponding linearized system can be bounded from above by c I ¯ M − / , see Theorem 2.4for details.Roughly speaking, the proof of our two results mentioned above is based on a key observationthat the deviation function η of MHD fluid particles enjoys the estimate k ¯ M ∂ η k c I , where c I depends on the initial total mechanical energy. The above estimate can be extendedto the case of higher-order derivatives of ∂ η . Moreover, if η additionally satisfies the odevityconditions, then we formally have an important inverse relation: ∇ η ∝ c I / ¯ M , (1.11)see Section 2.1 for a detailed discussion. This relation intuitively not only reveals that the(nonlinear) solutions of (1.1) in Lagrangian coordinates can be approximated by the (linear)solutions of the corresponding linearized equations for ¯ M ≫ c I , but also provides a convergencerate in ¯ M . Since the nonlinear solutions can be approximated by the linear solutions, wenaturally expect the existence of strong solutions under some class of large initial perturbationsas in [37]. In fact, in [37] Zhang first obtained the (linear) solution of (1.10), then proved theexistence of the small error solution ( v − v L , ψ − ψ L ) as ¯ M ≫ c I , and finally got the large solution( v, ψ ) by adding the linear solution and the small error solution together. It is worth to mentionthat the relation (1.11) allows us to directly establish the existence of solutions under some classof large initial perturbations by one-step procedure, rather than Zhang’s three-step procedure.We mention that recently, some authors studied the case of inviscid, non-resistive MHDfluids with zero resistivity, i.e., the viscosity term in the system (1.1) is replaced by the velocitydamping term κρv with κ being the damping coefficient. Wu–Wu–Xu first proved the existenceof a unique global solution with algebraic time-decay to the 2D Cauchy problem, provided thatthe initial perturbation ( v , b ) is small in H n ( R ) with n sufficiently large [34]. Recently, Du–Yang–Zhou also obtained the existence of a unique global solution with exponential time-decayto the initial-boundary value problem in a strip domain Ω, provided that the initial perturbation( v , b ) around some non-trivial equilibrium is small in H (Ω) [8]. Motivated by [8, 34], we canextend our aforementioned results in this paper on the asymptotic behavior of solutions in theviscous case to the inviscid case with the damping term κρv , and show that for the inviscid casewith damping, the decay in time is exponentially fast, just as in [8], while the convergence ratein ¯ M as ¯ M → ∞ of a strong solution of the original nonlinear system to the solution of thecorresponding linear system is in the form of c I ¯ M − , which is faster than that for the viscouscase, see Theorem 2.5 for the details.Finally, we mention that the asymptotic behaviors of solutions with respect to other pa-rameters, such as the Mach and Alfv´en numbers, in MHD fluids have been also extensivelyinvestigated, see, for example, [6] and the references cited therein.4he rest of this paper is organized as follows: In Section 2 we introduce our main resultsincluding the existence of a unique strong solution with some class of large initial data to the 2Dequations (1.1) in a periodic domain in Lagrangian coordinates, and the time-decay of the strongsolution, and the convergence rate as ¯ M → ∞ of the strong solution as well as the extensionto the inviscid case with damping term, i.e., Theorems 2.1, 2.2, 2.4 and 2.5, the proofs of whichare given in Sections 3–6, respectively. Finally, in Section 7, we provide the proof of the localwell-posedness for the equations of viscous, non-resistive MHD fluids and inviscid, non-resistiveMHD fluids with damping, respectively.
2. Main results
In this section we describe the main results in details. To begin with, we reformulate theequations (1.1) in Lagrangian coordinates. Recalling that (1.1) is considered with in a 2D periodicdomain, we see, without loss of generality, that it suffices to consider the periodic domain T with T := R / Z .Let ( v, M ) be the solution of the 2D system of equations (1.1), and the flow map ζ be thesolution to ( ∂ t ζ ( y, t ) = v ( ζ ( y, t ) , t ) in T × R + ,ζ ( y,
0) = ζ ( y ) in T , (2.1)where ζ ( y ) satisfies det ∇ ζ = 1 and “det” denotes the determinant.Since v is divergence-free, then det ∇ ζ = 1 (2.2)as well as det ∇ ζ = 1. Thus, we define A T := ( ∇ ζ ) − := ( ∂ j ζ i ) − × . In particular, by virtue of(2.2), A = (cid:18) ∂ ζ − ∂ ζ − ∂ ζ ∂ ζ (cid:19) . We temporarily introduce some differential operators involving A , which will be used later.The differential operators ∇ A , div A and ∆ A are defined by ∇ A f := ( A k ∂ k f, A k ∂ k f ) T , div A ( X , X ) T := A lk ∂ k X l , curl A f := ∂ A k ∂ k f − A k ∂ k f and ∆ A f := div A ∇ A f for a scalar function f and avector function X := ( X , X ) T , where A ij denotes the ( i, j )-th entry of the matrix A . It shouldbe remarked that we have used the Einstein convention of summation over repeated indices, and ∂ k = ∂ y k . In addition, thanks to (2.2), we have ∂ k A ik = 0 . (2.3)Let R + = (0 , ∞ ), ν = µ/ρ and( u, B, q )( y, t ) = ( v, M, p/ρ )( ζ ( y, t ) , t ) for ( y, t ) ∈ T × R + . By virtue of the equations (1.1) and (2.1) , the evolution equations for ( ζ , u, q ) in Lagrangiancoordinates read as follows. ζ t = u,u t + ∇ A q − ν ∆ A u = λB · ∇ A B/ πρ,B t − B · ∇ A u = 0 , div A u = 0 , div A B = 0 . (2.4)5e can derive from (2.4) the differential version of magnetic flux conservation [15]: A jl B j = A jl B j , which yields B = ∇ ζ A T0 B . (2.5)Here and in what follows, the notation f as well as f denote the value of the function f at t = 0. If we assume A T0 B = ¯ M (i.e., B = ∂ ¯ M ζ ) , (2.6)where ¯ M is defined by (1.2) with N = 2, then (2.5) reduces to B = ∂ ¯ M ζ . (2.7)Here we should point out that B given by (2.7) automatically satisfies (2.4) and (2.4) . Moreover,from (2.7) we see that the magnetic tension in Lagrangian coordinates has the relation B · ∇ A B = ∂ M ζ . Let I denote a 2 × m = λ ¯ M / πρ , η = ζ − y and˜ A = (cid:18) ∂ η − ∂ η − ∂ η ∂ η (cid:19) . Consequently, under the assumption (2.6), the equations (2.4) are equivalent to the followingsystem: η t = u,u t + ∇ A q − ν ∆ A u = m ∂ η, div A u = 0 (2.8)where B = m ( ∂ η + e ) and A = ˜ A + I .For the well-posedness of (2.8) defined in T , we impose the initial condition:( η, u ) | t =0 = ( η , u ) in T . (2.9)Before stating our main results, we introduce some notations which will be frequently usedthroughout this paper.(1) Basic notations: I ∞ := R +0 := [0 , ∞ ), I T := (0 , T ) for 0 < T ∞ , I T := [0 , T ] for T ∈ R + ,Ω T := T × I T , R := R ( − , , ( w ) T := R w d y , α = ( α , α ) denotes the multi-index withrespect to the variable y .(2) Simplified Banach spaces: L r := L r ( T ) = W ,r ( T ) , H i := W i, ( T ) , H := H ,H iσ := { u ∈ H i | div u = 0 } , H i +11 := { η ∈ H i +1 | det ∇ ( η + y ) = 1 } ,H i +12 := { u ∈ H i | ∂ u ∈ H i } , H i +11 , := H i ∩ H i +12 ,X := { w ∈ X ∩ L | ( w ) T = 0 } , where X denotes a Banach space, 1 < r ∞ and i > C ( I T , H i ) := { u ∈ L (Ω T ) | u ∈ C ( I T \ Z , H i ) for some zero-measurable set Z ⊂ I T } , U i +1 ,T := { u ∈ C ( I T , H i +2 ) | u t ∈ C ( I T , H i ) , ( u, u t ) ∈ L ( I T , H i +3 × H i +1 ) } , (2.10) U T := { u ∈ U ,T | ( u ) T = 0 } . (4) Simplified function classes: for integer i > H i ∗ := { ξ ∈ H i | ξ ( y ) + y : R → R is a C -diffeomorphism mapping } ,H i, ∗ ,T := { η ∈ C ( I T , H i ) | η ( t ) ∈ H i ∗ for each t ∈ I T } ,C B, weak ( I T , L ) := C ( I T , L ) ∩ L ∞ ( I T , L ) ∩ C ( I T , L ) , C ( I T , H i +12 ) := { η ∈ C ( I T , H i ) | ∂ α ∂ η ∈ C B, weak ( I T , L ) for any | α | = i } , H i +1 , ∗ , ,T := { η ∈ C ( I T , H i +12 ) | η ( t ) ∈ H i ∩ H i ∗ for each t ∈ I T } , U i +1 T := { u ∈ C ( I T , H i ) | ∂ α u ∈ C B, weak ( I T , L ) for any | α | = i + 1 } , U i +1 T := { u ∈ U i +1 T | ( u ) T = 0 } . (5) Simplified norms: for integers i > n > i , k · k i := k · k H i ( T ) , k∇ i · k := X | α | = i k ∂ α · k , k · k i +1 , := q k · k i + k∇ i ∂ · k , E n,i ( t ) := k ∂ i ( ∇ η, u, m∂ η )( t ) k n − i , E n,i := E n,i (0) . (6) General constants: c i (1 i
3) and c κi (1 i
4) are fixed constants which may dependon the parameters ν and κ respectively, but not on m . If not stated explicitly, c , c , c κ , C and C κ will denote generic positive constants, which may vary from one place to another.Moreover, • c is independent of any parameter; • c and c κ may depend on ν and κ respectively ( but not on m ); • C depends on ν and p E , , and increases with respect to p E , . In particular, C onlydepends on ν and the norm k u k for the case η = 0; • C κ depends on κ and k ( u , η , m∂ η ) k , and increases with respect to k ( u , η , m∂ η ) k .In addition, A . B , A . B and A . κ B mean that A c B , A cB and A c κ B , respectively. Before stating the global existence result of solutions to the initial-value problem (2.8)–(2.9)in some classes of large data under strong magnetic fields, let us fist mention the heuristic idea,which leads us to study this topic.First, multiplying (2.8) with u in L , we obtain the basic energy identity:12 dd t (cid:0) k u k + k m∂ η k (cid:1) + ν k∇ A u k = 0 , (2.11)7hich implies k u k + k m∂ η k + 2 ν Z t k∇ A u k d τ = k u k + k m∂ η k =: I . (2.12)We call I the initial total mechanical energy, which includes the kinetic energy, and the pertur-bation magnetic energy that could be regarded as the potential energy. We easily see from (2.12)that k ∂ η k → m → ∞ for fixed I . This basic relation motivates us to expect that thedeformation quantity ∇ η may be small, when m is sufficiently large. Fortunately, this is indeedthe case for ( η, u ) satisfying the additional odevity conditions imposed by Pan–Zhou–Zhu in [24],see (2.14) for details.We rewrite (2.8) –(2.8) as a nonhomogeneous system of the Stokes equations: ( u t + ∇ q − ν ∆ u = F , div u = div ˜ A u, (2.13)where we have defined F := m ∂ η + N , N := N ν − ∇ ˜ A q , N ν := ∂ l ( N ν ,l , N ν ,l ) T and N νj,l := ν ( A kl ˜ A km + ˜ A ml ) ∂ m u j . This formally reveals that the system (2.13) can be approximated by thecorresponding linear system, if ∇ η is sufficiently small. Since the linear system admits a globalsolution, the nonlinear system (2.13) may also admit a global solution in some classes of largedata under the strong magnetic fields. This result read as follows. Theorem 2.1.
There are positive constants c > , c > and a sufficiently small constant c ∈ (0 , , such that for any ( η , u ) ∈ ( H ∩ H ∗ ) × H and m satisfying the incompressiblecondition div A u = 0 , the odevity conditions ( η , u )( y , y ) = ( η , u )( y , − y ) and ( η , u )( y , y ) = − ( η , u )( y , − y ) , (2.14) and the condition of strong magnetic fields m > c max (cid:26)(cid:16) c E , e c E , (cid:17) / , c E , e c E , (cid:27) , (2.15) then the initial value problem (2.8) – (2.9) admits a unique global strong solution ( η, u, q ) ∈ H , ∗ , ∞ ×U ∞ × C ( R +0 , H ) . Moreover, the solution ( η, u ) enjoys the stability estimate: E , ( t ) + Z t ( k u k + k m∂ η k )d τ . E , e c E , (1 + E , ) for any t > , (2.16) where k η k + k u k + k mη k , . p E , . In addition, k∇ q k . k ∂ u k + m ( k ∂ η k k ∂ η k + k ∂ η k ) , (2.17) k u k i +1 , . ( k ∂ u k + k ∂ η k k ∂ u k for i = 0; k ∂ u k i for i = 1 and , (2.18) k η k i +1 , . k ∂ η k i , k η k L ∞ . p k ∂ η k k ∂ η k . (2.19) Remark 2.1.
We can easily construct a family of ( η , u ) satisfies all the assumptions in The-orem 2.1, where η = 0 and u = 0. In fact, let ¯ η = ¯ u = (sin x cos x , − cos x sin x ). Because8iv ¯ η = div ¯ u = 0, for sufficiently small ε , there exists a function pair ( η , u ) enjoying the form( η , u ) = ( ε ¯ η + ε η r , ¯ u + εu r ), where ( η r , u r ) satisfies k η r k + k u r k c , − ∆ η r + ∇ β = 0 , div η r = div (cid:0) (¯ η + εη r1 )( − ∂ (¯ η + εη r2 ) , ∂ (¯ η + εη r2 )) T (cid:1) , ( η r ) T = 0 , (2.20)and − ∆ u r + ∇ β = 0 , div u r = ε − div ˜ A ( u + εu r ) , ( u r ) T = 0 , for a proof of which we refer to [17, Proposition 5.1] . It is easy to check that for sufficientlysmall ε , ( η , u ) is non-zero, belongs to ( H ∩ H ∗ ) × H , and satisfies div A u = 0 and (2.14).We further take m = ε − to immediately see that ( η , u ) and m satisfy the condition of strongmagnetic fields (2.15) for sufficiently small ε . Furthermore, k ( ∇ η , u , m∂ η ) k c for someconstant c independent of ε and m . Remark 2.2.
Noting that the initial perturbation magnetic field “ B − ¯ M ” is equal to ¯ M ∂ η ,we see from (2.15) that the strength of the impressive magnetic field increasingly depends on the H -norm of the initial velocity and perturbation magnetic field. Remark 2.3.
In the above theorem, we have assumed the condition ( η ) T = ( u ) T = 0. If(( η ) T , ( u ) T ) = 0, we can define ¯ η := η − ( η ) T and ¯ u := u − ( u ) T . Then, by Theorem2.1, there exists a unique global strong solution (¯ η, ¯ u, ¯ q ) to the initial value problem (2.8)–(2.9)with initial data (¯ η , ¯ u ). It is easy to verify that ( η, u, q ) := (¯ η + t ( u ) T + ( η ) T , ¯ u + ( u ) T , ¯ q ) isjust the unique strong solution of (2.8)–(2.9) with initial data ( η , u ). Remark 2.4.
Since ( η , u ) satisfies the odevity conditions (2.14), the strong solution ( η, u ) of(2.8)–(2.9) with an associated pressure function q enjoys the same odevity conditions as ( η , u )does, i.e.,( η , u )( y , y , t ) = ( η , u )( y , − y , t ) and ( η , u )( y , y , t ) = − ( η , u )( y , − y , t ) . (2.21)Now, we briefly describe the proof idea of Theorem 2.1. Motivated by (2.12), we naturallyexcept that for given value E , , the solution ( η, u ) enjoys the estimate k ( u, m∂ η ) k C for sufficiently large m. (2.22)Thus, we want to derive the a priori estimate of ( η, u ) like k ( u, m∂ η ) k C/ a priori assumption (2.22).However, if we follow the above idea, we find that the assumption (2.22) does not suffice toestablish the a priori estimate (2.23), expect further requiring the additional assumption: k∇ η k C. (2.24)9ore precisely, we can conclude that there are constants K (depending on E , , E , and m ) and δ , such that sup t T E , ( t ) K / , (2.25)if sup t T ( k ( ∇ η, m∂ η )( t ) k + k ∂ u ( t ) k ) K for any given T > { K / , K } /m ∈ (0 , δ ] . (2.27)The above a priori stability estimate, together with a local well-posedness result on (2.8)–(2.9), immediately yields Theorem 2.1. The detailed proof will be presented in Section 3. Inaddition, the proof of the existence of a unique local solution will be provided in Section 7.2. Itshould be remarked that the odevity conditions (2.21) play an important role in the derivationof the a priori stability estimate, see the key Lemma 3.2. Now, we turn to stating the asymptotic behaviors of the global solution given in Theorem2.1. To begin with, we state the result of the asymptotic behavior with respect to the time.
Theorem 2.2.
Let ( η, u, q ) be the global solution of (2.8) – (2.9) established in Theorem 2.1, then X i =1 (cid:16) h t i i k ∂ i η ( t ) k − i + h t i i +1 k ∂ i ( u, m∂ η )( t ) k − i + Z t (cid:0) h τ i i +1 k ∂ i u k − i + h τ i i k m∂ i +12 η k − i (cid:1) d τ (cid:17) + h t ik ( u, m∂ η )( t ) k + Z t h τ ik u k d τ C (2.28) and X i =1 (cid:18) h t i i +1 k u ( t ) k − i + h t i i +1 Z t e ν ( τ − t ) / k u k − i d τ (cid:19) (1 + m ) C for any t > . (2.29) Remark 2.5.
In view of (2.18), (2.19), (2.28) and Poinc´are’s inequality, we get from (2.28) that h t i / k mη ( t ) k L ∞ ( T ) + X i =1 (cid:18) h t i i k η ( t ) k − i, + Z t h τ i i k m∂ η k − i, d τ + h t i i +1 k ( u, m∂ η )( t ) k − i, + Z t h τ i i +1 k u k − i, d τ (cid:19) C, ∀ t > . (2.30) From (2.26) we easily get
K/m δ . We should point out that the term K / in (2.26) can be replaced by K to establish Theorem 2.1 with “1 /
2” in place of “1 /
4” in (2.15). Here we further choose K / in order to makesure that m − does not appear on the right hand of (2.28). u enjoys the same time-decay rate as m∂ η does. Similarresults were obtained for the global solution with small perturbation, see (1.3)–(1.7). Next, webriefly explain the basic idea how to get the faster time-decay of the velocity (2.29).It is well-known that any solution of the following homogeneous Stokes equations ( u t + ∇ q − ν ∆ u = 0 , div u = 0 (2.31)decays exponentially in time. Thus, for the nonhomogeneous case in (2.13), the algebraic time-decay of k u k − i depends on F where i = 1 and 2. Since the decay rate of the linear term k ∂ η k − i is h t i − ( i +1) , we naturally except that k u ( t ) k − i also decays in the rate of h t i − ( i +1) . Fortunately,by employing carefully estimates, one sees that the nonlinear term N does not prevent us fromobtaining the desired decay rate, see Section 4.2.Recalling that the solution η in Theorems 2.1 satisfies ζ := η ( y, t ) + y : R → R is a C diffeomorphism mapping , (2.32)one can easily recover the decay result in Theorem 2.2 from Lagrangian coordinates to the onein Eulerian coordinates: Theorem 2.3.
Let ( v, η, q ) be the global solution given in Theorem 2.1, ζ = η + y , ζ − denotesthe inverse function and ( v, M, p ) = ( u, m∂ ζ , q ) | y = ζ − . Then ( v, M, p ) belongs to C ( R +0 , H σ × H × H ) , and is a unique global strong solution of theinitial value problem (1.1) with initial data ( v, M ) | t =0 = ( u , m∂ ζ ) | y = ζ − in T . Moreover, thesolution enjoys the decay estimates X i =0 (cid:18) h t i i +1 k ( v, b )( t ) k − i + Z t (cid:0) h τ i i +1 k v k − i + h τ i i k b k − i (cid:1) d τ (cid:19) C and X i =1 (cid:18) h t i i +1 k v ( t ) k − i + h t i i +1 Z t e ν ( τ − t ) / k v k − i d τ (cid:19) (1 + m ) C, where b := M − ¯ M . Remark 2.6.
