ANALYSIS OF A GENERAL FAMILY OF REGULARIZED NAVIER-STOKES AND MHD MODELS

ANALYSIS OF A GENERAL FAMILY OF REGULARIZED NAVIER-STOKES AND MHD MODELS MICHAEL HOLST, EVELYN LUNASIN, AND GANTUMUR TSOGTGEREL ABSTRACT. We consier a general family of regularize Navier-Stokes an Magnetohyroynamics (MHD) moels on n-imensional smooth compact Riemannian manifols with or without bounary, with n 2. This family captures most of the specific regularize moels that have been propose an analyze in the literature, incluing the Navier-Stokes equations, the Navier-Stokes-α moel, the Leray-α moel, the Moifie Leray-α moel, the Simplifie Barina moel, the Navier-Stokes-Voight moel, the Navier-Stokes-α-like moels, an certain MHD moels, in aition to representing a larger 3-parameter family of moels not previously analyze. This family of moels has become particularly important in the evelopment of mathematical an computational moels of turbulence. We give a unifie analysis of the entire three-parameter family of moels using only abstract mapping properties of the principal issipation an smoothing operators, an then use assumptions about the specific form of the parameterizations, leaing to specific moels, only when necessary to obtain the sharpest results. We first establish existence an regularity results, an uner appropriate assumptions show uniqueness an stability. We then establish some results for singular perturbations, which as special cases inclue the invisci limit of viscous moels an the α limit in α moels. Next we show existence of a global attractor for the general moel, an then give estimates for the imension of the global attractor an the number of egrees of freeom in terms of a generalize Grashof number. We then establish some results on etermining operators for the two istinct subfamilies of issipative an nonissipative moels. We finish by eriving some new length-scale estimates in terms of the Reynols number, which allows for recasting the Grashof number-base results into analogous statements involving the Reynols number. In aition to recovering most of the existing results on existence, regularity, uniqueness, stability, attractor existence an imension, an etermining operators for the well-known specific members of this family of regularize Navier-Stokes an MHD moels, the framework we evelop also makes possible a number of new results for all moels in the general family, incluing some new results for several of the well-stuie moels. Analyzing the more abstract generalize moel allows for a simpler analysis that helps bring out the core common structure of the various regularize Navier-Stokes an magnetohyroynamics moels, an also helps clarify the common features of many of the existing an new results. To make the paper reasonably self-containe, we inclue supporting material on spaces involving time, Sobolev spaces, an Gronwall-type inequalities. Date: December 19, 29. 1991 Mathematics Subject Classification. Primary 35Q3, 35Q35, 37L3, 76B3, 76D3; Seonary 76F2, 76F55, 76F65. Key wors an phrases. Navier-Stokes equations, Euler Equations, Regularize Navier-Stokes, Navier- Stokes-α, Leray-α, Moifie-Leray-α, Simplifie Barina, Navier-Stokes-Voight, Magnetohyroynamics, MHD. MH was supporte in part by NSF Awars 715146, 411723, an 511766, an DOE Awars DE- FG2-5ER2577 an DE-FG2-4ER2562. EL an GT were supporte in part by NSF Awars 715146 an 411723. 1

2 M. HOLST, E. LUNASIN, AND G. TSOGTGEREL CONTENTS 1. Introuction 2 2. Notation an preliminary material 7 3. Well-poseness results 11 3.1. Existence 11 3.2. Uniqueness an stability 15 3.3. Regularity 16 4. Singular perturbations 17 4.1. Perturbations to the linear part 18 4.2. Perturbations involving the nonlinear part 19 5. Global attractors 21 5.1. Existence of a global attractor 21 5.2. Estimates on the imension of the global attractor 23 6. Determining operators 24 6.1. Dissipative systems 26 6.2. Nonissipative systems 28 7. Length-scale estimates in terms of the Reynols number 3 8. Summary 32 9. Acknowlegements 33 Appenix A. Some key technical tools an some supporting results 34 A.1. A Grönwall type inequality 34 A.2. Spaces involving time 34 A.3. Multiplication in Sobolev spaces 34 References 34 1. INTRODUCTION The mathematical theory for global existence an regularity of solutions to the threeimensional Navier-Stokes equations (3D NSE) is consiere one of the most challenging unsolve mathematical problems of our time [16]. It is also well-known that irect numerical simulation of NSE for many physical applications with high Reynols number flows is intractable even using state-of-the-art numerical methos on the most avance supercomputers available. Over the last three ecaes, researchers have evelope turbulence moels as an attempt to sie-step this simulation barrier; the aim of turbulence moels is to capture the large, energetic eies without having to compute the smallest ynamically relevant eies, by instea moeling the effects of small eies in terms of the large scales in both NSE an magnetohyroynamics (MHD) flows. In 1998, the globally well-pose 3D Navier-Stokes-α (NS-α) equations (also known as the viscous Camassa-Holm equations an Lagrangian average Navier-Stokes-α moel) was propose as a sub-gri scale turbulence moel [8, 9, 1, 19, 2, 34, 51, 56] (see also [65] for n-imensional viscous Camassa-Holm equations in the whole space). The invisci an unforce version of 3D NS-α was introuce in [34] base on Hamilton s variational principle subject to the incompressibility constraint iv u = (see also [52]). By aing the correct viscous term an the forcing f in an a hoc fashion, the authors in [8, 9, 1] an [2] obtain the NS-α system. In references [8, 9, 1] it was foun that the analytical steay state solutions for the 3D NS-α moel compare well with average experimental ata from turbulent flows in channels an pipes for a wie range of large Reynols numbers. It was this fact which le [8, 9, 1] to suggest that the NS-α moel be use as a closure moel for the Reynols-average equations. Since then, it has been foun that there is in fact a whole family of globally well-pose α - moels which yiel similarly successful comparisons with empirical ata; among these are the Clark-α moel [4], the Leray-α moel [12], the moifie Leray-α moel [38], an the simplifie Barina moel [5, 46].