Since the periodic cell of T is bounded, we can also establish a result of almostexponential decay, where the decay rate is faster than Tan–Wang’s result (1.4) in [29] under thesame regularity on initial data. In fact, if the initial data ( η , u ) in Theorem 2.1 is in H n +1 × H n with n >
3, and m satisfies (2.15) with E n, in place of E , , then there exists a unique classicalsolution ( v, M, p ) ∈ C ( R +0 , H nσ × H n × H n ) to the initial value problem (1.1) with initial data( v, M ) | t =0 = ( u , m∂ ζ ) | y = ζ − . Moreover, n X i =0 (cid:18) h t i i +1 k ( v, b ) k n − i + Z t h τ i i +1 k v k n +1 − i d τ (cid:19) + n X i =1 Z t h τ i i k b k n − i d τ c I , n X i =1 (cid:18) h t i i +1 k v k n +1 − i + h t i i +1 Z t e ν ( τ − t ) / k v k n +2 − i d τ (cid:19) (1 + m ) c I , where b := M − ¯ M and the constant c I depends on ν and E n, .11ext, we further state the asymptotic behavior of solutions with respect to m . Noting thatthe inhomogeneous term N in F includes ∂ i η j for 1 i , j
2, and k∇ η k L ∞ . k ∂ η k K/ m by (2.25) and (3.9), we formally see that N → m → ∞ for fixed E , . Thus, the solution( η, u ) established in Theorem 2.1 converges in the rate m − / to the solution ( η L , u L ) of thecorresponding linearized system as m → ∞ . More precisely, we have Theorem 2.4.
Let ( η, u, q ) be the global solution of (2.8) – (2.9) given in Theorem 2.1. (1) Then, one can use the initial data of ( η, u ) to construct a function pair ( η r , u r ) ∈ H × H ,such that the following linear pressureless initial-value problem η L t = u L ,u L t − ν ∆ u L = m ∂ η L , div u L = 0 , ( η L , u L ) | t =0 = ( η + η r , u + u r ) (2.33) admits a unique strong solution ( η L , u L ) ∈ C ( R +0 , H σ ) × U ∞ . Moreover, • ( η L , u L ) also satisfies the odevity conditions (2.21) as ( η, u ) does; • the function pair ( η r , u r ) satisfies div( u + u r ) = div( η + η r ) = 0 , (2.34) k u r k . k ∂ η k k u k , and k ∂ j η r k − j . k ∂ η k k ∂ j η k − j , j = 0 , . (2.35)(2) Let ( η d , u d ) = ( η − η L , u − u L ) , then for any t ∈ R +0 , k η d ( t ) k + Z t k ( u d , m∂ η d ) k d τ Cm − , (2.36) X i =1 (cid:18) h t i i k ∂ i η d ( t ) k − i + h t i i +1 k ∂ i ( u d , m∂ η d )( t ) k − i + Z t ( h τ i i +1 k ∂ i u d k − i + h τ i i k m∂ i +12 η d k − i )d τ (cid:19) Cm − , (2.37) where the error function ( η d , u d ) enjoys the estimates (2.18) and (2.19) with ( η d , u d ) in placeof ( η, u ) . (3) If η further satisfies the additional regularity ∂ ( η , u ) ∈ H × H , (2.38) then for any t > , h t ik ( u d , m∂ η d )( t ) k + Z t h τ ik u d k d τ Cm − (1 + k ∂ η k + k ∂ u k ) . (2.39)We can follow the idea of deriving the estimate (2.28) to establish Theorem 2.4, the proof ofwhich will be presented in Section 5. Here we explain why one has to modify ( η , u ) to be initialdata of ( η L , u L ) in (2.33) , and why one imposes the additional regularity (2.38) in order to get(2.39). 121) Since the initial data u L | t =0 is divergence-free, i.e., div( u L | t =0 ) = 0, one has to adjust theinitial data u as in (2.33) .(2) If the initial data η of η is directly used to be initial data of the corresponding linearproblem, then we see that div η L = div η , and a time-decay of ∂ η d in (2.37) can not beexpected unless div η = 0. Hence, we have to modify η as in (2.33) , so that the obtainednew initial data “ η + η r ” is also divergence-free.(3) Subtracting (2.33) from (2.8)–(2.9), one obtains η d t = u d ,u d t + ∇ q − ν ∆ u d − m ∂ η d = N , div u d = − div ˜ A u, ( η d , u d ) | t =0 = − ( η r , u r ) . (2.40)Let α = ( α , α ), an application of ∂ α with | α | = 2 to (2.40) yields ∂ α u d t + ∂ α ∇ q − ν ∆ ∂ α u d − m ∂ ∂ α η d = ∂ α N ( | α | = 2) . (2.41)We multiply (2.40) by ∂ α η d and ∂ α u d in L , respectively, and integrate to obtain thefollowing two energy identities concerning derivatives of ( η d , u d ):dd t Z (cid:16) ∂ α η d · ∂ α u d + ν | ∂ α ∇ η d | (cid:17) d y + Z | m∂ ∂ α η d | d y − Z | ∂ α u d | d y = Z ∂ α N · ∂ α η d d y + Z ∂ α q∂ α div η d d y =: I , (2.42)and 12 dd t Z ( | ∂ α u d | + | m∂ α ∂ η d | )d y + ν Z |∇ ∂ α u d | d y = Z ∂ α N · ∂ α u d + Z ∂ α q∂ α div u d d y =: I . (2.43)It seems that the integral terms I for α = 0 and I could provide the convergence rate m − by directly employing the estimate k m∂ η k C . This idea, can not be directlyapplied to I for α = 0 due to the integral term R ∂ η ∂ u · ∂ u d d y hidden in I . Tocircumvent this difficult term, we rewrite it as follows. Z ∂ η ∂ u · ∂ u d d y = Z ∂ η d2 ∂ u · ∂ u d d y + Z ∂ η L2 ∂ u · ∂ u d d y. (2.44)Noting that ∂ η d2 can provide a convergence rate m − by using (2.42) and the energyidentity (2.43) with | α | = 1, thus the first term on the right hand of (2.44) also provide aconvergence rate m − . In addition, it is easy to formally derive from the odevity conditionof η L2 and the linear problem (2.33) that k ∂ η L2 k . k ∂ ∂ η L2 k Cm − (1 + k ∂ η k + k ∂ u k ) , (2.45)thus we need the regularity condition (2.38) to make the derivation procedure of (2.45)sense. Consequently, I also implies the convergence rate m − for α = 0 under theadditional condition (2.38). 13 .3. Extension to the inviscid case with damping We now describe how to extend the aforementioned results to the inviscid case with damp-ing. The equations of incompressible inviscid MHD fluids with zero resistivity and a low-orderdamping read as follows. ρv t + ρv · ∇ v + ∇ p − κρv = λM · ∇ M/ π,M t + v · ∇ M = M · ∇ v, div v = div M = 0 , (2.46)where κ > η t = u,u t + ∇ A q + κu = m ∂ η, div A u = 0 , (2.47)with initial data ( η, u ) | t =0 = ( η , u ) in T . (2.48)Then we have the following results, which can be regarded as an extension of Theorems 2.1, 2.2and 2.4. Theorem 2.5.
Let κ be a positive constant. There are constants c κ > and c κ ∈ (0 , , suchthat for any ( η , u ) ∈ ( H , ∩ H ∗ ) × H and κ satisfying div A u = 0 , the odevity conditions (2.14) and the condition of strong magnetic fields m > c κ max (cid:8) K / , K (cid:9) , (2.49) where A denotes the initial data of A and K := q c κ k ( η , u , m∂ η ) k e c κ k ( η ,u ,m∂ η ) k , the initial value problem (2.47) – (2.48) admits a unique global classical solution ( η, u, q ) ∈ H , ∗ , , ∞ × U ∞ × ( C ( R +0 , H ) ∩ C ( R + , H )) . Moreover, the solution ( η, u ) enjoys (1) Decay estimate: sup t > (cid:8) e c κ min { ,m } t ( k ( η, u )( t ) k + k mη ( t ) k , ) (cid:9) + Z ∞ e c κ min { ,m } τ ( k u k + k mη k , )d τ . κ K (2.50) and X i (cid:18) sup t > (cid:8) h t i i ( k ( η, u )( t ) k + k mη ( t ) k , ) (cid:9) + Z ∞ h τ i i ( k u k + k mη k , )d τ (cid:19) + Z t h τ ik η k , d τ C κ , (2.51) where we remark that the decay rates in (2.51) do not depend on m for fixed k ( η , u , m∂ η ) k . Stability around the solution ( η L , u L ) of the linear problem: sup t > (cid:8) e c κ min { ,m } t ( k ( η d , u d )( t ) k + k mη d ( t ) k , ) (cid:9) + Z ∞ e c κ min { ,m } τ ( k u d k + k mη d k , )d τ C κ m − (2.52) and sup t > (cid:8) h t i i ( k ( η d , u d )( t ) k + k mη d ( t ) k , ) (cid:9) + Z ∞ h τ i i ( k u d k + (1 + m ) k η d k , )d τ C κ m − , i = 1 , , (2.53) where the error function ( η d , u d ) := ( η − η L , u − u L ) and ( η L , u L ) ∈ C ( R + , H ) × U ∞ is the uniqueclassical solution of the following linear pressureless initial-value problem: η L t = u L ,u L t − κu L = m ∂ η L , div u L = 0 , ( η L , u L ) | t =0 = ( η + η r , u + u r ) , (2.54) for some ( η r , u r ) ∈ H × H satisfying (2.34) and k u r k . k ∂ η k k u k and k ∂ i η r k . k ∂ η k i k∇ ∂ i η k , i = 0 , . (2.55) Remark 2.7.
We have assumed ( η ) T = ( u ) T = 0 in Theorem 2.5. If (( η ) T , ( u ) T ) = 0,then the unique solution ( η, u, q ) of (2.47)–(2.48) enjoys the following form:( η, u, q ) = (¯ η + κ − ( u ) T (1 − e − κt ) + ( η ) T , ¯ u + ( u ) T e − t , ¯ q ) , where (¯ η, ¯ u, ¯ q ) is the unique solution, established in Theorem 2.5, of the following initial-valueproblem ¯ η t = ¯ u, ¯ u t + ∇ ¯ A ¯ q + κ ¯ u = m ∂ ¯ η, div ¯ A ¯ u = 0 , with initial data (¯ η, ¯ u ) | t =0 = ( η − ( η ) T , u − ( u ) T ) in T .The key idea in the proof of Theorem 2.5 is similar to that in the proof of Theorems 2.2–2.4.But there are remarkable differences between the decay results for the inviscid and viscous cases,which will be further explained in the proof process of Theorem 2.5, see all the footnotes inSection 6. We end this section by verifying the assertion in Remark 2.4. To this end, let ( η, u, q ) bea strong solution of (2.8)–(2.9), ψ = ( η , − η )( y , − y , t ), w = ( u , − u )( y , − y , t ) and p =( q , q )( y , − y , t ). Due to the uniqueness of strong solutions, we see that to get the desired15onclusion, it suffices to verify that ( ψ, w ) is also a strong solution of (2.8). It is obvious that( ψ, w ) satisfies (2.8) . Next, we show that ( ψ, w ) also satisfies (2.8) and (2.8) .Defining A = (cid:18) ∂ η + 1 − ∂ η − ∂ η ∂ η + 1 (cid:19) and B = (cid:18) ∂ ψ + 1 − ∂ ψ − ∂ ψ ∂ ψ + 1 (cid:19) , then ( B , B ) = ( A , A ) | y = − y and ( B , B ) = − ( A , A ) | y = − y . (2.56)Let χ be a function, and ˜ χ = χ ( y , − y ). Then, by (2.56), ∇ B ˜ χ = ( A i , −A i ) T ∂ i χ | y = − y (2.57)and div B w = div A u | y = − y = 0 , from which we see that ( η, u ) satisfies (2.8) .By (2.56) and (2.57), we further have∆ B ˜ χ = ∆ A χ | y = − y . (2.58)Thus, the identities (2.57) and (2.58) yield ∂ t w i + ∇ B p − ν ∆ B w i = ( ( ∂ t u + A i ∂ i q − ν ∆ A u ) | y = − y for i = 1 , − ( ∂ t u + A i ∂ i q − ν ∆ A u ) | y = − y for i = 2 , and ∂ ( ψ , ψ ) = ∂ ( η , − η ) | y = − y . Hence, we see that ( ψ, w ) satisfies (2.8) by the above two identity. This proves the property ofpreserving the odevity of strong solutions.
3. Proof of Theorem 2.1
This section is devoted to the proof of Theorem 2.1. First we derive some basic ( a priori )estimate for ( η, u, q ) under the a priori assumption (2.26) associated with the smallness condition(2.27) in Subsection 3.1, then further establish the stability estimate (2.16) in Subsection 3.2, andfinally, introduce a local well-posedness result for (2.8)–(2.9) and complete the proof of Theorem2.1 by a standard continuity argument in Subsection 3.3.
Let ( η, u, q ) be a solution of the initial-value problem (2.8)–(2.9) defined on Ω T for any given T >
0, where ( η , u ) belongs to H × H , and satisfies div A u = 0 and the odevity conditions(2.14). We recall here that the solution automatically satisfies ( η ) T = ( u ) T = 0 and the odevityconditions in (2.21) if ( η , u ) does. We further assume that ( η, u, q ) and K satisfy ( q ) T = 0,(2.26) and (2.27), where K > δ ∈ (0 ,
1] is a sufficiently smallconstant. It should be noted that the smallness of δ only depends on the parameter ν . Inaddition, by Young’s inequality, one easily finds from (2.26) and (2.27) thatsup t T k ∂ η ( t ) k Km m (2 K / + K ) δ. (3.1)Before deriving the energy estimates for ( η, u ), we introduce some basic inequalities andestablish some preliminary estimates of ( η, u ) by using the following two lemmas.16 emma 3.1. We have the following basic inequalities: (1)
Generalized Poinc´are’s inequalities: k f k . k ∂ f k for any f ∈ H satisfying Z − f ( y , y )d y = 0 , (3.2) k f k i ˜ c i k∇ i f k for any f ∈ H i , (3.3) where and the positive constant ˜ c i only depends on i . (2) Interpolation inequality: for any f ∈ H , k f k L ∞ . k f k k ( f, ∂ f ) k + k f k / k ( f, ∂ f ) k / k ∂ f k / k ∂ ( f, ∂ f ) k / . (3.4) In particularly, by Young’s inequality, k f k L ∞ . k ∂ ( f, ∂ f ) k + k ( f, ∂ f ) k . (3.5)(3) Product estimate: for any ( f, g, h ) ∈ H × H × L , Z | f gh | d y . q k f k k f k , k ( g, ∂ g ) k k h k . k ( f, ∂ f ) k k ( g, ∂ g ) k k h k . (3.6)(4) There is a constant δ ∈ (0 , , such that for any ( ξ − y ) ∈ H satisfying k∇ ( ξ − y ) k L ∞ δ , k∇ B f k . k∇ f k . k∇ B f k for any ∇ f ∈ L , (3.7) where B := ( ∇ ξ + I ) − T . Proof. (1) The inequalities (3.2) and (3.3) are obvious to get by virtue of classical Poinc´are’sinequality: k g k rL r (Ω) ˜ c r, Ω (cid:18) k∇ g k rL r (Ω) + (cid:12)(cid:12)(cid:12)(cid:12)Z Ω g d y (cid:12)(cid:12)(cid:12)(cid:12) r (cid:19) for any g ∈ W ,r (Ω) , where Ω = T n with n > r > c r, Ω depends only on r and Ω.(2) Now, we turn to the derivation of (3.4). Here and in what follows, we denote k f k L pyi := (cid:18)Z − | f | p d y i (cid:19) /p for f ∈ L p . Since C ( T ) is dense in H , it suffices to prove (3.4) for f ∈ C ( T ). Noting that, for any g ∈ C ( T ), sup y ∈ ( − , g ( y , y ) is a measurable function defined on ( − , , we use the Fubini theorem and the one-dimensional interpolation inequality (see [21, Theorem])to deduce that sup s ∈ ( − , | φ ( s ) | . k φ ( s ) k L ( − , k ( φ, φ ′ )( s ) k L ( − , for φ ∈ H ( − , . (3.8)17herefore, k f k L ∞ = sup y ∈ ( − , sup y ∈ ( − , | f ( y , y ) | ! . sup y ∈ ( − , (cid:16) k f k L y k ∂ f k L y + k f k L y (cid:17) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup y ∈ ( − , | f | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) / L y (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup y ∈ ( − , | ∂ f | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) / L y + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup y ∈ ( − , | f | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L y . k f k / k ( f, ∂ f ) k / k ∂ f k / k ∂ ( f, ∂ f ) k / + k f k k ( f, ∂ f ) k , which gives (3.4).(3) Finally, we prove (3.6). Recalling that C ( T ) is dense in H and H , it suffices to prove(3.6) for ( f, g ) ∈ C ( T ). Let ( f, g ) ∈ C ( T ), we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup y ∈ ( − , | f | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L y . q k f k k f k , , sup y ∈ ( − , k g k L y . p k g k k ( g, ∂ g ) k , which, together with the Fubini theorem and H¨older’s inequality, implies Z | f gh | d y . Z T sup y ∈ ( − , | f | Z T | gh | d y d y . sup y ∈ ( − , k g k L y Z T sup y ∈ ( − , | f |k h k L y d y . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup y ∈ ( − , | f | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L y sup y ∈ ( − , k g k L y k h k . q k f k k f k , k k ( g, ∂ g ) k k h k . Therefore, an application of Young’s inequality yields (3.6).(4) The equivalent estimate (3.7) holds obviously. (cid:3)
Lemma 3.2.
Under the condition (3.1) with sufficiently small δ ∈ (0 , , we have (1) Estimates for η : k ∂ i η j k L ∞ . ( k ∂ η k for i = 1 and j , k ∂ η k for i = 2 , (3.9) k η k i +1 , . k ∂ η k i for i , (3.10) k η k L ∞ . p k ∂ η k k ∂ η k . (3.11)(2) Estimates for u : k u k i +1 , . ( k ∂ u k + k ∂ η k k ∂ u k for i = 0 , k ∂ u k i for i = 1 and , (3.12) k ∂ i u k L ∞ . ( k ∂ u k for i = 1 , k ∂ u k for i = 2 . (3.13)183) Product estimates: Z | ∂ α ηgh | d y . ( k ∂ η k | α | k g k k h k for α = 0 , k ∂ ∂ α η k k g k k h k for α = 0 , (3.14) Z | ∂ β ugh | d y . ( k ∂ u k | β | k g k k h k for β = 0 , k ∂ ∂ β u k k g k k h k for β = 0 (3.15) for any ( g, h ) ∈ H × L , where α and β satisfy | α | and | β | = 2 . Remark 3.1.