A GENERAL FAMILY OF REGULARIZED NAVIER-STOKES AND MHD MODELS 3 In aition to the early success of the α-moels mentione above, the valiity of the original α-moel (namely, the NS-α moel) as a sub-gri scale turbulence moel was also teste numerically in [11, 56]. In the numerical simulation of the 3D NS-α moel, the authors of [11, 27, 28, 56] showe that the large scales of motion bigger than α (the length scale associate with the with of the filter which regularizes the velocity fiel) in a turbulent flow are capture (see also [49, 5] for the 2D case an [7] for the rate of convergence of the 2D α-moels to NSE). For scales of motion smaller than the length scale α, the energy spectra ecays faster in comparison to that of NSE. This numerical observation has been justifie analytically in [19]. In irect numerical simulation, the fast ecay of the energy spectra for scales of motion smaller than the supplie filter length represents reuce gri requirements in simulating a flow. The numerical stuy of [11] gives the same results. The same results hol as well in the stuy of the Leray-α moel in [12, 27]. The NS-α turbulence moel has also been implemente in a primitive equation ocean moel (see [32, 33, 6]). Their simulations with the NS-α in an iealize channel omain was shown to prouce statistics which resembles oubling of resolution. For other applications of α regularization techniques, see [1] for application to the quasi-geostrophic equations, [45] for application to Birkhoff-Rott approximation ynamics of vortex sheets of the 2D Euler equations, an [48, 54, 55] for applications to incompressible magnetohyroynamics equations. In [6], an α-regularize nonlinear Schröinger equation was propose for the purpose of a numerical regularization that is hope to she some light on the profile of the blow-up solutions to the nonlinear Schröinger equations. Also, in [2], the authors exten the erivation of the invisci version of NS-α (calle Euler-α, also known as Lagrangian average Euler-α) to barotropic compressible flows. Perhaps the newest aition to the family of α turbulence moel is the Navier-Stokes- Voight (NSV) equations propose in [5], which turn out to also moel the ynamics of Kelvin-Voight viscoelastic incompressible fluis as introuce in [58, 59]. The statistical properties of 3D NSV have been stuie in [47]. The long-term asymptotic behavior of solutions is stuie in [42, 43]. In [18], the NSV was use in the context of image inpainting. The numerical stuy of NSV in [18] suggests that the NSV, in comparison with NSE, can provie a more efficient numerical process when automating the inpainting proceure for certain classes of images. It is worthwhile to note that the invisci NSV coincie with the invisci simplifie Barina moel which is shown in [5] to be globally well-pose. This new regularization technique for Euler equations prevents the risk of amping too much energy in the small scales which coul lea to unrealistic numerical results. As a representative of the more general moel consiere in this paper (escribe in etail in Section 2), we consier first the following somewhat simpler constraine initial value problem on an 3-imensional flat torus T 3 : t u + Au + (Mu )(Nu) + χ (Mu) T (Nu) + p = f(x), u =, u() = u, (1.1) where A, M, an N are boune linear operators having certain mapping properties, an where χ is either 1 or. As in prior work on regularize moels of the Navier-Stokes, an Euler equations, we employ a single real parameter θ to control the strength of the issipation operator A. We then introuce two parameters which control the egree of smoothing in the operators M an N, namely θ 1 an θ 2, respectively, when χ =, an θ 2 an θ 1, respectively, when χ = 1. Some examples of operators A, M, an N which

4 M. HOLST, E. LUNASIN, AND G. TSOGTGEREL satisfy the mapping assumptions we will nee in this paper are A = ( ) θ, M = (I α 2 ) θ 1, N = (I α 2 ) θ 2, (1.2) for fixe positive real number α an for specific choices of the real parameters θ, θ 1, an θ 2. However, we emphasize that the abstract mapping assumptions we employ are more general, an as a result o not require any specific form of the parameterizations of A, M, an N; this abstraction allows (1.1) to recover most of the existing regularization moels that have been previously stuie, as well as to represent a much larger threeparameter family of moels that have not been explicitly stuie in etail. As a result, the system in (1.1) inclues the Navier-Stokes equations an the various previously stuie α turbulence moels as special cases, namely, the Navier-Stokes - α moel [8, 9, 1, 19, 2, 34, 56], Leray - α moel [12], moifie Leray - α moel (ML - α) [38], simplifie Barina moel (SBM) [5], Navier-Stokes-Voight (NSV) moel [42, 43, 58, 59], an the Navier-Stokes - α - like (NS - α - like) moels [57]. For clarity, some of the specific wellknown regularization moels recovere by (1.1) for particular choices of the operators A, M, N an χ are liste in Table 1. TABLE 1. Some special cases of the moel (1.1) with α >, an with S = (I α ) 1 an S θ2 = [I + ( α ) θ2 ] 1. Moel NSE Leray-α ML-α SBM NSV NS-α NS-α-like A ν ν ν ν ν S ν ν( ) θ M I S I S S S S θ2 N I I S S S I I χ 1 1 Our main goal in this paper is to evelop well-poseness an long-time ynamics results for the entire three-parameter family of moels, an then subsequently recover the existing results of this type for the specific regularization moels that have been previously stuie. Along these lines, we first establish a number of results for the entire three-parameter family, incluing results on existence, regularity, uniqueness, stability, linear an nonlinear perturbations (with the invisci an α limits as special cases), existence an finite imensionality of global attractors, an bouns on the number of etermining egrees of freeom. Elaborating on the latter a bit more, for θ >, we erive a lower boun for the number of egrees of freeom m given by m G n/θ, where G is the Grashof number an n is the spatial imension. A lower boun for the nonissipative case is also establishe. These results give necessary an/or sufficient conitions on the ranges of the three parameters for issipation an smoothing in orer to obtain each result, an we inicate where appropriate which particular regularization moels are covere in the allowable parameter ranges for each result. In the final section of the paper, we evelop some tools for relating the Grashof number-base results to analogous statements involving the Reynols number. Analyzing a generalize moel base on abstract mapping properties of the principal operators A, M, an N allows for a simpler analysis that helps bring out the core common structure of the various regularize NSE (as well as regularize magnetohyroynamics) moels, an also helps clarify the common features of many of the existing an new results. In [57], a two-parameter family of moels was stuie, corresponing to a subset of those stuie here, which we will call here the NS-α-like moels. In orer to escribe this subset of moels, let θ an θ 2 be two nonnegative parameters, an consier the following

system on T 3 : A GENERAL FAMILY OF REGULARIZED NAVIER-STOKES AND MHD MODELS 5 t u + ( ) θ u + (Mu )u + (Mu) T u + p = f, u =, (Mu) =, u() = u, (1.3) where M = (I + ( α 2 ) θ 2 ) 1. This family of NS-α-like moel equations interpolates between incompressible hyper-issipative equations an the NS-α moels when varying the two nonnegative parameters θ an θ 2. This is a special case of (1.1) with θ 1 = an χ = 1, with the egree of issipation controlle by the parameter θ an the egree of nonlinearity controlle by only one parameter θ 2. In this particular case, the NSE are obtaine when θ = 1 an θ 2 =, while the NS-α moel is obtaine when θ = θ 2 = 1. In [57], sufficient conitions on the relationship between θ an θ 2 are establishe to guarantee global well-poseness an global regularity of solutions. Our results here can be viewe as generalizing the global well-poseness an regularity results in [57] (see also [65]) to a larger three-parameter family using a more abstract framework that oes not impose a specific form for the parameterizations, an then also establishing a number of aitional new results for the larger three-parameter family, incluing results on stability, linear an nonlinear perturbations, existence an imension of global attractors, an on etermining operator bouns. As a subset of the results mentione above, we list some of the new results that we have obtaine for the family of α moels as a special case of the more generalize equation (1.1). As far as we know, these results have not been previously establishe in the literature. The global existence an uniqueness of solutions for the invisci α sub-gri scale turbulence moels has been establishe only for the SBM [5]. Here, as a consequence of a more general result, we have establishe the global existence of a weak solution to the invisci Leray-α-moel of turbulence. In [2], the convergence of weak solutions of the NS-α to a weak solution of the NSE as the parameter α was establishe. Here we have establishe this convergence result as well for the NS-α-like equations. In aition, we have establishe for the NS-α-like equations, the existence an finite imensionality of its global attractor, an etermining operator bouns. In the case of Leray-α, ML-α, an SBM, the results on etermining operator bouns also appear to be new. It is important to note that the general framework here allows for the evelopment of results for certain (regularize or un-regularize) magnetohyroynamics (MHD) moels. The basic MHD system has the form t u ν u + (u )u (h )h = π 1 2 h 2, t h η h + (u )h (h )u =, h = u =, (1.4) where the unknowns are the velocity fiel u, the magnetic fiel h, an the pressure π, an where ν > an η > enote the constant kinematic viscosity an constant magnetic iffusivity, respectively. Our global well-poseness results inclue for example a particular regularize MHD moel t u ν u + (Mu )u (Mh )h = π 1 2 h 2, t h η h + (Mu )h (Mh )u =, (1.5) supplemente with several ivergence-free an bounary conitions, where M = (I α 2 ) 1. Note that the system (1.5) is ifferent from the 3D Leray-α-MHD moels

6 M. HOLST, E. LUNASIN, AND G. TSOGTGEREL propose in [48], where global well-poseness result is still open as in the case of the original MHD equations. It is also ifferent in nature from the MHD moels propose in [63, 64]. For the 3D Leray-α-MHD moel propose in [48] the magnetic fiel h is not regularize. Another regularize moel whose global well-poseness result is covere here is the following moifie version of the MHD-α moel propose in [48], t u ν u + (Mu )u + (Mu) T u (Mh )h = π 1 2 h 2, t h η h + (Mu )h (Mh )u =, (1.6) supplemente with several ivergence-free an bounary conitions. Again, the above system is ifferent from the original version of the MHD-α system propose in [48], which oes not have a regularization on the magnetic fiel h. For the original MHD-α system, global well-poseness is establishe in [48]. Here we woul like to stress that our current framework is best suite to MHD moels where the velocity fiel u an the magnetic fiel h are treate on an equal footing as far as the regularization is concerne, so we cannot obtain sharpest results in our framework for the systems like the ones propose in [48]. However, our framework requires only minor moification to inclue these moels; the function spaces become prouct spaces, an the principal issipation an smoothing operators become block operators on these prouct spaces, typically with block iagonal form. It is worthwhile to note here that filtering the magnetic fiel, as is one in [54, 55], can be interprete as introucing hyperviscosity for the filtere magnetic fiel. This observation was first introuce in [48], an smoothing the magnetic fiel was thought to yiel unnecessary extra issipation ae to the system. Since the α moels of turbulence were intene as a basis for regularizing numerical schemes for simulating turbulence, it is important to verify whether the a hoc smoothe equations inherit some of the original properties of the 3D MHD equations. In particular, one woul like to see if the ieal (ν = η = ) quaratic invariants of the smoothe MHD system can be ientifie with the ieal invariants of the original 3D MHD system uner suitable bounary conitions. There are three ieal quaratic invariants in 3D MHD (e.g. uner rectangular perioic bounary conitions or in the whole space), namely, the energy E = 1 2 Ω u(x) 2 + h(x) 2 x, the cross helicity h c = 1 u(x) h(x) x, 2 Ω an the magnetic helicity h M = 1 a(x) h(x) x, where a(x) is the vector potential 2 Ω so that h(x) = a(x). In the case of the 3D MHD-α equations in [48], the three corresponing ieal invariants are the energy E α = 1 u(x) Mu(x) + 2 Ω h(x) 2 x (which reuce, as α, to the conserve energy of the 3D MHD equations), the cross helicity h α c = 1 u(x) h(x) x, an the magnetic helicity 2 Ω hα M = 1 a(x) 2 Ω h(x) x. For our system in (1.5) the corresponing ieal invariants are the energy E α = 1 2 Ω u(x) 2 + h(x) 2 x, an the cross helicity h α c = 1 u(x) h(x) x. Currently, we 2 Ω are unable to ientify a conserve quantity in the ieal version of (1.5) which correspon to the magnetic helicity in the 3D MHD system. Similar problems arise in the 3D MHD- Leray-α equations introuce in [48]. When both the magnetic fiel an the velocity fiel are filtere as it is one in [54], the corresponing ieal quaratic invariants are E α = 1 u(x) Mu(x)+h(x) Mh(x) x, the cross helicity 2 Ω hα c = 1 u(x) Mh(x) x, 2 Ω an the magnetic helicity h α M = 1 Ma(x) Mh(x) x. 2 Ω The remainer of the paper is structure as follows. In 2, we establish our notation an give some basic preliminary results for the operators appearing in the general regularize moel. In 3, we buil some well-poseness results for the general moel; in particular, we establish existence results ( 3.1), regularity results ( 3.3), an uniqueness an stability results ( 3.2). In 4 we establish some results for singular perturbations, which as special cases inclue the invisci limit of viscous moels an the α