Thanks to (3.1) and (3.9), one has k∇ η k L ∞ . k ∂ η k . δ, (3.16)which, together with (3.7), yields the following equivalent estimate: for sufficiently small δ , itholds that k∇ A f k . k∇ f k . k∇ A f k for any ∇ f ∈ L . (3.17) Proof. (1) By (3.5) we have that for 1 i, j k ∂ i η j k L ∞ . k ∂ ∂ i ( η j , ∂ η j ) k + k ∂ i ( η j , ∂ η j ) k . (3.18)Since η ( y , y ) is a odd function with respect to y for any given y , one sees that Z − ∂ k η d y = 0 for 0 k . In addition, η ( y , y ) is a periodic function with respect to y for any given y , then Z − ∂ l ∂ k η d y = 0 for 1 k + l l. Using (3.2) and the above two relations, we find that k ∂ k η k . k ∂ ∂ k η k for 0 k k ∂ l ∂ k η j k . k ∂ l +12 ∂ k η j k for 0 l + k l. (3.20)Putting (3.18)–(3.20) together, we get k ∂ i η j k L ∞ . ( k ∂ η j k for i = 1 and j = 2 , k ∂ η j k for i = 2 and j = 1 , . (3.21)To obtain (3.9), we next show that (3.21) also holds for ( i, j ) = (1 , I + ∇ η ) = 1, one gets from Sarrus’ rule thatdiv η = ∂ η ∂ η − ∂ η ∂ η . (3.22)19n particular, ∂ η = ∂ η ∂ η − ∂ η ∂ η − ∂ η . (3.23)Applying ∇ to (3.23), and then multiplying the resulting identity by ∇ ∂ η in L , we obtain k∇ ∂ η k = Z ∇ ( ∂ η ∂ η − ∂ η ∂ η − ∂ η ) · ∇ ∂ η d y. (3.24)Making use of H¨older’s inequality, (3.3), (3.6), (3.19) and (3.21), we infer from (3.24) that k ∂ η k . k ∂ η k ( k ∂ η k + k ∂ η k L ∞ k ∂ η k + k ∂ η k L ∞ k ∂ η k )+ Z |∇ ∂ η || ∂ η ||∇ ∂ η | d y + Z | ∂ η ||∇ ∂ η ||∇ ∂ η | d y . k ∂ η k ( k ∂ η k + k ∂ η k k ∂ η k + ( k ∂ η k L ∞ + k ∂ η k ) k ∂ η k ) . By (3.1), (3.21) and Young’s inequality, we further deduce from the above estimate that k ∂ η k . k ∂ η k . (3.25)We immediately see from (3.18) and (3.25) that k ∂ η k L ∞ . k ∂ η k . This completes the proofof (3.9).Thanks to (3.9), we can get from (3.23) that k ∂ η k . k ∂ η k . (3.26)Putting (3.19), (3.25) and (3.26) together, one concludes k∇ η k i − . k ∂ η k i for 1 i . (3.27)Thus, from (3.3), (3.19), (3.26) and (3.27), the estimate (3.10) follows immediately. Finally, theestimate (3.11) is obvious by virtue of (3.4) and (3.10).(2) Since u enjoys the same odevity and the same periodicity as η does, we obtain, similarlyto (3.18)–(3.21), that k ∂ k u k . k ∂ ∂ k u k for 0 k , (3.28) k ∂ l ∂ k u j k . k ∂ l +12 ∂ k u j k for 0 l + k l, (3.29) k ∂ u k L ∞ . k ∂ ∂ ( u , ∂ u ) k + k ∂ ( u , ∂ u ) k (3.30)and k ∂ i u j k L ∞ . ( k ∂ u j k for i = 1 and j = 2 , k ∂ u j k for i = 2 and j = 1 , . (3.31)By (2.8) and the definition of ˜ A , we find thatdiv u = − div ˜ A u, (3.32)where ˜ A = (cid:18) ∂ η − ∂ η − ∂ η ∂ η (cid:19) . (3.33)20f we apply the norm k · k to (3.32), we get k ∂ u k . k ∂ u k + k ∂ η k L ∞ k ∂ u k + δ k ∂ u k , which, together with (3.9) and (3.28), gives k ∂ u k . k ∂ u k + k ∂ η k k ∂ u k (3.34)for sufficiently small δ .In addition, following the same process as in the derivation of (3.25), we easily deduce from(3.32) that k ∂ u k . k ∂ u k . (3.35)Thus, the estimate (3.12) follows from (3.1), (3.3), (3.28), (3.34) and (3.35).Plugging (3.35) into (3.30), we get k ∂ u k L ∞ . k ∂ u k , which, together with (3.31), yields(3.13).(3) The estimate (3.14) follows from (3.6), (3.20) and (3.27). Similarly, in view of (3.6),(3.28), (3.29) and (3.35), we see that (3.15) holds. This completes the proof of Lemma 3.2. (cid:3) Now, we proceed to derive some basic energy estimates for ( η, u, q ). Lemma 3.3.
Under the condition (3.1) with sufficiently small δ , we have k∇ q k i . k ( u, ∂ u ) k k∇ u k + m k ∂ η k k ∂ η k for i = 0; k ∂ u k k∇ u k or k ∂ u k + m ( k ∂ η k k ∂ η k + k ∂ η k ) for i = 1 , (3.36) k∇ q t k . k ∂ u k k∇ u k (1 + k∇ η k + k∇ u k ) + m k ∂ η k ( k ∂ η k k∇ u k + k ∂ u k ) . (3.37) Remark 3.2.
The estimate (3.37) will be used in the derivation of the error estimate (2.39) inTheorem 2.4, see the last term in (5.15).
Proof. (1) By (2.3) and (2.8) , we havediv A u t = − div A t u = − div( A T t u ) . (3.38)Multiplying (2.8) by ∇ A q in L and using (3.22), (3.38), we integrate by parts and recall thefact div A ∆ A u = 0 (3.39)to infer that k∇ A q k = Z ( A T t u ) · ∇ q d y + m Z ∂ η · ∇ ˜ A q d y + m Z ∂ ( ∂ η ∂ η − ∂ η ∂ η ) ∂ q d y. (3.40)Exploiting (3.6) and (3.14), we deduce from the above identity that k∇ A q k . ( k ( u, ∂ u ) k kA t k + m ( k∇ η k L ∞ k ∂ η k + k ∂ η k k ∂ η k )) k∇ q k . , (3.9) and (3.17), we obtain (3.36) with i = 0.(2) Applying div A to (2.8) , using then (2.8) , (3.39) and the first identity in (3.38), we arriveat ∆ q = f, (3.41)where f :=( ∂ u ∂ − ∂ u ∂ ) u + ( ∂ u ∂ − ∂ u ∂ ) u − (div ˜ A ∇ A q + div ∇ ˜ A q )+ m ( ∂ div η + ( ∂ η ∂ − ∂ η ∂ ) ∂ η + ( ∂ η ∂ − ∂ η ∂ ) ∂ η ) . Multiplying (3.41) by ∆ q in L , we use the regularity theory of elliptic equations to get k∇ q k . k ∆ q k . Z | f ∆ q | d y, (3.42)where the integral term on the right hand side can be estimated as follows. Z | f ∆ q | d y . ( k ∂ u k k∇ u k or k ∂ u k + m ( k ∂ η k ( k∇ η k L ∞ + k ∂ η k ) + k ∂ η k L ∞ k ∂ η k )+ (1 + k∇ η k L ∞ )( k∇ η k L ∞ k∇ q k + k ∂ η k k∇ q k )) k ∆ q k . ( k ∂ u k k∇ u k or k ∂ u k + m ( k ∂ η k k ∂ η k + k ∂ η k ) + δ k∇ q k ) k ∆ q k , (3.43)where we have used (3.14), (3.15) and (3.22) in the first inequality, and (3.9) and (3.16) in thesecond inequality. Consequently, combining (3.42) with (3.43), one obtains (3.36) with i = 2.(3) Applying ∇ to (2.8) and multiplying the resulting identity by ∇ u t in L , we have Z |∇ u t | d y = Z ∇ ( m ∂ η + ν ∆ A u − ∇ A q ) : ∇ u t d y. It is easy to see from the above identity that k∇ u t k . m k ∂ η k + k∇ u k (1 + k∇ η k ) + k∇ q k , which, combined with (3.36) with i = 1, implies that for sufficiently small δ , k∇ u t k . m k ∂ η k + k∇ u k (1 + k∇ η k + k∇ u k ) . (3.44)An application of ∂ t to (3.41) yields ∆ q t = f t . Thus, multiplying this equation with ∆ q t in L and applying the regularity theory of elliptic equations, we get k∇ q t k . k ∆ q t k . Z | f t ∆ q t | d y. (3.45)Furthermore, it is easy to verify that Z | f t ∆ q t | d y . ( m k ∂ η k k ∂ u k + k ∂ u k k∇ u t k + k ∂ u k k∇ q k + δ k∇ q t k ) k ∆ q t k . (3.46)Consequently, if we making use of (3.36) with i = 1, and (3.44), (3.46) and Young’s inequality,we obtain (3.37) from (3.45). (cid:3) emma 3.4. Under the conditions of (2.26) – (2.27) with sufficiently small δ , we have dd t k∇ u ( t ) k + ν k u ( t ) k . (1 + m ) k m∂ η k , t > and dd t k∇ ( u, m∂ η )( t ) k + ν k u ( t ) k . F := k ∂ η k k m∂ η k + k∇ η k k ∂ u k + k∇ η k k ∂ u k ( m k ∂ η k k ∂ η k + k ∂ u k ) , t > . (3.48) Proof.
Let α satisfy 1 | α |
2. Applying ∂ α to (2.13) yields ∂ α ( u t + ∇ q − ν ∆ u − m ∂ η ) = ∂ α ( N ν − ∇ ˜ A q ) . If we multiply the above identity by ∂ α u in L , integrate by parts and make use of (3.32), we get12 dd t k ∂ α u k + ν k∇ ∂ α u k = I + I − m Z ∂ η · ∂ α u d y for | α | = 1;12 dd t k m∂ ∂ α η k for | α | = 2 , (3.49)where I := − Z ∂ α N νj,l ∂ l ∂ α u j d y, I := Z ( ∂ α − ∇ ˜ A q · ∂ α + u − ∂ α q∂ α div ˜ A u )d y, (3.50) α − := ( ( α − , α ) for α = 0;( α , α −
1) for α > α + := ( ( α + 1 , α ) for α = 0;( α , α + 1) for α > . (3.51)Next, we consider two cases.(1) Case | α | = 1 . In view of (3.14) and (3.16), we find that I . (1 + k∇ η k L ∞ ) k ∂ η k k u k k∇ u k . δ k u k . (3.52)Recalling (2.26) and (2.27), we see that k ∂ η k ( k ∂ u k + k ( ∇ η, m∂ η ) k ) + ( k∇ η k + k∇ η k ) /m . δ. (3.53)The integral I can be estimated as follows. I . k ∂ η k k∇ q k k∇ u k . k ∂ η k k u k ( k ∂ u k k u k + m k ∂ η k k ∂ η k ) . δ ( k m∂ η k + k u k ) , (3.54)where we have used (3.9) and (3.14) in the first inequality, (3.12) with i = 0 and (3.36) with i = 0 in the second inequality, and (3.53) and Young’s inequality in the last inequality.Finally, putting (3.52) and (3.54) into (3.49), and utilizing (3.3), (3.20) and Young’s inequality,we get (3.47).(2) Case i = 2 .
23e integrate by parts and use (3.15) to deduce Z ∂ ˜ A ∂ u · ∂ ∂ u d y = − Z ∂ ˜ A ∂ ∂ u · ∂ u d y + Z ∂ ∂ ˜ A ∂ ( ∂ u · ∂ u )d y . k∇ η k k∇ ∂ u k k ∂ u k + k∇ ∂ η k ( k∇ u k + k ∂ u k L ∞ k∇ u k ) . (3.55)Thus, the integral I can be estimated as follows. I . ((1 + k∇ η k L ∞ )( k∇ η k L ∞ k∇ u k + k∇ ∂ η k ( k∇ u k + k ∂ u k L ∞ ) + k∇ η k ( k ∂ u k L ∞ + k∇ ∂ u k )) + ( k ∂ η k L ∞ + k∇ ∂ η k )( k∇ ∂ η k k ∂ u k L ∞ + k∇ η k k∇ u k L ∞ )) k∇ u k . δ k u k + k∇ η k k ∂ u k k u k , (3.56)where we have used (3.14) and (3.55) in the first inequality, and (3.4), (3.13), (3.16) and (3.53)in the second inequality.Similarly, the second integral I can be estimated as follows, using (3.36) with i = 1. I . ( k ∂ η k k u k + k∇ η k k ∂ u k ) k∇ q k . ( k ∂ η k k u k + k∇ η k k ∂ u k )( k ∂ u k + m k ∂ η k k ∂ η k ) , (3.57)Consequently, plugging the above two estimates into (3.49), and using Young’s inequality, (3.1),(3.3) and (3.53), we get (3.48). (cid:3) Lemma 3.5.
Under the conditions of (2.26) – (2.27) with sufficiently small δ , we have dd t ν k∇ η k + X | α | =2 (cid:18)Z ∂ α η · ∂ α u d y + νF α (cid:19) + k m∂ η k k∇ u k + cδ k u k , (3.58) where F α := R ∂ α A ∂ η · ∂ ∂ α η d y . Proof.
Applying ∂ α ( | α | = 2) to (2.13) , and multiplying then the resulting identity by ∂ α η in L , one sees thatdd t (cid:18) ν k∇ ∂ α η k + Z ∂ α η · ∂ α u d y (cid:19) + k m∂ ∂ α η k = k ∂ α u k + I + I , (3.59)where I := − Z ∂ α N νj,l · ∂ l ∂ α η j d y, I := Z ( ∂ α − ∇ ˜ A q · ∂ α + η + ∂ α q∂ α div η )d y, and α − and α + are defined by (3.51).Noting that Z ∂ α A ∂ u · ∂ ∂ α η d y = dd t F α − Z ∂ t ( ∂ α A ∂ ∂ α η ) · ∂ η d y, arguing similarly to the derivation of (3.56) and (3.57), we find that | I | c ( k ∂ η k k u k + k∇ η k ( k ∂ η k k u k (1 + k∇ η k ) + k ∂ η k k ∂ u k )) − dd t F α , (3.60) | I | . k ∂ η k k∇ η k ( k ∂ u k + m k ∂ η k k ∂ η k ) . (3.61)Inserting (3.60) and (3.61) into (3.59), and using (3.1), (3.3), (3.53) and Young’s inequality, weobtain (3.58). (cid:3) emma 3.6. Under the assumptions of (2.26) – (2.27) with sufficiently small δ , we have dd t k∇ − i ∂ i ( u, m∂ η ) k + ν k ∂ i u k − i . ( m k ∂ η k ( k ∂ η k k ∂ η k + k ∂ η k ) for i = 1 ,F for i = 2 , (3.62) and dd t ν k∇ − i ∂ i η k + X | α | =2 − i (cid:18)Z ∂ α ∂ i η · ∂ α ∂ i u d y + νF α (cid:19) + ( i − F + k m∂ i +12 η k − i k ∂ i u k − i + ( c ( k ∂ η k k ∂ η k k ∂ u k + δ k ∂ u k ) for i = 1 ,F + k ∂ η k k ∂ η k k ∂ u k + δ k ∂ u k for i = 2 , (3.63) where F := k ∂ η k k ∂ u k k u k k ∂ u k + k ∂ η k ( k ∂ u k + k m∂ η k + m k ∂ η k k ∂ u k )+ k m∂ η k ( k ∂ η k k ∂ u k + k m∂ η k k ∂ η k ) ,F α := for α = 0 , Z ∂ α ∂ ˜ A ∂ η · ∂ ∂ ∂ α η d y for α = 1 ,F := Z ( ∂ ( A k ˜ A k + ˜ A ) ∂ η · ∂ ∂ η + ∂ ( A k ˜ A k + ˜ A ) ∂ η · ∂ η ))d y,F := m k ∂ η k ( k ∂ η k k ∂ η k + k ∂ η k ) . Proof.
Let α satisfy | α | = 2 − i . Similar to the derivation of (3.49) and (3.59), one gets12 dd t (cid:0) k ∂ α ∂ i u k + k m∂ α ∂ i +12 η k (cid:1) + ν k ∂ α ∇ ∂ i u k = I + I (3.64)and dd t (cid:18) ν k∇ ∂ α ∂ i η k + Z ∂ α ∂ i η · ∂ α ∂ i u d y (cid:19) + k m∂ α ∂ i +12 η k = k ∂ α ∂ i u k + I + I , (3.65)where I := − Z ∂ α ∂ i N νj,l · ∂ l ∂ α ∂ i u j d y, I := Z ( ∂ α ∂ i − ∇ ˜ A q · ∂ α ∂ i +12 u − ∂ α ∂ i q∂ α ∂ i div ˜ A u )d y,I := − Z ∂ α ∂ i N νj,l · ∂ l ∂ α ∂ i η j d y, I := Z ( ∂ α ∂ i − ∇ ˜ A q · ∂ α ∂ i +12 η + ∂ α ∂ i q∂ α ∂ i div η )d y. Next, we estimate I – I by considering the cases i = 1 and i = 2 respectively.(1) Case i = 1From (3.1), (3.13) and (3.15) we get I . k ∂ η k k ∂ u k . δ k ∂ u k . (3.66)The integral I can be estimated in the following way. I . k ∂ η k k ∂ u k k∇ q k . m k ∂ η k ( k ∂ η k k ∂ η k + k ∂ η k ) k ∂ u k + δ k ∂ u k , (3.67)25here we have used (3.9) and (3.13)–(3.15) in the first inequality, and (3.36) with i = 1, (3.53)and Young’s inequality in the second inequality.Noting that for α = 1, Z ∂ α ∂ A ∂ u · ∂ ∂ ∂ α η d y = dd t F α − Z ∂ t ( ∂ α ∂ A ∂ ∂ ∂ α η ) · ∂ η d y, we argue, similarly to (3.66) and (3.67), to infer that I c k ∂ η k k ∂ η k k ∂ u k − dd t F α , (3.68) I . k ∂ η k k ∂ η k k∇ q k . δ ( k m∂ η k + k ∂ u k ) . (3.69)Thanks to the four estimates (3.66)–(3.69) and Young’s inequality, we derive (3.62) and (3.63)from (3.64) and (3.65) with i = 1, respectively.(2) Case i = 2 . By employing a partial integration, (3.13) and (3.15), one can easily see that Z ∂ ˜ A ∂ u · ∂ u d y = Z ∂ ∂ η ∂ ∂ u · ∂ u d y − Z ∂ η ∂ ( ∂ u · ∂ u )d y . k ∂ η k k ∂ u k k ∂ u k + k ∂ η k k ∂ u k . Similarly, we can also obtain Z ∂ A k ˜ A k ∂ u · ∂ u d y . k ∂ η k k ∂ u k k ∂ u k + k ∂ η k k ∂ u k . Thanks to the above two estimates, we easily deduce that I . k ∂ η k k ∂ u k k ∂ u k + k ∂ η k k ∂ u k + Z ∂ ˜ A ∂ u · ∂ u d y + Z ∂ ˜ A k ˜ A k ∂ u · ∂ u d y . k ∂ η k k ∂ u k k ∂ u k + k ∂ η k k ∂ u k . Noting that Z ( ∂ ( A k ˜ A k + ˜ A ) ∂ u · ∂ ∂ η + ∂ ( A k ˜ A k + ˜ A ) ∂ u · ∂ η ))d y dd t F + c k ∂ η k k ∂ η k k ∂ u k , (3.70)making use of (3.9) and (3.13)–(3.15), we can control I – I as follows. I . ( k ∂ η k k ∂ u k + k ∂ η k k ∂ u k )( k ∂ u k k u k + m ( k ∂ η k k ∂ η k + k ∂ η k )) ,I − dd t F + c k ∂ η k k ∂ η k k ∂ u k ,I . ( k ∂ η k k ∂ η k + k ∂ η k )( k ∂ u k + m ( k ∂ η k k ∂ η k + k ∂ η k )) . Utilizing the estimates for I – I , (3.1), and (3.2), (3.53) and Young’s inequality, we get (3.62)and (3.63) from (3.64) and (3.65) with i = 2, respectively. (cid:3) .2. Stability estimates With the energy estimates in Lemmas 3.4–3.6 in hand, we are in a position to establish thestability estimate (2.16).By virtue of (3.16), it is easy to see that | F α | δ k∇ η k , (3.71) | F α | δ k ∂ η k , (3.72) | F α | δ k ∂ η k . (3.73)Thus, we can use (3.3), (3.53), (3.71) and Young’s inequality to derive from (3.48) and (3.58) thetwo-order energy inequality:dd t E + c ( k u k + k m∂ η k ) . k ( ∇ η, u, m∂ η ) k ( k ∂ u k + k m∂ η k ) (3.74)for sufficiently small δ , where E := c k∇ ( u, m∂ η ) k + ν k∇ η k + X | α | =2 (cid:18)Z ∂ α η · ∂ α u d y + νF α (cid:19) , satisfying E , . E . E , . (3.75)Furthermore, if one utilizes (3.3), (3.53), (3.72) and Young’s inequality, one gets from (3.62)and (3.63) that dd t E i + c ( k ∂ i u k − i + k m∂ i +12 η k − i ) . ( i = 1; F for i = 2 , (3.76)where F := cF + F + k ∂ η k k ∂ η k and E i := c k∇ − i ∂ i ( u, m∂ η ) k + ν k∇ − i ∂ i η k + X | α | =2 − i (cid:18)Z ∂ α ∂ i η · ∂ α ∂ i u d y + νF α (cid:19) + ( i − F , satisfying E ,i . E i . E ,i . (3.77)From (3.76) with i = 1 we find that E , ( t ) + c Z t ( k ∂ u k + k m∂ η k )d τ . E , . (3.78)Thanks to (3.75), an application of Gronwall’s lemma to (3.74) yields E , . E , e c R T ( k ∂ u k + k m∂ η k )d τ . δ ∈ (0 , δ δ , E , ( t ) + Z t ( k ∂ u k + k m∂ η k )d τ c E , e c E , / t ∈ I T , (3.79)where c >
4. In addition, due to (3.78) and (3.79), we can obtain from (3.74) that Z t ( k u k + k m∂ η k )d t . E , (1 + E , e c E , ) . (3.80)Finally, if we take K := q c E , e c E , > . (3.81)we complete the derivation of (2.25) under the conditions (2.26) and (2.27) with δ δ . We start with introducing a local (-in-time) well-posedness result for the initial value problem(2.8)–(2.9) and a result concerning diffeomorphism mappings.