A GENERAL FAMILY OF REGULARIZED NAVIER-STOKES AND MHD MODELS 7 limit in α moels; this involves a separate analysis of the linear ( 4.1) an nonlinear ( 4.2) terms. These well-poseness an perturbation results are base on energy methos. In 5, we show existence of a global attractor for the general moel by issipation arguments ( 5.1), an then by employing the classical approach from [62], give estimates for the imension of the global attractor ( 5.2). In 6, we establish some results on etermining operators for the two istinct subfamilies of issipative ( 6.1) an nonissipative ( 6.2) moels, with the help of certain generalizations of the techniques use e.g., in [21, 41, 37, 43]. Since the results in 6 are (naturally) given in terms of the generalize Grashof number, we finish in 7 by eveloping some new results on lengthscale estimates in terms of the Reynols number, which allows for relating the Grashof number-base results in the paper to analogous statements involving the Reynols number. To make the paper reasonably self-containe, in Appenix A we evelop some supporting material on Gronwall-type inequalities (Appenix A.1), spaces involving time (Appenix A.2), an Sobolev spaces (Appenix A.3). 2. NOTATION AND PRELIMINARY MATERIAL Let Ω be an n-imensional smooth compact manifol with or without bounary an equippe with a volume form, an let E Ω be a vector bunle over Ω equippe with a Riemannian metric. With C (E) enoting the space of smooth sections of E, let V C (E) be a linear subspace, let A : V V be a linear operator, an let B : V V V be a bilinear map. At this point V is conceive to be an arbitrary linear subspace of C (E); however, later on we will impose restrictions on V implicitly through various conitions on certain operators such as A. Assuming that the initial ata u V, an forcing term f C (, T ; V) with T >, consier the following equation t u + Au + B(u, u) = f, u() = u, (2.1) on the time interval [, T ]. Bearing in min the moel (1.1), we are mainly intereste in bilinear maps of the form B(v, w) = B(Mv, Nw), (2.2) where M an N are linear operators in V that are in some sense regularizing an are relatively flexible, an B is a bilinear map fixing the unerlying nonlinear structure of the equation. In the following, let P : C (E) V be the L 2 -orthogonal projector onto V. Example 2.1. a) When Ω is a close Riemannian manifol, an E = T Ω the tangent bunle, an example of V is V per {u C (T Ω) : iv u = }, a subspace of the ivergence-free functions. The space of ivergence free perioic functions with vanishing mean on a torus T n is a special case of this example. In this case, one typically has A = ( ) θ, M = (I α 2 ) θ 1, an N = (I α 2 ) θ 2. b) When Ω is a compact Riemannian manifol with bounary, an again E = T Ω the tangent bunle, a typical example of V is V hom = {u C (T Ω) : iv u = } the space of compactly supporte ivergence-free functions. In this case, we keep the operators A = ( P ) θ, M = (I α 2 P ) θ 1, an N = (I α 2 P ) θ 2, in min as the operators that one woul typically consier. c) In either of the above two examples, one can consier as V the prouct spaces V per V per an V hom V hom, which are encountere, e.g., in magnetohyroynamics moels, cf. Example 2.2 below. For the operators A, M, an N, we woul have the

8 M. HOLST, E. LUNASIN, AND G. TSOGTGEREL corresponing block operators on the prouct space V for the above two examples, acting iagonally. Example 2.2. a) In a) or b) of the Example 2.1 above, the bilinear map B can be taken to be B 1 (v, w) = P [(v )w], (2.3) which correspons to the moels with χ = as iscusse in 1. b) Again in a) or b) of Example 2.1 above, B can be taken to be B 2 (v, w) = P [(w )v + ( w T )v], (2.4) which correspons to the moels with χ = 1 as iscusse in 1. c) An example of B that cannot be written in the form (2.2) is the bilinear map for the Clark-α moel, which is B c (v, w) = B 1 (Nv, w) + B 1 (w, Nv) B 1 (Nv, Nw) α B 1 ( j Nv, j Nw), where N = (I α 2 P ) 1, an where the Einstein summation convention is assume. Note that for this bilinear map, one only has B c (v, v), Nv = for any v V per or v V hom, in contrast to the examples B 1 an B 2 which are well-known to have stronger antisymmetry properties, cf. Proposition 2.5(a). ) The MHD system (1.4) has the bilinear map B 3 (v, w) = ( B1 (v 1, w 1 ) B 1 (v 2, w 2 ), B 1 (v 1, w 2 ) B 1 (v 2, w 1 ) ), (2.5) where v = (v 1, v 2 ) an w = (w 1, w 2 ) are elements of either V per V per or V hom V hom. The bilinear map for (1.5) is a special case of B 3 (v, w) = B 3 (Mv, Nw), with M an N having the form M = iag(m 1, M 1 ) an N = I. Another special case of B 3 is the bilinear form for the Leray-α-MHD moel propose in [48], where M an N have the form M = iag(m 1, I) an N = I. e) The MHD-α moel (1.6) has the bilinear map of the form B 4 (v, w) = ( B2 (v 1, w 1 ) B 1 (v 2, w 2 ), B 1 (v 1, w 2 ) B 1 (v 2, w 1 ) ). (2.6) The bilinear map for (1.6) is a special case of B 4 (v, w) = B 4 (Mv, Nw), with M an N having the form M = iag(m 1, M 1 ) an N = I. Another special case of B 4 is the bilinear form for the MHD-α moel propose in [48], where M an N have the form M = iag(m 1, I) an N = I. f) More generally, one can consier the bilinear maps of the form B 5 (v, w) = B i,j,k (v, w) = ( Bi (v 1, w 1 ) B j (v 2, w 2 ), B k (v 1, w 2 ) B j (v 2, w 1 ) ), where i, j, k {1, 2}. This class inclues the above examples ) an e). To refer to the above examples later on, let us introuce the shorthan notation: B i (v, w) = B i (Mv, Nw), i = 1,..., 5. (2.7) As usual, we will stuy equation (2.1) by extening it to function spaces that have weaker ifferentiability properties. To this en, we interpret the equation (2.1) in istribution sense, an nee to continuously exten A an B to appropriate smoothness spaces. Namely, we employ the spaces V s = clos H sv, which will informally be calle Sobolev spaces in the following. The pair of spaces V s an V s are equippe with the uality pairing,, that is, the continuous extension of the L 2 -inner prouct on V. Moreover, we assume that there is a self-ajoint positive operator Λ such that Λ s : V s V

A GENERAL FAMILY OF REGULARIZED NAVIER-STOKES AND MHD MODELS 9 is an isometry for any s R. For arbitrary real s, assume that A, M, an N can be continuously extene so that A : V s V s 2θ, M : V s V s+2θ 1, an N : V s V s+2θ 2, (2.8) are boune operators. We will assume θ ; however, there will not be any a priori sign restriction on θ 1 an θ 2. We remark that s in (2.8) is assume to be arbitrary for the purposes of the iscussion in this section, an that it is of course sufficient to assume (2.8) for a limite range of s for most of the results in this paper. Remark 2.3. Note that in this framework the best value for θ 1 is θ 1 = for both the Leray-α-MHD moel an the MHD-α moel as propose in [48], since those moels have M = (M 1, I), cf. Example 2.2 ) an e). It is possible to refine our analysis by consiering spaces such as V s V r instea of V s V s. We assume that A an N are both self-ajoint, an coercive in the sense that for β R, Av, Λ 2β v c A v 2 θ+β C A v 2 β, v V θ+β, (2.9) with c A = c A (β) >, an C A = C A (β), an that Nv, v c N v 2 θ 2, v V θ 2, (2.1) with c N >. We also assume that (2.9) is vali for β = θ 2 with Λ 2β replace by N. Note that if θ =, (2.9) is strictly speaking not coercivity an follows from the bouneness of A, an note also that (2.1) implies the invertibility of N. For clarity, we list in Table 2 the corresponing values of the parameters an bilinear maps iscusse above for special cases liste in Table 1. TABLE 2. Values of the parameters θ, θ 1 an θ 2, an the particular form of the bilinear map B for some special cases of the moel (2.1). (The bilinear maps B 1 an B 2 are as in (2.3) (2.4), (2.7).) Each of the NSE moels has a corresponing MHD analogue. Moel NSE Leray-α ML-α SBM NSV NS-α NS-α-like θ 1 1 1 1 1 θ θ 1 1 1 1 θ 2 1 1 1 1 θ 2 B B 1 B 1 B 1 B 1 B 1 B 2 B 2 We enote the trilinear form b(u, v, w) = B(u, v), w, an similarly the forms b, b i, an b i, i = 1,..., 5, following Example 2.2 an (2.7). We consier the following weak formulation of the equation (2.1): Fin u L 1 loc (, T ; V s ) for some s such that ( u(t), ẇ(t) + Au(t), w(t) + b(u(t), u(t), w(t)) f(t), w(t) ) t = u() = u, (2.11) for any w C (, T ; V). Here the ot over a variable enotes the time erivative. The left han sie of the first equation in (2.11) is well efine if u L 2 (, T ; V s ), f L 1 (, T ; V s ), an b : V s V s V γ R is boune for some s, s, γ R. The secon equation makes sense if u C(, ε; V s ) for some ε > an s R. However, the following lemma shows that the latter conition is implie by the first equation.