Proposition 3.1.
Let ( η , u ) ∈ H × H satisfy k ( ∇ η , u ) k B and div A u = 0 , where B is a positive constant, ζ := η + y and A is defined by ζ . Then there is a constant δ ∈ (0 , ,such that for any ( η , u ) satisfying k∇ η k , δ , (3.82) there exist a local existence time T > (depending possibly on B , ν , m and δ ) and a unique localstrong solution ( η, u, q ) ∈ C ( I T , H ) × U ,T × C ( I T , H ) to the initial value problem (2.8) – (2.9) ,satisfying < inf ( y,t ) ∈ R × I T det( ∇ η + I ) and sup t ∈ I T k∇ η k , δ . Proof.
The proof of Proposition 3.1 will be given in Section 7. (cid:3)
Remark 3.3.
If ( η , u ) in Proposition 3.1 further satisfies ( η , u ) ∈ H × H and the odevityconditions (2.14), then ( η, u ) belongs to C ( I T , H ) × U T and satisfies the odevity conditions(2.21) as ( η , u ) does. Proposition 3.2.
There is a positive constant δ , such that for any ϕ ∈ H satisfying k∇ ϕ k , δ , we have (after possibly being redefined on a set of measure zero) det( ∇ ϕ + I ) > / and ψ : R → R is a C homeomorphism mapping , (3.83) where ψ := ϕ + y . Proof.
In view of (3.4), we see that det( ∇ ϕ + I ) > / k∇ ϕ k L ∞ . k∇ ϕ k , δ . Thus,we easily verify that for sufficiently small δ , ψ : R → R is a C homeomorphism mapping,please refer to [15, Lemma 4.2] for a detailed proof. (cid:3) Here the uniqueness means that if there is another solution (˜ η, ˜ u, ˜ q ) ∈ C ( I T , H ) ×U ,T × C ( I T , H ) satisfying0 < inf ( y,t ) ∈ R × I T det( ∇ ˜ η + I ), then (˜ η, ˜ u, ˜ q ) = ( η, u, q ) by virtue of the smallness condition “sup t ∈ I T k∇ η k , δ ”. a priori estimate (3.79) (under the assumptions (2.26) and (2.27) with δ δ ) andPropositions 3.1 and 3.2 in hand, we can easily establish Theorem 2.1. We briefly give the proofbelow.Let m and ( η , u ) ∈ ( H ∩ H ∗ ) × H satisfy the odevity conditions (2.14) andmax { K / , K } /m min { δ , δ /c , δ /c } =: c , (3.84)where K is defined by (3.81), and the constant c is the same as in (3.10). Thus we see that η satisfies (3.82) by (3.10) and (3.84). Thus, by virtue of Proposition 3.1 and Remark 3.3,there exists a unique local solution ( η, u, q ) of (2.8)–(2.9) with a maximal existence time T max ,satisfying • for any T ∈ I T max , the solution ( η, u, q ) belongs to C ( I T , H ) × U T × C ( I T , H ) andsup t ∈ I T k∇ η k , δ ; • lim sup t → T max k∇ η ( t ) k , > δ or lim sup t → T max k u ( t ) k = ∞ , if T max < ∞ .In addition, the solution enjoys the odevity conditions (2.21).Let T ∗ = sup (cid:8) T ∈ I T max (cid:12)(cid:12) E , ( t ) K for any t T (cid:9) . Recalling the definition of K and the condition c >
4, we easily see that the definition of T ∗ makes sense and T ∗ > k∇ η k , δ for all t ∈ I T ∗ , then η ( t ) ∈ H ∗ for all t ∈ I T ∗ by Proposition3.2. Thus, to obtain the existence of a global solution, it suffices to verify T ∗ = ∞ . Now, weshow this by contradiction.Assume T ∗ < ∞ . Keeping in mind that T max denotes the maximal existence time and K/m δ /c by virtue of (3.84), we apply Proposition 3.1 to find that T max > T ∗ and E , ( T ∗ ) = K . (3.85)Since max { K, K } /m δ and sup t T ∗ E , ( t ) K , we can still show that the solution ( η, u )enjoys the stability estimate (3.79) with T ∗ in place of T by the regularity of ( η, u, q ). Moreprecisely, one has sup t T ∗ E , ( t ) K / , which contradicts with (3.85). Hence, T ∗ = ∞ , andthus T max = ∞ .Obviously, the global solution ( η, u ) enjoys the stability estimate (2.16) by using (3.79) and(3.80), and the estimates (2.17)–(2.19) are easily obtained from (3.10)–(3.12) and (3.36) with i = 1. The uniqueness of the global solutions is obvious due to the uniqueness of the localsolutions in Proposition 3.1 and the fact sup t > k∇ η k , δ . This completes the proof ofTheorem 2.1.
4. Proof of Theorem 2.2
We now proceed to the derivation of the time-decay estimates stated in Theorem 2.2. Itshould be noted that, under the assumptions of Theorem 2.2, the solution ( η, u, q ) of (2.8)–(2.9),established in Theorem 2.1, satisfies the differential inequalities in Lemma 3.4, (3.62) and (3.76)for a.e. t >
0, and the estimates (3.77) for any t > .1. Decay estimates for ∂ η We begin with the derivation of the decay estimates for η . Multiplying (3.48), (3.62) and(3.76) by h t i , h t i i +1 and h t i i respectively, we obtain for a.e. t > t (cid:0) h t ik∇ ( u, m∂ η ) k (cid:1) + ν h t ik u k . k∇ ( u, m∂ η ) k + h t i F (4.1)dd t (cid:0) h t i i +1 k∇ − i ∂ i ( u, m∂ η ) k (cid:1) + ν h t i i +1 k ∂ i u k − i . h t i i k∇ − i ∂ i ( u, m∂ η ) k + ( t m k ∂ η k ( k ∂ η k k ∂ η k + k ∂ η k ) for i = 1 , h t i F for i = 2 , (4.2)and dd t ( h t i i E i ) + c h t i i ( k ∂ i u k − i + k m∂ i +12 η k − i ) . h t i i − E i + ( i = 1 , h t i F for i = 2 . (4.3)Integrating (4.3) with i = 1 over (0 , t ), and then using (2.15), (2.16) and (3.77), we have h t i E , + Z t h τ i ( k ∂ u k + k m∂ η k )d τ (cid:18)q E , e c E , + E , e c E , (cid:19) (1 + E , ) for any t > . (4.4)Denoting ˜ F = k u k + k ∂ η k (1 + m + m k ∂ η k ) , one gets by (2.15) and (2.16) that Z t ˜ F d τ C. Applying Gronwall’s lemma to (4.3) with i = 2, and utilizing (3.77), (4.4) and the above estimate,and the fact F . ( k ∂ η k + k ∂ u k + k m∂ η k ) ˜ F , one can infer from (4.3) with i = 2 that h t i E , + Z t h τ i ( k ∂ u k + k m∂ η k )d τ . E , + Z t h τ i E , d τ e c R t ˜ F d τ C for all t > . (4.5)Integrating (4.1) and (4.2) over (0 , t ), using (2.15), (2.16), (4.4) and (4.5), we conclude h t i i +1 k ∂ i ( u, m∂ η ) k − i + Z t h τ i i +1 k ∂ i u k − i d τ C for 0 i . (4.6) u We proceed to derive decay estimates of higher derivatives of the velocity. Multiplying (3.48)by e νt/ and using Young’s inequality, we obtaindd t ( e νt/ k∇ u k ) + c Z t e ντ/ k u k d τ . e ντ/ ( m k ∂ η k + F ) , which implies k u k + Z t e ν ( τ − t ) / k u k d τ . k∇ u k e − νt/ + Z t e ν ( τ − t ) / ( m k ∂ η k + F )d τ. (4.7)In addition, 30 for any given a , θ >
0, there is a positive constant c , depending only on a and θ , such that e − at c h t i − θ for any t > . (4.8) • for any given r > r ∈ [0 , r ], there is a positive constant c , depending only on r and r , such that (see Lemma 2.5 in [9]) Z t h t − τ i − r h τ i − r d τ c h t i − r . (4.9) • for any τ ∈ (0 , t ), h t i h t − τ ih τ i . (4.10)Therefore, we make use of (2.16), (4.6) with i = 1, (4.8)–(4.10) and H¨older’s inequality to deducefrom (4.7) that k u k + Z t e ν ( τ − t ) / k u k d τ (1 + m ) C h t i − . (4.11)Similarly to (4.11), one can utilize (4.6) with i = 3, (4.8) and (4.9) to get from (3.47) that k u k + Z t e ν ( τ − t ) / k u k d τ (1 + m ) C h t i − . Finally, from (4.4)–(4.6) the estimate (2.28) follows, while from the above two estimates weobtain (2.29).
5. Proof of Theorem 2.4
This section is devoted to the proof of Theorem 2.4. Let ( η , u ) satisfy all the assumptionsin Theorem 2.1 and ( η, u, q ) be the solution constructed by Theorem 2.1. Then, by the regularitytheory of the Stokes problem, there exists a unique solution ( η r , u r , Q , Q ) satisfying − ∆ η r + ∇ Q = 0 , div η r = − div η , ( η r ) T = 0 (5.1)and − ∆ u r + ∇ Q = 0 , div u r = div ˜ A u , ( u r ) T = 0 , (5.2)where ˜ A := A − I . Moreover, ( η r , u r ) satisfies (2.35), also cf. the derivation of (3.36).Let ˜ η = η + η r and ˜ u = u + u r . Thus, it is easy to see that (˜ η , ˜ u ) belongs to H σ × H σ andenjoys the odevity conditions as ( η , u ) does. Therefore, there exists a unique global solution( η L , u L , q L ) ∈ C ( R +0 , H σ ) × U ∞ × C ( R +0 , H ) to the initial-value problem (2.33). Moreover, thesolution enjoys the odevity conditions as ( η, u ) does.31imilarly to (2.28), we employ (2.15) and (2.35) to see that the solution ( η L , u L ) of thelinearized problem enjoys the following estimate: k∇ η L k + X i =1 (cid:18) h t i i k ∂ i η L k − i + Z t h τ i i k m∂ i +12 η L k − i d τ (cid:19) + X i =0 (cid:18) h t i i +1 k ∂ i ( u L , m∂ η L ) k − i + Z t h τ i i +1 k ∂ i u L k − i d τ (cid:19) . E , . (5.3)Let ( η d , u d ) = ( η − η L , u − u L ), then the error function ( η d , u d ) satisfies (2.40). It is easy tosee from (2.40) that ( u d ) T = ( η d ) T = 0 for any t >
0, since ( η r ) T = ( u r ) T = 0. Moreover,( η d , u d ) also enjoys the odevity conditions as ( η, u ) does.Recalling that ( η, u, q ) is constructed by Theorem 2.1, the solution ( η, u ) satisfies all theestimates in Lemma 3.2. Hence, we can follow the arguments in the proof of Lemmas 3.4–3.6with slight modifications to derive from (2.40) thatdd t k∇ ( u d , m∂ η d ) k + ν k u d k . F d1 , (5.4)dd t (cid:18) ν k∇ η d k + X | α | =2 Z ∂ α η d · ∂ α u d d y + Z ∂ ˜ A k ∂ η · ∂ η d d y (cid:19) + k m∂ η d k . k u d k + F d2 , (5.5)dd t k∇ − i ∂ i ( u d , m∂ η d ) k + ν k ∂ i u d k − i . F d ,i , (5.6)dd t (cid:16) ν k∇ − i ∂ i η d k + X | α | =2 − i Z ∂ α ∂ i η d · ∂ α ∂ i u d d y (cid:17) + k m∂ i +12 η d k − i . k ∂ i u d k − i + F d ,i (5.7)for i = 1 ,
2, where F d1 := k ∂ η k k u k + k ∂ η k k ∂ u k + k ∂ η k k∇ q k + k∇ q k ( k ∂ η k k u k + k ∂ η k k ∂ u k ) ,F d2 := k ∂ η k ( k∇ η d k k∇ u k + k∇ η k k∇ u d k ) + k ∂ η k k∇ ( η, η d ) k k∇ q k ,F d ,i := k ∂ η k ( k ∂ u k + k∇ q k ) for i = 1; k ∂ η k k ∂ u k + k ∂ η k k ∂ u k + k ∂ η k k ( ∂ u, ∇ q ) k + k∇ q k ( k ∂ η k k ∂ u k + k ∂ η k k ∂ u k ) for i = 2 , and F d ,i := k ∂ η d k ( k ∂ η k k ∂ u k + k ∂ η k k ∂ u k )+ k ∂ η k k ∂ ( η, η d ) k k∇ q k for i = 1; k ∂ η d k ( k ∂ η k k ∂ u k + k ∂ η k k ∂ u k )+ k∇ q k ( k ∂ ( η, η d ) k k ∂ η k + k ∂ η k ) for i = 2 . If one integrates by parts, one gets from (2.35), (5.4) and (5.5) that k η d k + k ( u d , m∂ η d ) k + Z t k ( u d , m∂ η d ) k d τ . Cm − + k ∂ η k k∇ η k k∇ η d k + Z t ( F d1 + F d2 )d τ. (5.8)32ecalling that ( η, u, q ) satisfies (3.36), we make use of (2.16), (3.36) and (5.3) to get (2.36) from(5.8).Following the arguments of deriving (4.1)–(4.3), using (5.6) and (5.7), we arrive atdd t ( h t i i E d i ) + c h t i i ( k ∂ i u d k − i + k m∂ i +12 η d k − i ) . h t i i − E d i + h t i i ( F d ,i + F d ,i ) (5.9)and dd t (cid:0) h t i i +1 k∇ − i ∂ i ( u d , m∂ η d ) k (cid:1) + ν h t i i +1 k ∂ i u d k − i . h t i i k∇ − i ∂ i ( u d , m∂ η d ) k + h t i i +1 F d ,i for i = 1 , , (5.10)where E d i := c k∇ − i ∂ i ( u d , m∂ η d ) k + ν k∇ − i ∂ i η d k + X | α | =2 − i Z ∂ α ∂ i η d · ∂ α ∂ i u d d y, and E d i satisfies k ∂ i ( ∇ η d , u d , m∂ η d ) k − i . E d i . k ∂ i ( ∇ η d , u d , m∂ η d ) k − i . Utilizing (2.28), (2.35), (3.36) with i = 1 and (5.3), we easily obtain (2.37) from (5.9) and (5.10).Next, we further assume that η satisfies the additional regularity condition ∂ ( η , u ) ∈ H × H . Thus ∂ ( η r , u r ) ∈ H × H satisfies the estimates: k ∂ u r k . k ∂ ∂ η k k u k + k ∂ η k k ∂ u k , (5.11) k ∂ ∂ η r k . k ∂ η k k ∂ ∂ η k . (5.12)Keeping in mind that ∂ (˜ η , ˜ u ) ∈ H × H , we can further get, using (5.11) and (5.12), that k ∂ ( u L2 , m∂ η L2 )( t ) k + ν Z t k ∂ ∇ u L2 k d τ = k ∂ ( u L2 , m∂ η L2 )( t ) k | t =0 . (1 + k ∂ η k + k ∂ u k ) . (5.13)If we make use of the estimate Z ∂ A k ∂ u k ∂ q d y = dd t Z ∂ A k ∂ η k ∂ q d y − Z ∂ η k ( ∂ t ∂ A k ∂ q + ∂ A k ∂ q t )d y, − Z ∂ ˜ A ∂ u · ∂ u d d y = Z ∂ ( η d2 + η L2 ) ∂ u · ∂ u d d y and k ∂ η L k . k ∂ ∂ η L k , we can deduce from (2.40) thatdd t (cid:18) k∇ ( u d , m∂ η d ) k + Z ∂ A k ∂ η k ∂ q d y (cid:19) + c k u d k . F d5 , (5.14)where F d5 := k ∂ η k k u k + k ∂ η k k ∂ u k + k ∂ η d k k ∂ u k + k ∂ ∂ η L2 k k ∂ u k + k ∂ η k k∇ q k + k∇ q k ( k ∂ η k k u k + k ∂ η k k ∂ u k ) + k ∂ η k k∇ η k k∇ q t k . (5.15)Consequently, with the help of (2.16), (2.28), and (2.36), (3.37) and (5.13), we easily obtain(2.39) from (5.14). This completes the proof of Theorem 2.4.33 . Proof of Theorem 2.5 This section is devoted to the proof of Theorem 2.5. The key ideas to establish Theorem 2.5are similar to those in the proof of Theorems 2.1, 2.2 and 2.4. We break up the proof of Theorem2.5 into two subsections.
We start with the derivation of the a priori stability estimates (2.50) and (2.51) in Theorem2.5. To this end, let ( η, u, q ) be a solution of the initial-value problem (2.47)–(2.48) defined onΩ T for any given T >
0, where ( η , u ) belongs to H , × H , and satisfies div A u = 0 andthe odevity conditions (2.14). It should be remarked that the solution automatically satisfies( η ) T = ( u ) T = 0 and the odevity conditions (2.21). We further assume that ( η, u, q ) and K κ satisfy ( q ) T = 0, sup t T q k ( η, u, m∂ η )( t ) k K κ for any given T > , (6.1)and max (cid:8) K / κ , K κ (cid:9) /m δ, (6.2)where K κ will be given in (6.29) and δ is a sufficiently small constant. We should point out herethat the smallness of δ depends only on ν and κ .By virtue of (6.1) and (6.2), it is easy to see thatsup t T k ∂ η ( t ) k δ (6.3)and sup t T ( m − k η ( t ) k + k u ( t ) k k ∂ η ( t ) k ) . δ. (6.4)Now, we establish some basic energy estimates for ( η, u, q ) which will be used later. Lemma 6.1.
Under the condition (6.3) with sufficiently small δ , one has k η k i +2 , . k ∂ η k i +1 , (6.5) k∇ q k . k u k + m k ∂ η k k ∂ η k , (6.6) k χ k i +1 . k∇ i curl A χ k . k χ k i +1 , i , (6.7) where χ denotes η , or ∂ η , or u . Proof.
We can argue in the same manner as in the derivation of (3.10) and (3.36) with i = 1to deduce (6.5) and (6.6). To show (6.7), keeping in mind that ∆ χ = ∇ ⊥ curl χ + ∇ div χ wherecurl χ = ∂ χ − ∂ χ and ∇ ⊥ := ( − ∂ , ∂ ), we easily obtain k∇ χ k i = k curl χ k i + k div χ k i . (6.8)Recalling the product estimate k f g k j . k f k k g k for j = 0 , k f k j k g k for 0 j , k f k k g k j + k f k j k g k for 3 j , (6.9)34nd noting that ( η, u ) satisfies (3.22) and (3.32), we find that k div χ k i . k η k k∇ χ k i , i . (6.10)Thus, with the help of (3.3), (6.3) and (6.5), we get from (6.8) and (6.10) that k χ k i +1 . k∇ i curl χ k . (6.11)In addition, by (6.3), (6.5) and (6.9), k∇ i curl χ k . k∇ i curl A χ k . k χ k i +1 , (6.12)which, together with (6.11), yields (6.7). (cid:3) Lemma 6.2.
Under the condition (6.3) with sufficiently small δ , one has dd t k∇ curl A ( u, m∂ η ) k + κ k u k . k ( u, m∂ η ) k k ( u, m∂ η ) k (6.13) and dd t (cid:16) X | α | =3 Z ∂ α curl A η∂ α curl A u d y + κ k∇ curl A η k (cid:17) + k mη k , . κ k u k . (6.14) Proof.
One can not directly apply ∂ β to (2.47) to derive (6.13) and (6.14) for the case | β | = 4,since q does not have the fifth-order derivatives. Instead, we apply ∂ α curl A to (2.47) to get ∂ α curl A u t + κ∂ α curl A u = m ∂ α curl A ∂ η, (6.15)where | α | = 3 . Multiplying (6.15) by ∂ α curl A u , resp. ∂ α curl A η , in L , we have12 dd t k ∂ α curl A ( u, m∂ η ) k + κ k ∂ α curl A u k = Z ∂ α curl A t u∂ α curl A u d y + m Z ( ∂ α curl A ∂ η∂ α curl A t ∂ η − ∂ α curl ∂ A ∂ η∂ α curl A u − ∂ α curl A ∂ η∂ α curl ∂ A u )d y =: I , (6.16)resp. dd t (cid:18)Z ∂ α curl A u∂ α curl A η d y + κ k ∂ α curl A η k (cid:19) + k m∂ α curl A ∂ η k = Z ( ∂ α curl A u∂ α curl A t η + ( ∂ α curl A u ) + ∂ α curl A t u∂ α curl A η )d y − m Z ∂ α curl A ∂ η∂ α curl ∂ A η d y + κ Z ∂ α curl A η∂ α curl A t η d y =: I , (6.17) We explain why to take α satisfying | α | = 3. Let us consider the first term R ∂ α curl A t u∂ α curl A u d y on theright hand of (6.16). Obviously, R ∂ α curl A t u∂ α curl A u d y . k∇ u k L ∞ k ∂ α u k . Since H ֒ → L ∞ , obviously, weneed at least | α | >
3. However, for | α | = 3, it seems difficult to close the energy estimates by (6.16) and (6.17).To overcome this difficulty, we adopt the two-layers energy method, i.e., we close the lower-order energy estimate k∇ ( u, m∂ η ) k by a lower-order energy inequality, and then further close the highest-order energy estimate bya highest-order energy inequality. Thus apparently, we need at least | α | >
4. Here we remark that the initial-value problem (2.47) with small initial data ( η , u ) ∈ ( H , ∩ H ∗ ) × H admits a unique global strong solution( η, u, q ) ∈ H , ∗ , , ∞ × C ( R +0 , H × H ). I and I can be bounded as follows. I . k∇ ( u, m∂ η ) k k∇ ( u, m∂ η ) k ,I . κ (1 + k∇ η k ) k∇ u k + m k∇ η k k∇ ∂ η k + k∇ u k k∇ η k . Consequently, inserting the above two estimates into (6.16) and (6.17) respectively, using (6.3)–(6.5) and Young’s inequality, we obtain (6.13) and (6.14). (cid:3)
Lemma 6.3.
Under the conditions (6.3) – (6.4) with sufficiently small δ , we have dd t k∇ ( u, m∂ η ) k + κ k u k . δ k mη k , (6.18) and dd t (cid:16) X | α | =3 Z ∂ α η · ∂ α u d y + κ k∇ η k (cid:17) + k mη k , . k u k . (6.19) Proof.
It is easy to see from the proof of Lemma 6.2 that we can not directly establish theestimates (6.18) and (6.19) by the curl A -estimate method, since we shall use some smallnessproperties which are hidden in some integral terms involving ∇ A q .To exploit the smallness properties hidden in the pressure term ∇ A q , we apply ∂ α ( | α | = 3)to (2.47) to get ∂ α u t + ∂ α ∇ A q + κ∂ α u = m ∂ α ∂ η. ( | α | = 3)Thus, multiplying the above identity by ∂ α u and ∂ α η in L respectively, we have12 dd t k ∂ α ( u, m∂ η ) k + κ k ∂ α u k = I := − Z ∂ α ∇ A q · ∂ α u d y, (6.20)and dd t (cid:18)Z ∂ α η · ∂ α u d y + κ k ∂ α η k (cid:19) + k m∂ α ∂ η k − k ∂ α u k = I := − Z ∂ α ∇ A q · ∂ α η d y, (6.21)respectively, where the right-hand sides can be bounded as follows, using (3.22), (3.32) and (6.6). I . k∇ η k k∇ u k ( k u k + m k ∂ η k k ∂ η k ) ,I . k∇ η k k∇ η k ( k u k + m k ∂ η k k ∂ η k ) , which, together with (6.20), (6.21) and (6.3)–(6.5), implies (6.18) and (6.19). (cid:3) Now, we are in a position to show the a priori stability estimate (2.50). Firstly, using (6.7),we derive from Lemmas 6.2–6.3 thatdd t E κ + c κ ( k u k + k mη k , ) . κ k ( u, m∂ η ) k k ( u, m∂ η ) k , (6.22)dd t E κl + c κ ( k u k + k mη k , ) , (6.23)36here E κ := c k∇ curl A ( u, m∂ η ) k + X | α | =3 Z ∂ α curl A η∂ α curl A u d y + κ k∇ curl A η k , (6.24) E κl := c k∇ ( u, m∂ η ) k + X | α | =3 Z ∂ α η · ∂ α u d y + κ k∇ η k , k ( η, u ) k + k mη k , . κ E κ . κ k ( η, u, m∂ η ) k , k ( η, u ) k + k mη k , . κ E κl . κ k ( η, u, m∂ η ) k . Thus, we further obtaindd t e c κ min { ,m } t E κ + c κ e c κ min { ,m } t ( k u k + k mη k , ) . κ e c κ min { ,m } t k ( u, m∂ η ) k k ( u, m∂ η ) k , (6.25)dd t ( h t i i E κ ) + c κ h t i i ( k u k + k mη k , ) . κ h t i i − E κ + h t i i k ( u, m∂ η ) k k ( u, m∂ η ) k for any i > , (6.26) k ( η, u ) k + k mη k , + Z t ( k u k + k mη k , )d τ . κ k ( η , u , m∂ η k . (6.27)Let c κ be the positive constant in (6.41). In view of (6.25) and (6.27), we find that e c κ min { ,m } t ( k ( η, u ) k + k mη k , ) c κ k ( η , u , m∂ η ) k e c κ k ( η ,u ,m∂ η ) k /c κ , (6.28)where c κ >
4. From now on, we take K κ = q c κ k ( η , u , m∂ η ) k e c κ k ( η ,u ,m∂ η ) k . (6.29)Thus, there is a δ κ ∈ (0 , δ δ κ , k ( η, u ) k + k mη k , e c κ min { ,m } t ( k ( η, u ) k + k mη k , ) K κ /c κ . (6.30)Besides, we get from (6.25) and (6.30) that Z t e c κ min { ,m } τ ( k u k + k mη k , )d τ . κ K κ /c κ , (6.31)which, together with (6.30), yields the desired stability estimate: e c κ min { ,m } t ( k ( η, u ) k + k mη k , ) + Z t e c κ min { ,m } τ ( k u k + k mη k , )d τ . κ K κ . (6.32) We can get the exponential decay in the fourth-order energy inequality (6.32), since the energy functional E κ is equivalent to the dissipative functional D κ in (6.22) where D κ := k u k + k mη k , . Obviously, for theviscous fluid case, a similar equivalent relation does not hold in the second-order energy inequality (3.74), inwhich one sees that the second-order energy functional E only controls the second-order dissipative functional D := k u k + k m∂ η k . Similarly, if we further derive the first-order energy inequality, then the first-order energyfunctional, denoted by E , also controls the the first-order dissipative functional. However, the first-order energyfunctional E can be controlled by the second-order dissipative functional D ; and this relation admits us to expectat most algebraic decay rates in the viscous fluid case. Z t k η k , d τ . K κ . (6.33)Applying Gronwall’s lemma to (6.26) and using (6.27), we conclude h t i i ( k ( η, u ) k + k mη k , ) . κ (cid:18) k ( η , u , m∂ η ) k + Z t h τ i i − k ( η, u, m∂ η ) k d τ (cid:19) e c κ ( k ( η ,u ,m∂ η ) k . (6.34)Thanks to (6.27) and (6.34), we further deduce from (6.26) that Z t h τ i i ( k u k + k mη k , )d τ . κ (1 + ( k ( η , u , m∂ η ) k ) × (cid:18) k ( η , u , m∂ η ) k + Z t h τ i i − k ( η, u, m∂ η ) k d τ (cid:19) e c κ k ( η ,u ,m∂ η ) k . (6.35)Consequently, we conclude from (6.2), (6.31), (6.33) and (6.35) that for i = 0 and 1, Z t h τ i i (cid:0) k u k + (1 + m ) k η k , (cid:1) d τ C κ . (6.36)Finally, we get from (6.34) and (6.36) that for any i = 1 and 2, h t i i ( k ( u, η ) k + k mη k , ) C κ , which, together with (6.36), yields X i (cid:18) h t i i ( k ( η, u ) k + k mη k , ) + Z t h τ i i ( k u k + k mη k , )d τ (cid:19) + Z t h τ ik η k , d τ C κ . (6.37)Next, we introduce a global well-posedness result for the linear initial-value problem (2.54)and a local well-posedness result for the nonlinear initial-value problem (2.47)–(2.48). Proposition 6.1.
Let i > be an integer, κ and m be positive constants. If (˜ η , ˜ u ) ∈ ( H i +12 ∩ H iσ ) × H iσ , then there exists a unique strong solution ( η L , u L ) ∈ C ( R + , H i +12 ) × U i ∞ to the followinglinear initial-value problem: η L t = u L ,u L t − κu L = m ∂ η L , div u L = 0 , ( η L , u L ) | t =0 = (˜ η , ˜ u ) . (6.38) Proof.
The proof of Proposition 6.1 is trivial, and hence we omit it here.
Proposition 6.2.
Let B κ > and ( η L , u L ) ∈ C ( R + , H ) × U ∞ be the unique global solution of (6.38) with (˜ η , ˜ u ) ∈ ( H ∩ H σ ) × H σ . Assume that ( η , u ) ∈ H × H , k ( u , ∂ η ) k B κ and A u = 0 , where A is defined by ζ and ζ = η + y . Then, there is a constant δ κ ∈ (0 , ,such that if, in addition, k∇ η k δ κ , (6.39) the initial value problem (2.47) – (2.48) possesses a unique local classical solution ( η, u, q ) ∈ C ( I T , H ) × U T × ( C ( I T , H ) ∩ C ( I T , H )) for some T > dependent of B κ , κ , m and δ κ .Moreover, ( η, u ) satisfies < inf ( y,t ) ∈ R × I T det( ∇ η + I ) , sup t ∈ I T k∇ η k δ κ ; (6.40)sup t ∈ I T k ( η, u, m∂ η ) k c κ k ( η , u , m∂ η ) k ; (6.41) and sup τ ∈ I t k (curl A η d , curl A u d , curl A m∂ η d )( τ ) k . κ e t k ( η ,u ,m∂ η ) k k ( η − ˜ η , u − ˜ u , m∂ ( η − ˜ η ) k + t (cid:16) k ( u , m∂ η , ˜ u , m∂ ˜ η ) k + k ( u , m∂ η ) k k ( u , m∂ η , ˜ u , m∂ ˜ η ) k (cid:17) , ∀ t ∈ I T (6.42) where the constant c κ > depends on κ at least, and ( η d , u d ) := ( η − η L , u − u L ) . Proof.
We postpone the proof of Proposition 6.2 to Section 7.3. (cid:3)
Remark 6.1.
If ( η , u ) in Proposition 6.2 further satisfies ( η , u ) ∈ H , × H and the odevityconditions (2.14), then for each fixed t ∈ (0 , T ], ( η, u )( t ) also belongs to H , × H and satisfiesthe odevity conditions (2.21). Remark 6.2.
Since ( u, ∂ η ) may do not belong to C ( I T , H ), one needs the additional estimates(6.41) and (6.42) to further establish the existence result of a global solution and its asymptoticbehavior with respect to m , repspectively.With the priori estimate (6.30) and Propositions 3.2 and 6.2 in hand, we can easily establishthe existence and uniqueness of a global time-decay smooth solution stated in Theorem 2.5. Next,we sketch the proof.Let m and ( η , u ) ∈ ( H , ∩ H ∗ ) × H satisfy the odevity conditions (2.14) andmax { K / κ , K κ } /m min { δ κ , δ κ /c , δ /c } =: c κ , (6.43)where K κ is defined by (6.29), and the constant c is the same as in (6.5). Thus, η satisfies(6.39) by virtue of (6.5) and (6.43). Moreover, by Proposition 6.2 and Remark 6.1, one sees thatthere is a unique local solution ( η, u, q ) of (2.47)–(2.48) defined on a maximal existence timeinterval [0 , T max ), such that • for any T ∈ I T max , the solution ( η, u, q ) belongs to H , ∗ , ,T × U T × ( C ( I T , H ) ∩ C ( I T , H ))and satisfies sup t ∈ I T k∇ η k δ κ ; • lim sup t → T max k∇ η ( t ) k > δ κ , or lim sup t → T max k ( u, ∂ η )( t ) k = ∞ .39n addition, the solution enjoys the odevity conditions (2.21).Let T ∗ = sup { T ∈ I T max | k ( η, u, m∂ η k K κ for any t T } . Recalling the definition of K κ and the condition c κ >
4, we see that the definition of T ∗ makessense and T ∗ >
0. Moreover, η ( t ) ∈ H ∗ for any t ∈ I T max by Proposition 3.2 and (6.5). To obtainthe existence of a global solution, we prove T ∗ = ∞ by contradiction.Let us assume T ∗ < ∞ . Then, for any given T ∗∗ ∈ I T ∗ , it holds thatsup t T ∗∗ k ( η, u, m∂ η k K κ . (6.44)Thanks to (6.41) and (6.44), the condition “max { K / κ , K κ } /m δ κ ” and the factsup t ∈ I T k f k = k f k L ∞ ( I T ,L ) for any f ∈ C ( I T , L ) , (6.45)we can verify that the solution ( η, u ) indeed satisfies the stability estimate (6.30) by a standardregularity method. More precisely, we havesup t T ∗ k ( η, u, m∂ η )( t ) k K κ / p c κ . Now, we take ( η ( T ∗∗ ) , u ( T ∗∗ )) as an initial data. Noting that k ( u, ∂ η )( T ∗∗ ) k B κ := 2 max { K κ / p c κ , K κ /m p c κ } and k∇ η ( T ∗∗ ) k δ κ , we apply Proposition 3.1 to see that there exists a unique local classical solution, denoted by( η ∗ , u ∗ , q ∗ ), to the initial-value problem (2.47)–(2.48) with ( η ( T ∗∗ ) , u ( T ∗∗ )) in place of ( η , u ).Moreover, sup t ∈ [ T ∗∗ ,T ] k ( η ∗ , u ∗ , m∂ η ∗ ) k K κ and sup t ∈ [ T ∗∗ ,T ] k∇ η ∗ k δ κ , (6.46)where the local existence time T > T ∗∗ depends possibly on B κ , κ , m and δ κ .In view of the uniqueness in Proposition 6.2 and the fact that T max is the maximal existencetime, we immediately see that T max > T ∗ + T / t ∈ [0 ,T ∗ + T/ k ( η, u, m∂ η ) k K κ . Thiscontradicts with the definition of T ∗ . Hence, T ∗ = ∞ and thus T max = ∞ .In addition, we also verify that the global solution ( η, u ) indeed enjoys the time-decay esti-mates (2.50) and (2.51) by the a priori estimates (6.32) and (6.37). Finally, the uniqueness ofthe global solutions is obvious due to the uniqueness of the local solutions in Proposition 6.2 andthe fact sup t > k∇ η k δ κ . This completes the proof of the existence and uniqueness of globaltime-decay solutions stated in Theorem 2.5. We now turn to the proof of stability around the solutions of (2.54). Thanks to the regularityof ( η , u ) in Theorem 2.5, there exists a unique solution ( η r , u r , Q , Q ) satisfying (5.1) and (5.2).Furthermore, ( η r , u r ) satisfies (2.34) and (2.55).Let ˜ η = η + η r and ˜ u = u + u r . Then it is easy to see that (˜ η , ˜ u ) is in ( H ∩ H σ ) × H σ and enjoys the odevity conditions as ( η , u ) does. Hence, there is a unique global solution( η L , u L ) ∈ C ( I T , H ) × U ∞ to the problem (2.54). Furthermore, the solution enjoys the odevityconditions as ( η, u ) does. 40et ( η d , u d ) = ( η − η L , u − u L ), then the error function ( η d , u d ) satisfies η d t = u d ,u d t + ∇ A q + κu d = m ∂ η d , div u d = − div ˜ A u, ( η d , u d ) | t =0 = − ( η r , u r ) . (6.47)Moreover, ( u d ) T = ( η d ) T = 0, div η d = div η , and ( η d , u d ) also satisfies the odevity conditions as( η, u ) does.Employing the arguments used for (6.25) and (6.26), together with a standard regularitymethod, we deduce from (6.47) that for a.e. t > t e c κ min { ,m } t E κ d + c κ e c κ min { ,m } t ( k u d k + k mη d k , ) . κ e c κ min { ,m } t k ( η, u, m∂ η ) k ( E κ d + k∇ η k k ( u, m∂ η ) k ) , (6.48)dd t ( h t i i E κ d ) + c κ h t i i ( k u d k + k mη d k , ) . κ h t i i − E κ d + h t i i k ( η, u, m∂ η ) k ( E κ d + k∇ η k k ( u, m∂ η ) k ) , i = 1 , , (6.49)where E κ d is defined by (6.24) with ( u d , η d ) in place of ( u, η ), in W , ∞ ( R + ), and satisfies that forany t > k ( η d , u d ) k + k mη d k , . κ E κ d + k∇ η k k ( u, m∂ η ) k , E κ d . κ k ( η d , u d , m∂ η d ) k . (6.50)In view of (6.42), one sees that there is a T >
0, such that for any t < T ,sup τ ∈ I t E κ d . κ (1 + t ) e t k ( η ,u ,m∂ η ) k k ( η r , u r , m∂ η r ) k + t ( k ( u , m∂ η , ˜ u , m∂ ˜ η ) k + t k ( u , m∂ η ) k k ( u , m∂ η , ˜ u , m∂ ˜ η ) k ) , which, together with (2.55), yields thatlim sup t → sup τ ∈ I t E κ d . κ k ( η r , u r , m∂ η r ) k . κ k ∂ η k k ( η , u , m∂ η ) k . (6.51)Thus, if we apply Gronwall’s lemma to (6.48), and make use of (2.49), (2.50), (6.5), (6.50) and(6.51), we obtainsup t > ( e c κ min { ,m } t ( k ( η d , u d ) k + k mη d k , )) + Z t e c κ min { ,m } τ ( k u d k + k mη d k , )d τ . κ (1 + K ) k ( η , u , m∂ η ) k e c κ K m − , which gives (2.52). Finally, following the same process as in the derivation of (6.37) with necessarymodifications in arguments, we obtain (2.53). This completes the proof of Theorem 2.5 . It is easy to see from the proof of (2.52) and (2.53) that due to the special structure of energy inequalitiesof ( η d , u d ), we can use (6.51) and k m ∇ η k C κ to obtain the convergence rate m − . For the viscous fluidcase, however, we mainly exploit k m∂ η k C to get the convergence rate m − . This consequently yieldssome nonlinear terms of third-order derivatives, such as k ∂ η k k∇ η k k∇ q k , on the right-hand side of the energyinequalities for ( η d , u d ) which can only provide the convergence rate m − . . Local well-posedness This section is devoted to the proof of the local well-posedness results in Propositions 3.1 and6.2, and is organized as follows.First we establish the existence of both strong and classical solutions to the following linearinitial-value problem in Section 7.1: u t + ∇ A q − ν ∆ A u + κu = f, div A u = 0 ,u | t =0 = u in T , (7.1)where ν > κ >
0, ( η , u , w ) are given, A = ( ∇ ζ ) − T and ζ = Z t w d y + η + y, (7.2)see Propositions 7.1 and 7.2 for the details. Then we give the proof of Proposition 3.1 based onProposition 7.1 by a standard iterative method in Section 7.2. Similarly, we also give the localexistence of a unique classical solution to the following initial-value problem: η t = u,u t + ∇ A q − ν ∆ A u + κu = f, div A u = 0 , ( η, u ) | t =0 = ( η , u ) in T , (7.3)see Proposition 7.2 in Section 7.2 for the details. Finally, thanks to Proposition 7.2, we completethe proof of Proposition 6.2 by a standard method of vanishing viscosity limit in Section 7.3.Finally, we introduce new notations appearing in this section. H − = the dual space of H, H − σ = the dual space of H σ ,< · , · > X − ,X denotes dual product , U T := { u ∈ U ,T | u tt ∈ L ( I T , H ) } , G T := { f ∈ C ( I T , L ) | ( f, f t ) ∈ L ( I T , H × H − ) } , k v k U ,T := s X j (cid:16) k ∂ jt v k C ( I T ,H − j ) ) + k ∂ jt v k L ( I T ,H − j )+1 ) (cid:17) , k v k U T := s X j k ∂ jt v k C ( I T ,H − j ) ) + X j k ∂ jt v k L ( I T ,H − j )+1 ) ,A . L B means A c L B, where U i,T for i = 1, 2 is defined by (2.10), and c L denotes a generic positive constant dependingon ν , κ and m , and may vary from one place to another (if not stated explicitly). (7.1)This section is devoted to establishing the existence and uniqueness of both strong and clas-sical solutions to (7.1). We start with the existence of a unique strong solution.42 roposition 7.1. Let B > , δ > , ( η , u ) ∈ H ∩ H , A = ( ∇ η + I ) − T , w ∈ U ,T , f ∈ G T , A and ζ be defined by (7.2) , η = ζ − y and T := min { , ( δ/B ) } . (7.4) Assume that k∇ η k , δ, div A u = 0 , w | t =0 = u , q k∇ w k C ( I T ,H ) + k∇ w k L ( I T ,H ) + k∇ w t k L ( I T ,L ) B , (7.5) then there is a sufficiently small constant δ L1 ∈ (0 , independent of m and ν , such that for any δ δ L1 , there exists a unique local strong solution ( u, q ) ∈ U ,T × ( C ( I T , H ) ∩ L ( I T , H )) to theinitial-value problem (7.1) . Moreover, the solution enjoys the following estimate: ζ , k∇ η k , δ, k u k U ,T + k q k C ( I T ,H ) + k q k L ( I T ,H ) . L p B ( u , f ) , (7.6) where B ( u , f ) := (1 + k∇ η k ) ˜ B ( u , f ) , ˜ B ( u , f ) := k u k + k u k + k f k C ( I T ,L ) + k f k L ( I T ,H ) + k f t k L ( I I ,H − ) + k∇ w k C ( I T ,H ) (1 + k u k )( k u k + k f k L ( Q T ) ) . Moreover, if f = ∂ η , then q ˜ B ( u , ∂ η ) . k u k + q k∇ w k C ( I T ,H ) (1 + k u k / ) , (7.7)∆ A q = m div A ∂ η + div A t u holds for any t ∈ I T , (7.8) k q k C ( I T ,H ) . L k∇ w k C ( I T ,H ) (cid:16) k u k + q k∇ w k C ( I T ,H ) (cid:16) k u k / (cid:17)(cid:17) , (7.9) k q t k L ( I T ,H ) . L (1 + k u k + k∇ w k C ( I T ,H ) )(1 + k∇ u k + k∇ ( w, w t ) k L ( I T ,H × L ) ) . (7.10) Remark 7.1.