1 M. HOLST, E. LUNASIN, AND G. TSOGTGEREL Lemma 2.4. Let X L 2 (, T ; V s ) be the set of functions that satisfy the first equation in (2.11). Let f L 1 (, T ; V s 2θ ), an let b : V s V s V 2θ s R be boune. Then we have X C(, T ; V s 2θ ). Proof. If u X, we have u L 1 (, T ; V s 2θ ); an so [61, Lemma 3.1.1] implies that u C(, T ; V s 2θ ). In concluing the preliminary material in this section, we state the following result on the trilinear forms b i, i = 1,..., 5, which are the main concrete examples for the trilinear form b. Recall that b(u, v, w) = b(mu, Nv, w). Proposition 2.5. a) The trilinear forms b i, i = 1,..., 5, are antisymmetric in their secon an thir variables: bi (w, v, v) =, w, v V, i = 1,..., 5, (2.12) where V is one of the appropriate spaces in Example 2.1. b) The trilinear form b 1 : V σ 1 V σ 2 V σ 3 R is boune provie that an that for some k {, 1}, σ 1 + σ 2 + σ 3 > n+2 2, (2.13) σ 2 + σ 3 1, σ 1 + σ 3 k, σ 1 + σ 2 1 k. (2.14) If the above three conitions are satisfie, an if σ i is a non-positive integer for some i {1, 2, 3}, then the inequality in (2.13) can be replace by the non-strict version of the inequality. The non-strict inequality is also allowe if for some k {, 1}, σ 1, σ 2 k, σ 3 1 k. (2.15) c) The trilinear form b 2 : V σ 1 V σ 2 V σ 3 R is boune provie that (2.13) hols an in aition that σ 2 + σ 3, σ 1 + σ 3 1, σ 1 + σ 2 1. (2.16) If the above three conitions are satisfie, an if σ i is a non-positive integer for some i {1, 2, 3}, then the inequality in (2.13) can be replace by the non-strict version of the inequality. The non-strict inequality is also allowe if σ 1 1, σ 2, σ 3. (2.17) ) The trilinear form b 3 : V σ 1 V σ 2 V σ 3 R is boune uner the same conitions on σ 1, σ 2, an σ 3 that are given in b). e) The trilinear forms b 4, b 5 : V σ 1 V σ 2 V σ 3 R are boune provie that (2.13) hols an in aition that σ 2 + σ 3 1, σ 1 + σ 3 1, σ 1 + σ 2 1. (2.18) If the above three conitions are satisfie, an if σ i is a non-positive integer for some i {1, 2, 3}, then the inequality in (2.13) can be replace by the non-strict version of the inequality. The non-strict inequality is also allowe if for some k {, 1}, σ 1 1, σ 2 k, σ 3 1 k. (2.19) Proof. The antisymmetricity of b 1, an b 2 is well known, an the bouneness of B 1 is immeiate from Lemma A.3. The antisymmetricity of b 5 (which a fortiori implies that of b 3 an b 4 ) can be seen from b5 (w, v, v) = b i (w 1, v 1, v 1 ) b j (w 2, v 2, v 1 ) + b k (w 1, v 2, v 2 ) b j (w 2, v 1, v 2 ),

A GENERAL FAMILY OF REGULARIZED NAVIER-STOKES AND MHD MODELS 11 where i, j, k {1, 2}. Part b) is proven by applying Lemma A.3 to each of the following two representations b1 (u, v, w) = u i i v k, w k = u i v k, i w k. Part c) is proven by applying Lemma A.3 to b2 (u, v, w) = v i i u k + u i k v i, w k = v i i u k + k (u i v i ) v i k u i, w k = v i i u k v i k u i, w k. To complete the proof, ) an e) follow from parts b) an c). 3. WELL-POSEDNESS RESULTS Similar to the Leray theory of NSE, we begin the evelopment of a solution theory for the general 3-parameter family of regularize Navier-Stokes an MHD moels with clear energy estimates that will be use to establish existence an regularity results, an uner appropriate assumptions show uniqueness an stability. To reinforce which existing results we recover in this general unifie analysis, we give the corresponing simplifie results that have been establishe previously in the literature for the special cases liste in Table 1, at the en of the proof of every theorem. 3.1. Existence. In this subsection, we establish sufficient conitions for the existence of weak solutions to the problem (2.11). By a weak solution, we mean a solution satisfying u L 2 (, T ; V θ θ 2 ) an u L 1 (, T ; V γ ) for some γ R. Theorem 3.1. a) Let the following conitions hol. i) b : V σ 1 V σ 2 V γ R is boune for some σ i [ θ 2, θ θ 2 ], i = 1, 2, an γ [θ + θ 2, ) (θ 2, ); ii) b(v, v, Nv) = for any v V θ θ 2 ; iii) b : V σ 1 V σ 2 V γ R is boune for some σ i < θ θ 2, i = 1, 2, an γ γ; iv) u V θ 2, an f L 2 (, T ; V θ θ 2 ), T >. Then, there exists a solution u L (, T ; V θ 2 ) L 2 (, T ; V θ θ 2 ) to (2.11) satisfying { u L p (, T ; V γ 2θ min{2, σ ), p = 1 +σ 2 +2θ 2 }, if θ > 2, if θ =. b) With some β θ 2, let the following conitions hol. i) b : V β V β V θ β R is boune; ii) u V β, an f L 2 (, T ; V θ+β ), T >. iii) b : V σ V σ V γ R is boune for some σ < θ + β, an γ θ β; Then, there exist T (u ) = T ( u β ) > an a local solution u L (, T ; V β ) L 2 (, T ; V θ+β ) to the equation (2.11). Remark 3.2. a) Let θ + θ 1 > 1. Then from Proposition 2.5 the trilinear forms b 2 1 an b 3 fulfill the hypotheses of a) for γ θ θ 2 1 with γ < min{2θ + 2θ 1 n+2, θ θ 2 2 + 2θ 1, θ + θ 2 1}. Note that in particular this gives the global existence of a weak solution for the invisci Leray-α moel. As far as we know this result has not been reporte previously. b) Let θ + 2θ 1 k, θ + 2θ 2 1, β > n+2 θ 2(θ 2 1 + θ 2 ), an β 1 k θ 2 1 θ 2, for some k {, 1}. Then the trilinear forms b 1 an b 3 satisfy the hypothesis of b).