For the case f = ∂ η , from (7.6) and (7.7) we get k u k U ,T + k q k C ( I T ,H ) + k q k L ( I T ,H ) c L (1 + k u k ) + k∇ w k C ( I T ,H ) / . (7.11) Proof.
We shall break up the proof into three steps.(1)
Existence of local strong solutions
Recalling that k∇ η k , δ , the definition (7.4) and η = Z t w d τ + η , (7.12)we make use of the regularity of w , (7.4) and (7.5) to find that η ∈ C ( I T , H ), and k∇ η ( t ) k k∇ η k + √ t k∇ w k L ( I T ,H ) k∇ η k , (7.13)and k∇ η ( t ) k , δ + √ t k∇ w k L ( I T ,H ) δ for all t ∈ I T .
43y (3.5) one has k∇ η ( t ) k L ∞ . k∇ η ( t ) k , . δ for any t ∈ I T . (7.14)Thanks to the estimate (7.14), we have for sufficiently small δ that 1 J
3, where and inwhat follows, J := det ∇ ζ . Therefore, A makes sense and is given by the following formula: A = J − (cid:18) ∂ η + 1 − ∂ η − ∂ η ∂ η + 1 (cid:19) . We remark that the smallness of δ (independent of ν ) will be often used in the derivation ofsome estimates and conclusions later, and we shall omit to mention it for the sake of simplicity. Inspired by the proof in [12, Theorem 4.3], we next solve the linear problem (7.1) by applyingthe Galerkin method. Let { ϕ i } ∞ i =1 be a countable orthogonal basis in H σ . For each i > ψ i = ψ i ( t ) := ∇ ζ ϕ i . Let H ( t ) = { v ∈ H | div A v = 0 } . Then ψ i ( t ) ∈ H ( t ) and { ψ i ( t ) } ∞ i =1 is a basis of H ( t ) for each t ∈ I T . Moreover, ψ it = Rψ i , (7.15)where R := ∇ w A T .For any integer n >
1, we define the finite-dimensional space H n ( t ) := span { ψ , . . . , ψ n } ⊂ H ( t ), and write P n ( t ) : H ( t ) → H n ( t ) for the H orthogonal projection onto H n ( t ). Clearly, foreach v ∈ H ( t ), P n ( t ) v → v as n → ∞ and kP n ( t ) v k k v k .Now, we define an approximate solution u n ( t ) = a nj ( t ) ψ j with a nj : I T → R for j = 1 , . . . , n, where n > a nj , so that for any 1 i n , Z u nt · ψ i d y + ν Z ∇ A u n : ∇ A ψ i d y + κ Z u n : ψ i d y = Z f · ψ i d y (7.16)with initial data u n (0) = P n u ∈ H n .Let X = ( a ni ) n × , N = (cid:18)Z f · ψ i d y (cid:19) n × , C = (cid:18)Z ψ i · ψ j d y (cid:19) n × n , C = (cid:18)Z Rψ i · ψ j d y + ν Z ∇ A ψ i : ∇ A ψ j d y + κ Z ψ i · ψ j d y (cid:19) n × n . Recalling the regularity of w , we easily verify that C ∈ C , / ( I T ) , C ∈ C , / ( I T ) , N ∈ C ( I T ) and N t ∈ L ( I T ) . (7.17)Noting that C is invertible, we can rewrite (7.16) as follows. X t + ( C ) − ( C X − N ) = 0 (7.18)with initial data X | t =0 = (cid:18)Z P n u · ψ i d y (cid:19) n × , where ( C ) − denotes the inverse matrix of C . By virtue of the well-posedness theory of ODEs(see [32, Section 6 in Chapter II]), the equation (7.18) has exactly one solution X ∈ C ( I T ).44hus, one has established the existence of the approximate solution u n ( t ) = a nj ( t ) ψ j . Next, wederive uniform-in- n estimates for u n .Due to (7.14), we easily get from (7.16) with u n in place of ψ that for sufficiently small δ ,dd t k u n k + c L k u n k . L k f k . (7.19)By (7.15), u nt − Ru n = ˙ a ni ψ i . (7.20)Obviously, we can replace ψ by ˙ a ni ψ i in (7.16) and use (7.20) to deduce that k u nt k + ν Z ∇ A u n : ∇ A u nt d y + κ Z u n : u nt d y = Z ( u nt + κu n ) · ( Ru n )d y + ν Z ∇ A u n : ∇ A ( Ru n )d y + Z f · ( u nt − Ru n )d y. (7.21)Thanks to (3.4), (3.6) and (7.14), one can further obtain from (7.21) thatdd t (cid:0) κ k u n k + ν k∇ A u n k (cid:1) + k u nt k . L k∇ u n k ( k R k L ∞ k∇ u n k + kA t k L ∞ k∇ u n k ) + Z |∇ u n ||∇ R || u n | d y + k ( f, Ru n ) k + k R k L ∞ k u n k . L k∇ w k , ( k u n k + k∇ A u n k ) + k∇ w k k∇ w k , k u n k + k f k . (7.22)With the help of Gronwall’s lemma, (7.4) and (7.14), we infer from (7.19) and (7.22) that forany t ∈ I T , k u n k + Z t k u τ k d τ . L (cid:18) kP n u k + Z t k f k d τ (cid:19) e R t c ( k∇ w k + k∇ w k k∇ w k , )d τ . L k u k + k f k L ( Q t ) . (7.23)Recall (see [22, Theorem 1.67]) Z f ( τ ) · ψ i ( τ )d y (cid:12)(cid:12)(cid:12)(cid:12) τ = tτ =0 = Z ts (cid:18) < f τ , ψ i > H − ,H + Z f · ψ iτ d y (cid:19) d τ, which gives dd t Z f ( t ) · ψ i ( t )d y = < f τ , ψ i > H − ,H + Z f · ψ iτ d y for a.e. t ∈ I T . (7.24)Hence, C t ∈ L ( I T ). In view of (7.17) and (7.18), we have ¨ a nj ( t ) ∈ L ( I T ). This means that u ntt makes sense. So, with the help of (7.15) and (7.24), we get from (7.16) that Z u ntt · ψ i d y + ν Z ∇ A u nt : ∇ A ψ i d y + κ Z u nt · ψ i d y = < f t · ψ i > H − ,H + Z ( f − u nt − κu n ) · ( Rψ i )d y − ν Z ( ∇ A t u n : ∇ A ψ i + ∇ A u n : ( ∇ A t ψ i + ∇ A ( Rψ i ))d y a.e. in I T . (7.25)45oting that (also see [22, Theorem 1.67])12 k u nt k − Z u nt · ( Ru n )d y − (cid:18) k u nt k − Z u nt · ( Ru n )d y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) t =0 = Z t (cid:18)Z u nττ · ( u nτ − Ru n )d y − Z u nτ · ( Ru n ) τ d y (cid:19) d τ and Z f ( τ ) · ( Ru n )( τ )d y (cid:12)(cid:12)(cid:12)(cid:12) τ = tτ =0 = Z t (cid:18) < f τ , Ru n > H − ,H + Z f · ( Ru n ) τ d y (cid:19) d τ, we utilize (7.20) and the above two identities to infer from (7.25) with ψ i replaced by ( u nt − Ru n )that 12 k u nt k − Z u nt · ( Ru n )d y + Z f · ( Ru n )d y + Z t ( κ k u nτ k + ν k∇ A u nτ k )d τ = (cid:18) k u nt k − Z u nt · ( Ru n )d y + Z f · ( Ru n )d y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) t =0 + I , (7.26)where I := Z t (cid:18) < f τ , u nτ > H − ,H + Z (cid:16) f · (2 Ru nτ + R τ u n − R u n ) + ( u nτ + κu n ) · ( R ( u nτ − Ru n )) (cid:17) d y + Z ( κu nτ ( Ru n ) − u nτ · ( Ru n ) τ )d y − ν Z (cid:0) ∇ A u n : ( ∇ A τ ( u nτ − Ru n )+ ∇ A ( R ( u nτ − Ru n )) (cid:1) + ∇ A τ u n : ∇ A ( u nτ − Ru n ) − ∇ A u nτ : ∇ A ( Ru n ))d y (cid:19) d τ. Keeping in mind that k∇ w k Z t k∇ w τ k d τ + k∇ u k , (7.27)we get from (7.26) that k u nt k + Z t ( κ k u τ k + ν k∇ u nτ k )d τ . L k∇ w k k∇ w k k u n k + k u k + k f k C ( I T ,L ) + k u nt | t =0 k + I . L k∇ w k (1 + k u k ) (cid:16) k u k + k f k L ( Q t ) (cid:17) + k u k + k f k C ( I T ,L ) + k u nt | t =0 k + I , (7.28)where we have used (3.6) in the first inequality, (7.4), (7.23) and (7.27) in the second inequality.Below, we shall bound the the last two terms in (7.28).Replacing ψ i by ( u nt − Ru n ) in (7.16), one sees that k u nt k = Z f · ( u nt − Ru n )d y + ν Z ∆ A u n : ( u nt − Ru n )d y + Z u nt · ( Ru n )d y − κ Z u n · ( u nt − Ru n )d y, (7.29)46hich implies k u nt k . L k f k + k u n k + k∇ w k k u n k , ∀ t ∈ [0 , T ) . In particular, k u nt | t =0 k . L k u k + k u k + k f k . (7.30)Thus, the last term on the right-hand side of (7.26) can be estimated as follows, using thefirst inequality in (3.6) and (7.4). I . L Z t k f τ k − k u nτ k + k u n k (cid:16) k f k k∇ w k L ∞ k∇ w k + p k f k k f k k∇ w τ k (cid:17) + k f k k∇ w k k u nτ k + k u n k k∇ w k L ∞ (cid:18) k∇ w k L ∞ + q k∇ w k k∇ w k , (cid:19) + k u nτ k ( k∇ w k k u nτ k + (1 + k∇ w k L ∞ ) k∇ w k k u n k ) + k∇ w τ k k u n k p k u nτ k k u nτ k + k u n k k u nτ k (cid:18) k∇ w k L ∞ + q k∇ w k k∇ w k , (cid:19) ! d τ . L k u k + k f k C ( I t ,L ) + k f k L ( I t ,H ) + Z t k f τ k H − k u nτ k d τ + (cid:0) k u k + k f k L ( I t ,H ) (cid:1) ( k u nt k C ( I t ,L ) + k∇ u nτ k L ( I t ,L ) ) + δ k u nt k C ( I t ,L ) . (7.31)Substituting (7.30) and (7.31) into (7.28), and applying Young’s inequality, we arrive at k u nt k C ( I T ,L ) + k u nt k L ( I T ,H ) . L ˜ B ( u , f ) . (7.32)Summing up (7.23) and (7.32), we conclude k ( u n , u nt ) k C ( I T ,H × L ) + k ( u n , u nt ) k L ( I T ,H ) . L ˜ B ( u , f ) . (7.33)In view of (7.33), the Banach–Alaoglu and Arzel`a–Ascoli theorems, up to the extraction of asubsequence (still labelled by u n ), we have, as n → ∞ , that( u n , u nt ) → ( u, u t ) weakly- ∗ in L ∞ ( I T , H × L ) , ( u n , u nt ) → ( u, u t ) weakly in L ( I T , H × H ) ,u n → u strongly in C ( I T , L ) , div A u = 0 a.e. in Ω T and u (0) = u , where u and u t are measurable functions defined on Ω T . Moreover, k ( u, u t ) k L ∞ ( I T ,L ) + k ( u, u t ) k L ( I T ,H ) . L q ˜ B ( u , f ) . (7.34)Therefore, we can take to the limit in (7.16) as n → ∞ , and obtain Z u t · ζ d y + ν Z ∇ A u : ∇ A ζ d y + κ Z u · ζ d y = Z f · ζ d y a.e. in I T , ∀ ζ ∈ H . (7.35)Now, we begin to show spatial regularity of u . Let us further assume that δ is so small that η satisfies (2.32) by virtue of Proposition 3.2. Denoting F := f − κu − u t , ˜ F := F ( ζ − , t ) and˜ J := J ( ζ − , t ), we see that ˜ F has the same regularity as that of F , i.e., k ˜ F k L ∞ ( I T ,L ) + k ˜ F k L ( I T ,H ) < ∞ . (7.36)47oreover, Z F ( y, t )d y = Z ˜ F ˜ J − d x. Applying the regularity theory of the Stokes problem, we see that there is a unique strong so-lution α ∈ L ∞ ( I T , H ) ∩ L ( I T , H ) with a unique associated function p ∈ L ∞ ( I T , H ) ∩ L ( I T , H ),such that ( ∇ p − ν ∆ α = ˜ F , div v = 0 . (7.37)Let ̟ = α ( ζ , t ) and q = p ( ζ , t ) − ( p ( ζ , t )) T , then ( ̟, q ) ∈ ( L ∞ ( I T , H ) ∩ L ( I T , H )) × ( L ∞ ( I T , H ) ∩ L ( I T , H )) satisfies the following system: ( ∇ A q − ν ∆ A ̟ = F, div A w = 0 for a.e. t ∈ I T . (7.38)By a density argument, the identity (7.35) also holds for ζ ∈ H with div A ζ = 0. This fact,together with (7.38), implies ∇ u = ∇ ̟ .Following the derivation of (3.36) with slight modification, we can get from (7.38) that fora.e. t ∈ I T , k q k i +1 . k∇ q k i . k ( f, u t ) k i for i = 0 , . (7.39)Similarly to the derivation of (7.39) with i = 1, we can derive from (7.38) that k ̟ k . k ( u, u t , ∇ q, f ) k , (7.40) k ̟ k , . k ( u, u t , ∇ q, f ) k . (7.41) k ̟ k . k ( u, u t , ∇ q, f ) k + k∇ η k k ̟ k , . (7.42)So, it follows from (7.13), (7.34) and (7.39)–(7.42) that k ( u, u t , q ) k L ∞ ( I T ,H × L × H ) . k ( u, u t , f ) k L ∞ ( I T ,L ) . L q ˜ B ( u , f ) (7.43)and k ( u, u t , q ) k L ( I T ,H × H × H ) . (1 + k∇ η k ) k ( u, u t , f ) k L ( I T ,H ) . L p B ( u , f ) . (7.44)Combining (7.43) with (7.44), one obtains k ( u, u t , q ) k L ∞ ( I T ,H × L × H ) + k ( u, u t , q ) k L ( I T ,H × H × H ) . L p B ( u , f ) . (7.45)This completes the existence of local strong solutions. Moreover, a strong solution, which enjoysthe regularity of ( η, u ) constructed above, is obviously unique.(2) Strong continuity of ( u t , u, q ) on I T with values in L × H × H .For any given ϕ ∈ H , let ψ = ϕ ( ζ ( y, t )). Noting J t = J div A w , we can derive from (7.38) with v in place of w that for any φ ∈ C ∞ ( I T ), − Z t φ t Z u · ψJ d y d τ = Z t φ Z ( f + ν ∆ A u − κu − ∇ A q ) ψJ d y d τ − Z t φ Z w · ∇ A u · ψJ d y d τ. v = u ( ζ − ( x, t ) , t ), ˜ w = w ( ζ − ( x, t ) , t ) and g = f ( ζ − ( x, t ) , t ), then from the above identitywe can get that − Z t φ t Z v · ϕ d x d τ = Z t φ Z ( f + ν ∆ A u − κu − ∇ A q + w · ∇ A u ) · ψJ d y d τ = Z t φ Z ( g + ν ∆ v − κv − ∇ p − ˜ w · ∇ v ) · ϕ d x d τ, (7.46)which immediately results in v t = g + ν ∆ v − κv − ∇ p − ˜ w · ∇ v ∈ L ∞ ( I T , L ) (7.47)and Z v t · ϕ d x = Z ( f + ν ∆ A u − κu − ∇ A q − w · ∇ A u ) · ψJ d y. (7.48)We find by (7.38) and (7.47) that u t = ( v t + ˜ w · ∇ v ) | x = ζ = v t | x = ζ + w · ∇ A u. (7.49)Now let us further assume ϕ ∈ H σ , then div A ψ = 0. Recalling ∂ j ( J A ij ) = 0, the identity(7.48) impliesdd t Z v t · ϕ d y = < f t , ψJ > H − ,H + Z ( f ( ψJ ) t − ∂ t ((( κu + w · ∇ A u ) · ψ + ν ∇ A u : ∇ A ψ ) J ))d y =: < χ, ϕ > H − σ ,H σ . (7.50)Noting that k ψ k . k ϕ k . k ψ k for any t ∈ I T , and recalling the definition of <χ, ϕ > H − σ ,H σ , we have χ ∈ L ( I T , H − σ ). Therefore, v tt = χ and v t ∈ C ( I T , L ). Conse-quently, the identity (7.49) implies u t ∈ C ( I T , L ). Since u ∈ L ( I T , H ) and u t ∈ L ( I T , H ), u ∈ C ( I T , H ). Hence, u ∈ U ,T . In addition, we can derive from (7.38) that q ∈ C ( I T , H ) forsufficiently small δ . Thanks to the strong continuity of ( u t , u, q ) on I T with values in L × H × H ,we immediately get (7.6) from (7.45).(3) More regularities of q under the case “ f = ∂ η ”. Obviously, (7.7) holds for f = ∂ η . Keeping in mind that p is in C ( I T , H ), we obtain from(7.37) that Z ∇ p · ∇ ϕ d x = Z ( g − ˜ w · ∇ v ) · ∇ ϕ d x for any t ∈ I T , (7.51)which implies that p ( t ) ∈ H ( t ∈ I T ) satisfies∆ p = div g − ∇ ˜ w : ∇ v T for any t ∈ I T . (7.52)Then, q ( t ) ∈ H satisfies (7.8). Thus, the estimate (7.9) follows from (7.6)–(7.8).Recalling the derivation of (3.36) with i = 1 and the fact k u ( τ ) | τ = tτ = s k R ts k u τ k d τ , we seethat q ∈ AC ( I T , H ), i.e., q is absolutely contiguous in I T with respect to the norm H . Next, weproceed to show that q t exists and enjoys the estimate (7.10).By virtue of the Riesz representation theorem, it is easy to show that there is a unique function χ ∈ L ( I T , H ), such that − Z ∇ A χ · ∇ A ς d y = Z (cid:0) ∂ t ( m A T ∂ η + A T t u ) + A T ∇ A t q + A T t ∇ A q (cid:1) · ∇ ς d y, ς ∈ H . (7.53)49oreover, w enjoys the following estimate k χ k L ( I T ,H ) . L k ( u, u t , q ) k C ( I T ,H × L × H ) (cid:0) k∇ u k + k∇ ( w, w t ) k L ( I T ,H × L ) (cid:1) + k∇ w k C ( I T ,H ) . (7.54)Let t ∈ I T and D h ϑ = (cid:0) ϑ ( y, t + h ) − ϑ ( y, t ) (cid:1) /h where t + h ∈ I T . Multiplying (7.8) by ς in L and applying then D h to the resulting equation, we get − Z ∇ A D h q · ∇ A ς d y = Z ( D h ( m A T ∂ η + A T t u ) + A T ∇ D h A q ( t + h )+ D h ( A T ) ∇ A ( y,t + h ) q ( y, t + h )) · ∇ ς d y. (7.55)Subtracting (7.53) from (7.55) and denoting ς = D h q − χ , we have k D h q − χ k . L k ( D h − ∂ t )( m A T ∂ η + A T t u ) k + kA T ∇ D h A−A t q ( y, t + h ) k + kA T ∇ A t ( q ( t + h ) − q ( t )) k + k ( D h − ∂ t ) A T ∇ A ( y,t + h ) q ( y, t + h ) k + kA T t ( ∇ A ( y,t + h ) q ( y, t + h ) − ∇ A q ) k =: Θ( t ) for a.e. t ∈ I T . Noting that the generalized derivative with respect to t is automatically strong derivative,we easily see that Θ( t ) → t ∈ I T . So, k ( D h q − χ ) k → h → t ∈ I T .This means that the strong derivative of q with respect to t is equal to that of χ . Because of q ∈ AC ( I T , H ), q t = χ , where q t denotes the generalized derivative of q . Hence, q t ∈ L ( I T , H )satisfies the estimate (7.10) by (7.7), (7.43) and (7.54). This completes the proof of Proposition7.1. (cid:3) Now, we turn to establishing the existence and uniqueness of classical solutions to the initial-value problem (7.3).
Proposition 7.2.
Let δ > . Under the assumptions of Proposition 7.1 with f = ∂ η , assumefurther that ( η , u ) ∈ H × H , k∇ η k δ and q k∇ ( w, w t ) k C ( I T ,H × H ) + k w tt k C ( I T ,L ) + k∇ ( w t , w tt ) k L ( I T ,H × L ) B . (7.56) Then, there is a sufficiently small constant δ L2 δ L1 , such that for any δ δ L2 , the solution ( u, q ) constructed by Proposition 7.1 belongs to U T × C ( I T , H ) , where δ L2 is independent of m and ν .Moreover, k u k U ,T + k q t k C ( I T ,H ) . κ (cid:0) k∇ η k (cid:1) (1 + k u k + k u k k∇ w t | t =0 k ) , (7.57) k q k C ( I T ,H ) . κ k∇ η k + k w k C ( I T ,H ) k u k C ( I T ,H ) , (7.58) k∇ η k δ, and k u t | t =0 k . k u k . (7.59) Proof.
Let ( u, q ) be constructed in Proposition 7.1 and k∇ η k , δ δ L1 . Recalling T = min { , ( δ/B ) } , (7.60)we see that k∇ η k δ for any t ∈ I T . (7.61)50e remark here that the smallness of δ will be used in the derivation of some estimates later.From now on, we denote D t σ := σ t − Rσ .By (7.8) we see that q ∈ H satisfies∆ A q = div A ∂ η + div A t | t =0 u . Thanks to (7.61), we can apply a standard difference quotient method to deduce that q ∈ H .Moreover, k∇ q k . k ∂ η k + k u k . k u k . (7.62)Recalling u t + ∇ A q − ν ∆ A u + κu = ∂ η , we have k u t | t =0 k . k∇ q k + k u k . k u k , (7.63)which gives k D t u | t =0 k . k u k . (7.64)Noting that k∇ w k Z t k∇ w t k d τ + k u k , (7.65)we make use of (7.7), (7.56), (7.60) and (7.65) to deduce from the definition of B ( u , ∂ η ) that B ( u , ∂ η ) . L k u k . (7.66)Similarly, we can obtain by using (7.6), (7.9) and (7.10) that k u k U ,T + k q k C ( I T ,H ) + k q t k L ( I T ,H ) . L k u k . (7.67)Let F = ν (div A t ∇ A u T + div A ∇ A t u T + ∆ A ( Ru ) − R ∆ A u ) + R ∇ A q − ∇ A t q − R t u . One mayuse (7.56), (7.60), and (7.61), (7.64) and (7.65) to verify that B ( D t u | t =0 , D t ∂ η + F ) . k u k + kF k C ( I T ,L ) + kF k L ( I T ,H ) + kF t k L ( I T ,H − ) + (1 + k u k ) (cid:16) k u k + kF k L ( Q T ) (cid:17) . (7.68)Thanks to (7.10), (7.56), (7.60), (7.65) and (7.67), we have the following upper bounds for F and F t : kF k L ( I T ,H ) . L T sup t ∈ I T ( k u k ( k∇ w k + k∇ w k + k∇ w t k ))+ k∇ w k ( k u k L ( I T ,H ) + k∇ q k L ( I T ,H ) ) . L k u k and kF t k L ( I T ,H − ) . L T sup t ∈ I T (cid:0) ( k∇ w k + k∇ w k + k∇ w t k + k∇ w k k∇ w t k )( k u k + k u t k + k∇ q k ) (cid:1) + sup t ∈ I T k∇ w k ( k u t k L ( I T ,H ) + k∇ q t k L ( I T ,L ) )+ Z T (cid:16)p k w tt k k w tt k k u k + p k∇ w t k k∇ w t k ( k u k + k u t k ) (cid:17) d t . L k u k . kF k C ( I T ,L ) . kF k L ( I T ,H ) + kF k L ( I T ,H − ) . L k u k . Substitution of the above three estimates into (7.68) yields q B ( D t u | t =0 , D t ∂ η + F ) . L k u k . (7.69)Now, let us consider the problem U t + ∇ A Q − ν ∆ A U + κU = D t ∂ η + F , div A U = 0 ,w | t =0 = D t u | t =0 . (7.70)Recalling div A ( D t u ) | t =0 = 0, and using (7.64), (7.65) and (7.69), we can apply Proposition 7.1to (7.70) to see that the initial-value problem (7.70) admits a unique strong solution ( U, Q ) ∈U ,T × ( C ( I T , H ) ∩ L ( I T , H )) , which satisfies k U k U T + k Q k C ( I T ,H ) . L k u k . (7.71)Let ˜ U = U ( ζ − ( x, t ) , t ) and ˜ U ( x,
0) = ( D t u | y = ζ − ( x,t ) ) | t =0 . Similarly to (7.46), we can obtainfrom (7.70) that − Z T φ Z ˜ U · ξ d x d t + Z φ Z ( ν ∇ ˜ U : ∇ ξ + κ ˜ U · ξ )d x d t = Z T φ Z (( D t ∂ η + F ) | y = ζ − ( x,t ) − ˜ w · ∇ ˜ U ) · ξ d x d t, φ ∈ C ∞ ( I T ) , ξ ∈ H σ , which implies dd t Z ˜ U · ξ d x + Z ( ν ∇ ˜ U : ∇ ξ + κ ˜ U · ξ )d x = Z (( D t ∂ η + F ) | y = ζ − ( x,t ) − ˜ w · ∇ v ) · ξ d x for a.e. t ∈ I T . (7.72)Since u solves (7.3) with f = ∂ η , we utilize the regularity of u to find thatdd t Z g Du t · ξ d x + Z ( ν ∇ g Du t : ∇ ξ + κ g Du t · ξ )d x = Z (( D t ∂ η + F ) | y = ζ − ( x,t ) − ˜ w · ∇ v ) · ξ d x for a.e. t ∈ I T , (7.73)where g Du t := Du t | x = ζ − ( y,t ) and g Du t | t =0 = ˜ U ( x, U = g Du t , which givest U = D t u . Thus, in view of (7.6), (7.66) and (7.71), we have k u t k U T . L k u k + k u k k∇ w t | t =0 k . (7.74)Taking into account the regularity of u , we get from (7.3) that( D t u ) t + ∇ A q t − ν ∆ A D t u + κD t u = D t ∂ η + F . (7.75)52hus, Q = q t by virtue of (7.70) and (7.75).Similarly to the derivation of (7.6), we can obtain k u k C ( I T ,H ) + k u k L ( I T ,H ) . L (cid:0) k∇ η k (cid:1) (1 + k u k + k u k k∇ w t | t =0 k ) , (7.76)which, together with (7.71) with q t in place of Q and (7.74), yields (7.57). Employing the samearguments as in the proof of (7.9), one gets (7.58). Finally, the two estimates in (7.59) areobvious to get by using (7.61) and (7.63). This completes the proof of Proposition 7.2. (cid:3) Now we are in a position to show Proposition 3.1. To start with, let ( η , u ) satisfy all theassumptions in Proposition 3.1 and k∇ η k , δ δ L1 . We should remark here that the smallnessof δ (independent of m and ν ) will be frequently used in the calculations that follow.Denote B := 2 c L (1 + B ) , (7.77)where B comes from Proposition 3.1 and the constant c L is the same as in (7.11). By Proposition7.1 with B defined by (7.77) and Remark 7.11, one can easily construct a function sequence { u k , q k } ∞ k =1 defined on Ω T with T satisfying (7.4). Moreover, • for k >
1, ( u k +1 , q k +1 ) ∈ U ,T × C ( I T , H ) and η k = R t u k d τ + η ,u k +1 t + ∇ A k q k +1 − ν ∆ A k u k +1 = m ∂ η k , div A k u k +1 = 0 (7.78)with initial condition u k +1 | t =0 = u , where A k is defined by ζ k := η k + y ; • ( u , q ) is constructed by Proposition 7.1 with w = 0 and ∂ η in place of f ; • the solution sequence { u k , q k } ∞ k =1 satisfies the following uniform estimates: for all k > ∇ η k + I ) , k∇ η k k , δ for all t ∈ I T , (7.79) k u k k U ,T B and k∇ η k k + k q k k C ( I T ,H ) . L B . (7.80)In order to take limits in (7.78) as k → ∞ , we have to show that { u k , q k } ∞ k =1 is a Cauchysequence. To this end, we define for k > η k , ¯ u k +1 , ¯ A k , ¯ q k +1 ) := ( η k − η k − , u k +1 − u k , ˜ A k − ˜ A k − , q k +1 − q k ) , which satisfies ¯ η k = R t ¯ u k d τ, ∆¯ q k +1 = M k , ¯ u k +1 t + ∇ ¯ q k +1 − ν ∆¯ u k +1 − m ∂ ¯ η k = N k , div¯ u k +1 = − (div ¯ A k u k +1 + div ˜ A k − ¯ u k +1 ) , ¯ u k +1 | t =0 = 0 , (7.81)53here M k := m ∂ (div ¯ η k + div ¯ A k η k + div ˜ A k − ¯ η k )+ div ¯ A kt u k +1 + div A k − t ¯ u k − (div ¯ A k ∇ A k q k +1 + div ˜ A k − ∇ ¯ A k q k +1 + div ˜ A k − ∇ ˜ A k − ¯ q k +1 + div( ∇ ¯ A k q k +1 + ∇ ˜ A k − ¯ q k +1 )) , N k := ν (div ¯ A k ∇ A k u k +1 + div ˜ A k − ∇ ¯ A k u k +1 + div ˜ A k − ∇ A k − ¯ u k +1 + div ∇ ¯ A k u k +1 + div ∇ ˜ A k − ¯ u k +1 ) − ( ∇ ¯ A k q k +1 + ∇ ˜ A k − ¯ q k +1 ) . Keeping in mind that k ¯ A k k . (1 + B ) k∇ ¯ η k k . T / (1 + B ) k∇ ¯ u k k L ( I T ,H ) , k ¯ A kt k i . k∇ ¯ u k k i + B k∇ ¯ η k k , , i = 0 , , Z | ¯ A kt ||∇ u k +1 || ∆¯ q | d y . q k ¯ A kt k k ¯ A kt k k u k +1 k k ∆¯ q k , Z |A k − t ||∇ ¯ u k || ∆¯ q | d y . kA kt k p k∇ ¯ u k k k∇ ¯ u k k k ∆¯ q k , k∇ ¯ u k k T / k∇ ¯ u kt k L ( I T ,L ) , we make use of (3.4), (3.6), (7.79), (7.80) and the above five estimates to deduce from (7.81) –(7.81) that k∇ ¯ q k +1 k C ( I T ,H ) . L T / (1 + B )( k∇ ¯ u k k L ( I T ,H ) + k∇ ¯ u kt k L ( I T ,L ) + k∇ ¯ u k k C ( I T ,H ) ) , (7.82) k∇ ¯ u k +1 k C ( I T ,L ) + k∇ ¯ u k +1 k L ( I T ,L ) . L T (1 + B ) k∇ ¯ u k k L ( I T ,H ) + (1 + B )( k∇ ¯ q k +1 k C ( I T ,H ) + T / ( k∇ ¯ u k +1 k C ( I T ,L ) + k∇ ¯ u k +1 k L ( I T ) ,L ) )) , (7.83) k ¯ u k +1 t k C ( I T ,L ) + k ¯ u k +1 t k L ( I T ,H ) . L (1 + T / (1 + B )) k∇ ¯ u k +1 k C ( I T ,L ) + (1 + T / (1 + B )) k∇ ¯ u k +1 k C ( I T ,H ) + k∇ ¯ q k +1 k C ( I T ,H ) + T / (1 + B ) k∇ ¯ u k k L ( I T ,H ) . (7.84)Recalling (¯ u k +1 ) T = 0, we put (7.82)–(7.84) together to conclude that for sufficiently small T (depending possibly on B , ν and m ), k ¯ u k +1 k U ,T + k ¯ q k +1 k C ( I T ,H ) k ¯ u k k U ,T / k > , which implies ∞ X k =2 ( k ¯ u k k U T + k ¯ q k k C ( I T ,H ) ) < ∞ . Hence, { u k , q k } ∞ k =1 is a Cauchy sequence in U ,T × C ( I T , H ) and( η k , u k , q k ) → ( η, u, q ) strongly in C ( I T , H ) × U ,T × C ( I T , H ) , (7.85)where η := Z t u d τ + η . (7.86)54emembering that (7.86) implies η t = u , we infer from (7.78) and (7.85) that the limit ( η, u, q )is a solution to the initial-value problem (2.8)–(2.9). The uniqueness of solutions to (2.8)–(2.9) inthe function class C ( I T , H ) × U ,T × C ( I T , H ) is easily verified by a standard energy method,and its proof will be omitted here. The proof of Proposition 3.1 is complete.Similar to Proposition 3.1, we can use Proposition 7.2 to establish the following existence anduniqueness of a classical solution to the problem (7.3) with an additional damping term. Proposition 7.3.
Let δ > , ( η , u ) ∈ H × H and k∇ η k δ . Then, there is a sufficientlysmall constant δ a ∈ (0 , δ L2 ] , independent of m and ν , such that for any δ δ a , the initial-valueproblem (7.1) admits a unique local strong solution ( η ν , u ν , q ν ) ∈ C ( I T , H ) × U T × C ( I T , H ) ,where T := min { (2 c κ k ( u , m∂ η , √ ν ∇ η ) k ) − , δ ( √ c κ k ( u , m∂ η , √ ν ∇ η ) k ) − } , and the con-stant c κ is the same as in the definition of T . Moreover, ∇ η + I ) , k∇ η ν k δ, (7.87) k η ν k . κ k η k + t k ( u , m∂ η , √ ν ∇ η ) k for any t ∈ I T , (7.88) k ( u ν , m∂ η ν ) k C ( I T ,H ) + √ ν k u ν k L ( I T ,H ) . κ k ( u , m∂ η , √ ν ∇ η ) k , (7.89) k ( u νt , q ν , q νt ) k C ( I T ,H × H × H ) . κ I , (7.90) where I := (1 + k ( u , m∂ η , √ ν ∇ η ) k )( k ( u , m∂ η , √ ν ∇ η ) k + (1 + ν + m ) k ( u , m∂ η , √ ν ∇ η ) k ) . Proof.