12 M. HOLST, E. LUNASIN, AND G. TSOGTGEREL Remark 3.3. a) Let θ + θ 1 > 1. Then the trilinear form b 2 2 fulfills the hypotheses of a) for γ θ θ 2 1 with γ < min{2θ + 2θ 1 n+2, θ θ 2 2 + 2θ 1 1, θ + θ 2 }. b) The trilinear form b 2 satisfies the hypothesis of b) for β > n+2 θ 2(θ 2 1 + θ 2 ) with β 1 θ 2 1 θ 2 provie θ + 2θ 1 1 an θ + 2θ 2. Remark 3.4. a) Let θ + θ 1 > 1. Then the trilinear forms b 2 4 an b 5 fulfill the hypotheses of a) for γ θ θ 2 1 with γ < min{2θ + 2θ 1 n+2, θ θ 2 2 + 2θ 1 1, θ + θ 2 1}. b) Let θ + 2θ 1 1, θ + 2θ 2 1, β > n+2 θ 2(θ 2 1 + θ 2 ), an β 1 θ 2 1 θ 2. Then the trilinear forms b 1 an b 3 satisfy the hypothesis of b). Proof of Theorem 3.1. Let {V m : m N} V θ θ 2 be a sequence of finite imensional subspaces of V θ θ 2 such that (1) V m V m+1 for all m N; (2) m N V m is ense in V θ θ 2 ; (3) For m N, with W m = NV m V θ+θ 2, the projector P m : V θ θ 2 V m efine by P m v, w m = v, w m, w m W m, v V, is uniformly boune as a map V γ V γ. Such a sequence can be constructe e.g., by using the eigenfunctions of the isometry Λ 1+θ : V 1+θ θ 2 V θ 2. Consier the problem of fining u m C 1 (, T ; V m ) such that for all w m W m u m, w m + Au m, w m + b(u m, u m, w m ) = f, w m, u m (), w m = u, w m. (3.1) Upon choosing a basis for V m, the above becomes an initial value problem for a system of ODE s, an moreover since N is invertible by (2.1), the stanar ODE theory gives a unique local-in-time solution. Furthermore, this solution is global if its norm is finite at any finite time instance. The secon equality in (3.1) gives c N u m () 2 θ 2 u m (), Nu m () = u(), Nu m () u() θ2 Nu m () θ2, so that u m () θ2 N θ 2 ;θ 2 c N u() θ2. (3.2) Now in the first equality of (3.1), taking w m = Nu m, an using the conition ii) on b, we get t u m, Nu m + 2 Au m, Nu m = 2 f m, Nu m ε 1 f 2 θ θ 2 + ε N 2 θ 2 ;θ 2 u m 2 θ θ 2, for any ε >. Since by choosing ε > small enough we can ensure 2 Au m, Nu m + ε N 2 θ 2 ;θ 2 u m 2 θ θ 2 u m 2 θ 2, by Grönwall s inequality we have ( u m (t) 2 θ 2 u m () 2 θ 2 + t (3.3) f 2 θ θ 2 ) e Ct, (3.4)

A GENERAL FAMILY OF REGULARIZED NAVIER-STOKES AND MHD MODELS 13 for some C R. For any fixe T >, this gives u m L (, T ; V θ 2 ) with uniformly (in m) boune norm. Moreover, integrating (3.3), an taking into account (3.4), we infer t Au m, Nu m ψ(t), t [, ), (3.5) where ψ : [, ) (, ) is a continuous function. If θ >, by the coerciveness of A, the above boun implies u m L 2 (, T ; V θ θ 2 ) with uniformly boune norm. So in any case, u m is uniformly boune in L (, T ; V θ 2 ) L 2 (, T ; V θ θ 2 ), an passing to a subsequence, there exists u L (, T ; V θ 2 ) L 2 (, T ; V θ θ 2 ) such that u m u weak-star in L (, T ; V θ 2 ), u m u weakly in L 2 (, T ; V θ θ 2 ). (3.6) For passing to the limit m in (3.1) we shall nee a strong convergence result, which is obtaine by a compactness argument. We procee by eriving a boun on u m um. Note that (3.1) can be written as t Therefore u m + P m Au m + P m B(u m, u m ) = P m f, u m () = P m u(). (3.7) u m γ C u m θ θ2 + P m B(u m, u m ) γ + P m f γ =: I 1 + I 2 + I 3. (3.8) By the bouneness of B, we have I 2 u m σ1 u m σ2. (3.9) If θ =, then the norms in the right han sie are the V θ 2 -norm which is uniformly boune. If θ >, since u m σi u m 1 λ i θ 2 u m λ i θ θ 2, λ i = σ i + θ 2, i = 1, 2, (3.1) θ by the uniform bouneness of u m in L (V θ 2 ) we have Hence, with λ := λ 1 + λ 2 = σ 1 + σ 2 + 2θ 2 θ I 2 u m 2 λ 1 λ 2 θ 2 u m λ 1+λ 2 θ θ 2 u m λ 1+λ 2 θ θ 2. (3.11) if θ >, an with λ = 1 if θ =, we get u m p L p (V γ ) u m p L p (V θ θ 2 ) + u m p L pλ (V θ θ 2 ) + f p L p (V θ θ 2 ). (3.12) The first term on the right-han sie is boune uniformly when p 2. The secon term is boune if pλ 2, that is p 2/λ. We conclue that u m is uniformly boune in L p (, T ; V θ θ 2), with p = min{2, 2/λ}. By employing Theorem A.2, we can now improve (3.6) as follows. There exists u C(, T ; V θ θ 2 ) L (, T ; V θ 2 ) L 2 (, T ; V θ θ 2 ) such that u m u weak-star in L (, T ; V θ 2 ), u m u weakly in L 2 (, T ; V θ θ 2 ), u m u strongly in L 2 (, T ; V s ) for any s < θ θ 2. (3.13) Now we will show that this limit u inee satisfies the equation (2.11). To this en, let w C (, T ; V) be an arbitrary function with w(t ) =, an let w m C 1 (, T ; W m )