We divided the proof into three steps.(1) Let δ ∈ (0 , δ L2 ], ( η , u ) ∈ H × H and k∇ η k δ . Let B κ > B κ > k ( ∇ η , u ) k and will be defined in (7.103). Thanks to Proposition7.2, we can follow the same arguments as in Section 7.2 to deduce that there is a constant δ a1 independent of ν and B κ , such that for any δ δ a1 , • there are a function sequence { η k , u k , q k } ∞ k =1 ∈ C ( I T ν , H ) × U T ν × C ( I T ν , H ) and a limitfunction ( η ν , u ν , q ν ), such that as k → ∞ ,( η k , u k , q k , q kt ) → ( η ν , u ν , q ν , q νt )in C ( I T ν , H ) × U T ν × C ( I T ν , H ) × C ( I I Tν , H ) , (7.91) η νt = u ν ,u νt + ∇ A q ν − ν ∆ A u ν + κu ν = m ∂ η ν , div A ν u ν = 0 , ( η ν , u ν ) | t =0 = ( η , u ) , (7.92)1 ∇ η ν + I ) , sup t ∈ I Tν k∇ η ν k k∇ η k + δ < δ ′ := 4 δ, (7.93)where the local existence time T νδ ∈ (0 ,
1] may depend on B κ , ν , m and δ . • the function ( η ν , u ν , q ν ) is just the unique solution of (7.92), i.e., if there is another solution(˜ η ν , ˜ u ν , ˜ q ν ) in C ( I T ν , H ) × U T ν × C ( I T ν , H ) satisfying 0 < inf ( y,t ) ∈ R × I T det( ∇ ˜ η ν + I ),then ( η ν , u ν , q ν ) = (˜ η ν , ˜ u ν , ˜ q ν ) by using the smallness condition “sup t ∈ I T k∇ η ν k δ ”.55rom now on, we further take δ δ a1 /
2, then the definition T ν min := min { T νδ , T ν δ } makes sense.(2) Noting that (6.8) holds for any χ ∈ L , anddiv η ν = ∂ η ν ∂ η ν − ∂ η ν ∂ η ν + det( ∇ η + I ) | t =0 − , we have the inequality: k ( u ν , m∂ η ν ) k . k ( u ν , ∇ curl A ν ( u ν , m∂ η ν )) k + k η ν k k ( u ν , m∂ η ν ) k + k η ν k k u ν k + k η ν k k m∂ η ν k + k m∂ η k . (7.94)On the one hand, remembering that sup t ∈ I Tν min k∇ η k δ ′ in (7.93), we can use (7.92) , (7.92) and (7.94), and follow the same process (under slight modifications) as in the derivation of (6.13),to deduce that there is a constant δ a2 independent of ν and B κ , such that for any δ ′ δ a2 , thesolution ( η ν , u ν ) satisfiesdd t k ( u ν , m∂ η ν , ∇ curl A ν ( u ν , m∂ η ν )) k + ν k u ν k + κ k u ν k / . κ k ( u ν , m∂ η ν ) k + ν k u ν k k∇ η ν k , (7.95)where k ( u ν , m∂ η ν ) k . k ( u ν , ∇ curl A ν ( u ν , m∂ η ν )) k + k m∂ η k , k ( u ν , ∇ curl A ν ( u ν , m∂ η ν )) k . k ( u ν , m∂ η ν ) k . On the other hand, by (7.92) we find that k∇ η ν ( t ) k k u ν k L ( I t ,H ) + k∇ η k . (7.96)Thus, one concludes from (7.95)–(7.96) that for any t ∈ (0 , T ν min ], k ( u ν , m∂ η ν ) k + ν Z t k u ν k d τ c κ k ( u , m∂ η , √ ν ∇ η ) k − c κ t k ( u , m∂ η , √ ν ∇ η ) k . (7.97)In particular, taking ˜ T = 1 / c κ k ( u , m∂ η , √ ν ∇ η ) k , we derive from (7.97) that for any t min { ˜ T , T ν min } , k ( u ν , m∂ η ν ) k + √ ν k u ν k L ( I t ,H ) √ c κ k ( u , m∂ η , √ ν ∇ η ) k =: B κ . (7.98)Let T min = min { ˜ T , T ν min , δ/B κ } .With the help of (7.97)–(7.98), we then infer from (7.92) that for any t T min , k η ν k k η k + tB κ , (7.99) k∇ η ν k k∇ η k + tB κ δ, (7.100) k∇ η ν k . κ k∇ η k + k ( u , m∂ η , √ ν ∇ η ) k p c κ /ν =: B κ . (7.101)Furthermore, by (7.100), (7.92) and (7.92) , we find that there is a constant δ a3 independent ofany parameters, such that for any δ δ a3 , k ( u νt , q ν , q νt ) k C ( I T min ,H × H × H ) . κ I . (7.102)563) Now, if we take B κ := max { B κ / , B κ / , k ( ∇ η , u ) k } , (7.103)then we have by (7.98) and (7.101) that k ( ∇ η, u ν ) k B κ for any t T min . (7.104)Denote ˜ T min = { ˜ T , δ/B κ } . If ˜ T min T ν , we easily see that the conclusion in Proposition 7.3holds for any given δ ∈ (0 , δ min ], where δ min := min { δ a1 , δ a2 / , δ a3 } . Next, we consider the case˜ T min > T ν min .For the case ˜ T min > T ν min , we can take ( η ( T ν min ) , u ( T ν min )) as initial data. By (7.100), (7.104)and Step (1), we see that for any δ δ min /
2, there exists a unique local solution ( η ∗ , u ∗ , q ∗ )defined on T × ( T ν min , T ν min ] of the initial-value problem: η ∗ t = u ∗ t ,u ∗ t + ∇ A q ∗ − ν ∆ A u ∗ + κu ∗ = m ∂ η ∗ , div A ∗ u ∗ = 0 , ( η ∗ , u ∗ ) | t = T ν = ( η ν ( T ν ) , u ν ( T ν )) . Moreover, 1 ∇ η ∗ + I ) k∇ η ∗ ( t ) k δ ′ for any t ∈ ( T ν min , T ν min ]. Due tothe uniqueness, we can get a new local solution defined on T × (0 , T ν min ], still denoted thisnew solution by ( η ν , u ν , q ν ). By Step (2) we find that for any δ δ min /
2, the new solutionsatisfies (7.98)–(7.100), (7.102) and (7.104) for any t ∈ (0 , min { ˜ T min , T ν min } ]. Let n = [( ˜ T min − T ν min ) /T ν min ] + 1, where [ · ] means the integer part. Therefore, by performing n -times extensionwith respect to time, we obtain the desired conclusion in Proposition 7.3. (cid:3) With the help of Proposition 7.3, we are able to prove Proposition 6.2 by the method ofvanishing viscosity limit.Let B κ > δ ∈ (0 , δ a ], ( η , u ) satisfy the assumptions in Proposition 6.2 and k∇ η k δ ,where δ a is the same constant as in Proposition 7.3.Let ε ∈ (0 ,
1) and S ε be a standard mollifier (or a regularizing operator, see [22, Section1.3.4.4] for definition). It is well-known that S ε ( ϑ ) ∈ C ∞ ( T ), and k S ε ( ϑ ) k i ˜ c i k ϑ k i for ϑ ∈ H i and i >
0, where the positive constant ˜ c i depends on i only. Let ( η ε , u ε , ˜ η ε , ˜ u ε ) = S ε ( η , u , ˜ η , ˜ u ).We now fix ε >
0. By virtue of Proposition 7.3, there is a sufficiently small ν (dependingpossibly on k S ε ( η ) k , k ( u , m∂ η ) k and k∇ η k ), such that for any ν < ν , there exists aunique solution ( η ν , u ν , q ν ) ∈ C ( I T , H ) × U T × C ( I T , H ) to the initial-value problem: η ν = R t u ν d τ + η ε ,u νt + ∇ A ν q ν + κu ν − ν ∆ A ν u ν = m ∂ η ν , div A ν u ν = 0 ,u ν | t =0 = u ε in T , (7.105)where A ν = ( ∇ η ν + I ) − T , T := min { (1 + 2 c κ k ( u , m∂ η ) k ) − , δ (1 + √ c κ k ( u , m∂ η ) k ) − } .57oreover, the solution satisfies the uniform estimates:1 ∇ η ν + I ) , k∇ η ν k . δ, (7.106) k η ν k . κ k ( η , u , m∂ η , √ ν ∇ η ) k for each t ∈ I T , (7.107) k ( u ν , m∂ η ν ) k C ( I T ,H ) + √ ν k u ν k L ( I T ,H ) . κ k ( u , m∂ η ) k , (7.108) k ( u νt , q ν , q νt ) k C ( I T ,H × H × H ) . κ ((1 + m )(1 + k ( u , m∂ η ) k )) , (7.109) ν k η ν k . ν k η ε k + tν Z t k u ν k d τ . ν k η ε k + t k ( u , m∂ η ) k . (7.110)From now on, we take ν = ν n = 1 /n with n > /ν , and renew to define ( η ν n , u ν n , q ν n ) and A ν n by ( η n , u n , q n ) and A n , respectively.Let ( η L , u L ) ∈ C ( R + , H ) × U ∞ be the unique global solution of the linear initial-value problem(6.38). In view of Proposition 6.1, we see that there is a unique strong solution ( η ε, L , u ε, L ) to(6.38) with (˜ η ε , ˜ u ε ) in place of (˜ η , ˜ u ). Moreover, • for i >
0, ( η ε, L , u ε, L ) ∈ C ( R + , H i +12 ) × U i ∞ , and for any t > k ( u ε, L , m∂ η ε, L ) k i + Z t k u ε, L k i d τ . κ k (˜ u ε , m∂ ˜ η ε ) k i ; (7.111) • for any t ∈ I T ,( η ε, L , u ε, L , ∂ η ε, L ) → ( η L , u L , ∂ η L ) weakly- ∗ in L ∞ ( I t , H ) as ε → . (7.112)Defining ( η n,ε, d , u n,ε, d ) := ( η n , u n ) − ( η ε, L , u ε, L ), then η n,ε, d t = u n,ε, d ,u n,ε, d t + ∇ A n q n + κu n,ε, d = m ∂ η n,ε, d + ν ∆ A n u ν , div u n,ε, d = − div ˜ A n u, ( η n,ε, d , u n,ε, d ) | t =0 = ( η ε − ˜ η ε , u − ˜ u ε ) . (7.113)It is easy to see from (7.105) , (7.106) and (7.113) that for any t ∈ I T , k curl A n ( u n,ε, d , m∂ η n,ε, d ) k . e t k ( u n ,m∂ η n ) k C It,H × H k ( u n,ε, d , m∂ η n,ε, d ) | t =0 k + t sup τ ∈ I t ( k ( u n , m∂ η n , m∂ η ε, L k )+ νt sup τ ∈ I t ( k η n k k ( u ε, L , u n ) k + k η n k ( k u ε, L k + k u n k )) + ν Z t k u ε, L k d τ ! . and k curl A n η n,ε, d k . k curl A n η n,ε, d | t =0 k + t sup τ ∈ I t k curl A n u n,ε, d k .
58f we making use of (7.108), (7.110), (7.111) and the above two estimates, we further obtainsup τ ∈ I t k curl A n ( η n,ε, d , u n,ε, d , m∂ η n,ε, d )( τ ) k . κ e t k ( η ,u ,m∂ η ) k ( k ( η − ˜ η , u − ˜ u , m∂ ( η − ˜ η )) k + t ( k ( u , m∂ η , ˜ u , m∂ ˜ η ) k + ( t k ( u , m∂ η ) k + ν k η ε k ) k ( u , m∂ η , ˜ u , m∂ ˜ η ) k + √ ν ( √ t k ( u , m∂ η ) k + √ ν k η ε k )( k (˜ u ε , m∂ ˜ η ε ) k + k ( u , m∂ η ) k ) + ν k (˜ u ε , m∂ ˜ η ε ) k ) . (7.114)Thanks to the uniform estimates (7.106)–(7.109) and (7.114), we can choose a sequence of { η n , u n , q n } (still labelled by { η n , u n , q n } for the sake of simplicity), such that as n → ∞ ,( η n , ∂ η n , u n , q n ) → ( η ε , ∂ η, u ε , q ε ) strongly in C ( I T , H × H × H × H ) , ( η n , u n , u nt , q n , q nt ) → ( η ε , u ε , u εt , q ε , q εt ) weakly- ∗ in L ∞ ( I T , H × H × H × H × H ) , (curl A n η n,ε, d , curl A n u n,ε, d , curl A n ∂ η n,ε, d ) → (curl A ε η ε, d , curl A ε u ε, d , curl A ε ∂ η ε, d )weakly- ∗ in L ∞ ( I t , H ) for any t ∈ (0 , T ] , where ( η ε, d , u ε, d ) := ( η ε − η ε, L , u ε − u ε, L ). Moreover, the limit functions η ε , u ε and q ε solve theinitial-value problem η εt = u ε ,u εt + ∇ A q ε + κu ε = m ∂ η ε , div A u ε = 0 ,u ε | t =0 = u , (7.115)and satisfies1 ∇ η ε + I ) , k∇ η ε k L ∞ ( I T ,H ) δ, (7.116)ess sup t ∈ I T k ( η ε , u ε , m∂ η ε ) k . κ k ( η , u , m∂ η ) k , (7.117) k ( u εt , q ε , q εt ) k L ∞ ( I T ,H × H × H ) . κ ((1 + m )(1 + k ( η , u , m∂ η ) k )) , (7.118)ess sup τ ∈ I t k (curl A η ε,d , curl A u ε,d , curl A m∂ η ε,d )( τ ) k . e t k ( η ,u ,m∂ η ) k k ( η − ˜ η , u − ˜ u , m∂ ( η − ˜ η ) k + t ( k ( u , m∂ η , ˜ u , m∂ ˜ η ) k + k ( u , m∂ η ) k k ( u , m∂ η , ˜ u , m∂ ˜ η ) k ) . (7.119)In addition, by u ε ∈ L ∞ ( I T , H ), one has η ε ∈ C ( I T , H ) . (7.120)Let α and β satisfy | α | = 4 and | β | = 3. With the help of the uniform estimates (7.106)–(7.109),we can use (7.105) and (7.105) to deduce that the sequences { ∂ α u n } , { ∂ β (curl A n u n ) } , { ∂ α ∂ η n } and { ∂ β curl A n ∂ η n } are uniformly continuous in H − on I . Besides, they are also uni-formly bounded in L . Therefore, there is a sequence of { η n , u n , q n } (still denoted by { η n , u n , q n } ),such that (see [22, Lemma 6.2] for example), ∂ α ( u n , ∂ η n ) → ∂ α ( u ε , ∂ η ε ) in C ( I T , L ) , ∂ α ( u ε , ∂ η ε ) on a set of zero-measure in I T . Thus, by (6.45), (7.120) andthe above result of weak continuity, the notation “ess” can be removed in (7.117) and (7.119).Thanks to the uniform estimates (7.116)–(7.118) and the limit behavior (7.112), we can againtake to the limit as ε → η ε , u ε , q ε ), andthus obtain a limit function ( η, u, q ) which is a solution of the initial-value problem (2.47)–(2.48)and satisfies the estimates (7.118) with ( η, u, q ) in place of ( η ε , u ε , q ε ), (6.40)–(6.42), and the sameregularity as thst of ( η ε , u ε , q ε ). Moreover, the obtained solution ( η, u, q ) is unique, provided that δ is sufficiently small. To complete the proof of Proposition 7.3, obviously, it suffices to show thestrong continuity of ∂ α ( u, q ) in L with respect to time for any α satisfying | α | = 4. Next we willverify this fact.To begin with, we easily see by a straightforward calculation that ∂ t S ε ( ∂ β curl A u ) + κS ε ( ∂ β curl A u ) − m ∂ S ε ( ∂ β curl A ∂ η )= S ε ( ∂ β curl A t u ) − m S ε ( ∂ β curl ∂ A ∂ η ) , (7.121)and ∂ t S ε ( ∂ β curl A ∂ η ) = ∂ S ε ( ∂ β curl A u ) + S ε ( ∂ β curl A t ∂ η ) − S ε ( ∂ β curl ∂ A u ) . Thanks to the above two identities, we further get that for a.e. t ∈ I T , Z ∂ t S ε ( ∂ β curl A u ) S ε ( ∂ β curl A u )d y = 12 dd t Z | S ε ( ∂ β curl A u ) | d y, − Z ∂ S ε ( ∂ β curl A ∂ η ) S ε ( ∂ β curl A u )d y = 12 dd t Z | S ε ( ∂ β curl A ∂ η ) | d y + Z S ε ( ∂ β curl A ∂ η )( S ε ( ∂ β curl ∂ A u ) − S ε ( ∂ β curl A t ∂ η ))d y, where k S ε ( ∂ β curl A u ) k and k S ε ( ∂ β curl A ∂ η ) k ∈ AC ( I T \ ˜ Z ) for some zero-measurable set ˜ Z .For φ ∈ C ∞ ( I T ), we multiply (7.121) by S ε ( ∂ β curl A u ) φ in L (Ω T ), and take then to the limitsas ε → − Z T Z ( | ∂ β curl A u | + m ∂ β curl A ∂ η | )d yφ τ d τ + κ Z T Z | ∂ β curl A u | d yφ d τ = Z T Z ( ∂ β curl A t u∂ β curl A u d y + m ( ∂ β curl A ∂ η ( ∂ β curl A t ∂ η − ∂ η∂ β curl ∂ A u ) − ∂ β curl ∂ A ∂ η∂ β curl A u ))d yφ d τ, which implies that for any β satisfying | β | = 3,12 dd t k ∂ β (curl A u, m∂ β curl A ∂ η ) k + κ k ∂ β curl A u k = Z ( ∂ β curl A t u∂ β curl A u d y + m ( ∂ β curl A ∂ η∂ β curl A t ∂ η − ∂ β curl ∂ A ∂ η∂ β curl A u − ∂ β curl A ∂ η∂ β curl ∂ A u ))d y for a.e. t ∈ I T , whence, k ∂ β (curl A u, m curl A ∂ η ) k ∈ AC ( I T \ Z ) for some zero-measurable set Z. (7.122)60y (7.122) and the fact “ ∂ β (curl A u, curl A ∂ η ) ∈ C ( I T , L )”, we immediately get ∂ α (curl A u, curl A ∂ η ) ∈ C ( I T \ Z, L ) . (7.123)Since ∂ α ( u, ∂ η ) ∈ C ( I T , L ), one has by (6.45) thatsup t ∈ I T k ∂ α ( u, ∂ η ) k = ess sup t ∈ I T k ∂ α ( u, ∂ α ∂ η ) k . Keeping in mind that for any t and t ∈ I T , k ∂ β ∇ ( u ( t ) − u ( t )) k = q k ∂ β curl( u ( t ) − u ( t )) k + k ∂ β div( u ( t ) − u ( t )) k . k ∂ β (curl A u ( t ) − curl A u ( t )) k + k ∂ β (curl A u ( t ) − curl A u ( t )) k + k ∂ β (div ˜ A u ( t ) − div ˜ A u ( t )) k + k ∂ β (div ˜ A u ( t ) − div ˜ A u ( t )) k and k ∂ β ∇ ∂ ( η ( t ) − η ( t )) k = k ∂ β curl ∂ ( η ( t ) − η ( t )) k + k ∂ β div ∂ ( η ( t ) − η ( t )) k . k ∂ β (curl A ∂ η ( t ) − curl A ∂ η ( t )) k + k ∂ β (curl A ∂ η ( t ) − curl A ∂ η ( t )) k + k ∂ β ∂ (( ∂ η ∂ η − ∂ η ∂ η )( t ) − ( ∂ η ∂ η − ∂ η ∂ η )( t )) k , we employ the regularity ( η, u ) ∈ C ( I T , H × H ) to arrive at ∂ α ( u, ∂ η ) ∈ C ( I T \ Z, L ).Consequently, ∂ α q ∈ C ( I T \ Z, L ). The proof of Proposition 7.3 is complete. Acknowledgements.
The research of Fei Jiang was supported by NSFC (Grant Nos. 11671086and 12022102) and the Natural Science Foundation of Fujian Province of China (2020J02013),and the research of Song Jiang by National Key R&D Program (2020YFA0712200), NationalKey Project (GJXM92579), and NSFC (Grant No. 11631008), the Sino-German Science Center(Grant No. GZ 1465) and the ISF-NSFC joint research program (Grant No. 11761141008).
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