14 M. HOLST, E. LUNASIN, AND G. TSOGTGEREL be such that w m (T ) = an w m w in C 1 (, T ; V γ ). We have u m (t), ẇ m (t) t + Au m (t), w m (t) t + = u m (), w m () + b(u m (t), u m (t), w m (t))t f(t), w m (t) t. (3.14) We woul like to show that each term in the above equation converges to the corresponing term in u(t), ẇ(t) t + Au(t), w(t) t + b(u(t), u(t), w(t))t = u, w() + f(t), w(t) t, (3.15) which woul imply that u satisfies (2.11). Here we show this only for the nonlinear term. We have b(u m (t), u m (t), w m (t)) b(u(t), u(t), w(t)) t I m + II m + III m, (3.16) where the terms I m, II m, an III m are efine below. Firstly, it hols that I m = b(u m (t), u m (t), w m (t) w(t)) t u m (t) σ1 u m (t) σ2 w m (t) w(t) γ t u m L 2 (V σ 1 ) u m L 2 (V σ 2 ) w m w C(V γ ). thus, we get lim m I m = since a fortiori σ i θ θ 2, i = 1, 2. For II m we have II m = b(u m (t) u(t), u m (t), w(t)) t u m (t) u(t) σ1 u m (t) σ2 w(t) θ+θ2 t u m u L 2 (V σ 1 ) u m L 2 (V σ 2 ) w C(V γ ), so lim m II m = since σ 1 < θ θ 2 an σ 2 θ θ 2. Similarly, we have III m = b(u(t), u m (t) u(t), w(t)) t u L 2 (V σ 1 ) u m u L 2 (V σ 2 ) w C(V γ ), (3.17) (3.18) (3.19) so lim m III m = since σ 1 θ θ 2 an σ 2 < θ θ 2. For the proof of b), we choose the neste subspaces {V m : m N} V θ+β an W m = Λ 2β V m V θ β, in (3.1). Then substituting w m = Λ 2β u m () in the secon equality in (3.1) gives u m () β u β. Now in the first equality of (3.1), taking w m = Λ 2β u m, an using the bouneness of b, we get t u m 2 β f 2 β θ + u m 2 β + u m 4 β,

A GENERAL FAMILY OF REGULARIZED NAVIER-STOKES AND MHD MODELS 15 an thus u m (t) β (T t) 1/4, for some T >. The rest of the proof procees similarly to that of a). For clarity, the corresponing conitions an results of Theorem 3.1 above are liste in Table 3 below for the special stanar moel cases of the general three-parameter regularize moel liste in Table 1. For the NS-α-like case in the table, the allowe values for β are β > 5 2 θ 2θ 2 with β 1 2 θ 2 provie θ 1 an θ + 2θ 2. TABLE 3. Existence results for some special cases of the moel (1.1). The table gives values of (a, b) an (γ, p) for our recovere existence results for the stanar moels, giving existence of a solution u L (V a ) L 2 (V b ), with u L p (V γ ). (In the NSV case, ε >, an in the NS-α-like case, γ max{ θ 2 1/2, θ 2 + 1/2, n/2}.) The last row inicates the local existence of solutions u L (V β ) L 2 (V β+θ ). The result for each NSE moel has a corresponing MHD analogue. Moel NSE Leray-α ML-α SBM NSV NS-α NS-α-like Existence a, b, 1, 1-1, -1, -1, -1-1, θ 2, θ θ 2 γ, p 1, 4 3 1, 2 2, 2 2, 2 1 + ε, 2 2, 2 γ, 2 Local β > 3 2 β β > 1 2 β 1 β 1 β > 1 2 β 3.2. Uniqueness an stability. Now we shall provie sufficient conitions for uniqueness an continuous epenence of on initial ata for weak solutions of the general threeparameter family of regularize moels. Theorem 3.5. Let β θ 2, an let u 1, u 2 L (, T ; V β ) L 2 (, T ; V β+θ ) be two solutions of (2.11) with initial conitions u 1 (), u 2 () V β, respectively. a) Let b : V σ 1 V θ θ 2 V σ 2 R be boune for some σ 1 θ θ 2 an σ 2 θ + θ 2 with σ 1 + σ 2 θ. Moreover, let b(v, w, Nw) = for any v V σ 1 an w V σ 2. Then we have u 1 (t) u 2 (t) θ2 φ(t) u 1 () u 2 () θ2, t [, T ], (3.2) where φ C([, T ]) an φ() = 1. b) Let b : V β V β V θ β R be boune. Then we have u 1 (t) u 2 (t) β φ(t) u 1 () u 2 () β, t [, T ], (3.21) where φ C([, T ]) an φ() = 1. Remark 3.6. The trilinear forms b 1 an b 3 satisfy the hypotheses of a) provie θ + θ 1 1 k, θ + 2θ 2 1 k, θ + θ 2 1, 2θ + 2θ 2 1 + θ 2 > n+2, an 3θ + 2θ 2 1 + 2θ 2 2 k, for some k {, 1}. The forms b 1 an b 3 satisfy the hypothesis of b) for β > n+2 2(θ 2 1 + θ 2 ) θ with β 1 k (θ 2 1 + θ 2 ) provie 2θ 2 + θ 1 an 2θ 1 + θ k, for some k {, 1}. Remark 3.7. The trilinear form b 2 satisfies the hypotheses of a) for θ+2θ 1 1, θ+θ 1 1, θ + θ 2 2, 2θ + 2θ 1 + θ 2 > n+2, an 3θ + 2θ 2 1 + 2θ 2 1. The trilinear form b 2 satisfies the hypothesis of b) for β > n+2 2(θ 2 1 +θ 2 ) θ with β 1 (θ 2 1 +θ 2 ) provie 2θ 2 + θ an 2θ 1 + θ 1. Remark 3.8. The trilinear forms b 4 an b 5 satisfy the hypotheses of a) provie θ + θ 1 1, θ + 2θ 2 1 1, θ + θ 2 1, 2θ + 2θ 2 1 + θ 2 > n+2, an 3θ + 2θ 2 1 + 2θ 2 2. The forms b 4 an b 5 satisfy the hypothesis of b) for β > n+2 2(θ 2 1 + θ 2 ) θ with β 1 (θ 2 1 + θ 2 ) provie 2θ 2 + θ 1 an 2θ 1 + θ 1.

16 M. HOLST, E. LUNASIN, AND G. TSOGTGEREL Proof of Theorem 3.5. Let v = u 1 u 2. Then subtracting the equations for u 1 an u 2 we have v, w + Av, w + b(v, u 1, w) + b(u 2, v, w) =. (3.22) Taking w = Nv, we infer t v 2 θ 2 + c v 2 θ θ 2 C v σ1 u 1 θ θ2 v σ2 2θ 2 C v 2 λ 1 λ 2 θ 2 v λ 1+λ 2 θ θ 2 u 1 θ θ2, where λ 1 = σ 1+θ 2 an λ θ 2 = σ 2 θ 2. By applying Young s inequality we get θ t v 2 θ 2 C v 2 θ 2 u 1 2 θ θ 2. Now Grönwall s inequality gives v(t) 2 θ 2 v() 2 θ 2 exp t The part b) is proven similarly, taking, e.g. w = (I ) β v. C u 1 2 θ θ 2. (3.23) To clarify these results in the case of specific moels, the corresponing conitions an results of Theorem 3.5 above are liste in the Table 4 below for the special case moels liste in Table 1. TABLE 4. Uniqueness results for some special cases of the moel (1.1). The table gives values of β for our recovere uniqueness results for the stanar moels, where u V β an where u L (V β ) L 2 (V β+θ ). (In the NS-α-like case, the requirement on β is that β > max{ θ 2, 1/2 θ 2, 5/2 θ 2θ 2 }). The result for each NSE moel has a corresponing MHD analogue. Moel NSE Leray-α ML-α SBM NSV NS-α NS-α-like Uniqueness β > 3 2 β β > 1 2 β 1 β 1 β > 1 2 β or β = 1 or β = 1 3.3. Regularity. In this subsection, we evelop a regularity result on weak solutions for the general family of regularize moels. Theorem 3.9. Let u L 2 (, T ; V θ θ 2 ) be a solution to (2.11), an with some β > θ 2, let the following conitions hol. i) b : V α V α V θ β R is boune, where α = min{β, θ θ 2 }; ii) b(v, w, Nw) = for any v, w V; iii) u() V β, an f L 2 (, T ; V β θ ). Then we have u L (, T ; V β ) L 2 (, T ; V β+θ ). (3.24) Remark 3.1. Let 4θ + 4θ 1 + 2θ 2 > n + 2, 2θ + 2θ 1 1 k, θ + 2θ 2 1, 3θ + 4θ 1 1, θ + 2θ 1 l, an 3θ + 2θ 1 + 2θ 2 2 l, for some k, l {, 1}. Let β ( n+2 2(θ 2 1+θ 2 ) θ, 3θ+2θ 1 n+2 1 l ) [ θ 2 2 1 θ 2, min{2θ+θ 2 1, 2θ θ 2 +2θ 1 k}]. Then the trilinear forms b 1 an b 3 satisfy the hypotheses of the above theorem.

A GENERAL FAMILY OF REGULARIZED NAVIER-STOKES AND MHD MODELS 17 Remark 3.11. Let 4θ + 4θ 1 + 2θ 2 > n + 2, θ + 2θ 2, an θ + 2θ 1 1. Let β ( n+2 2 2(θ 1 +θ 2 ) θ, 3θ+2θ 1 n+2 2 ) [ 1 2 θ 1 θ 2, min{2θ+θ 2, 2θ θ 2 +2θ 1 1}]. Then the trilinear form b 2 satisfies the hypotheses of the above theorem. Remark 3.12. Let 4θ + 4θ 1 + 2θ 2 > n + 2, θ + 2θ 2 1, an θ + 2θ 1 1. Let β ( n+2 2 2(θ 1+θ 2 ) θ, 3θ+2θ 1 n+2 2 ) [ 1 2 θ 1 θ 2, min{2θ+θ 2 1, 2θ θ 2 +2θ 1 1}]. Then the trilinear forms b 4 an b 5 satisfy the hypotheses of the above theorem. Proof of Theorem 3.9. By Theorems 3.1.b) an 3.5.a), there is s > epening on u() β such that u L (, s; V β ) L 2 (, s; V β+θ ). With I = [, s), we have u, Λ 2β u + Au, Λ 2β u + b(u, u, Λ 2β u) = f, Λ 2β u, a.e. in I. (3.25) By employing the bouneness of b an the coercivity of A, we infer t u 2 β f 2 β θ + u 2 θ θ 2 u 2 β, a.e. in I, an using Grönwall s inequality, we conclue u(t) 2 β t f 2 β θ + u() 2 β exp t C u 2 θ θ 2, a.e. in I, (3.26) where the integral in the exponent is uniformly boune since u L 2 (, T ; V θ θ 2 ). Therefore we have u L (, T ; V β ), which transfers to u L 2 (, T ; V θ+β ) by the coercivity of A. Again for clarity, the corresponing conitions an results of Theorem 3.9 above are liste in the Table 5 below for the special case moels liste in Table 1. For the NS-α-like case in the table, the allowe values for β are β 2θ θ 2 1 with β < 3θ 5 2, provie that θ 1 2 an 4θ + 2θ 2 > 5. TABLE 5. Regularity results for some special cases of the moel (1.1). The table gives values of β for our recovere local an global regularity results for the stanar moels, where u V β an where u L (V β ) L 2 (V β+θ ). (In the NS-α-like case, see the text for the allowe values of β.) Again, the result for each NSE moel has a corresponing MHD analogue. Moel NSE Leray-α ML-α SBM NSV NS-α NS-α-like Regularity β 1 β 1 2 β 2 β 1 2 β β 4. SINGULAR PERTURBATIONS In this section, we will consier the situation where the operators A an B in the general three-parameter family of regularize moels represente by problem (2.1) have values from a convergent (in a certain sense) sequence, an stuy the limiting behavior of the corresponing sequence of solutions. As special cases we have invisci limits in viscous equations an α limits in the α-moels.

18 M. HOLST, E. LUNASIN, AND G. TSOGTGEREL 4.1. Perturbations to the linear part. Consier the problem an its perturbation u + Au + B(u, u) = f, (4.1) u i + A i u i + B(u i, u i ) = f, (i N), (4.2) where A, B, an N (that will appear below) satisfy the assumptions state in Section 2, an for i N, A i : V s V s 2ε is a boune linear operator satisfying A i v 2 ε θ 2 + v 2 θ θ 2 A i v, Nv + v 2 θ 2, v V ε θ 2. (4.3) Assuming that both problems (4.1) an (4.2) have the same initial conition u, an that A i A in some topology, we are concerne with the behavior of u i as i. We will also assume that ε θ. Theorem 4.1. Assume the above setting, an in aition let the following conitions hol. i) b : V σ 1 V σ 2 V γ R is boune for some σ j [ θ 2, θ θ 2 ], j = 1, 2, an γ [ε + θ 2, ) (θ 2, ); ii) b(v, v, Nv) = for any v V; iii) b : V σ 1 V σ 2 V γ R is boune for some σ j < θ θ 2, j = 1, 2, an γ γ; iv) u V θ 2, an f L 2 (, T ; V θ θ 2 ), T > ; v) A i converge weakly to A as i. Then, there exists a solution u L (, T ; V θ 2 ) L 2 (, T ; V θ θ 2 ) to (4.1) such that up to a subsequence, as i. u i u weak-star in L (, T ; V θ 2 ), u i u weakly in L 2 (, T ; V θ θ 2 ), u i u strongly in L 2 (, T ; V s ) for any s < θ θ 2, (4.4) Proof. Firstly, from Theorem 3.1, we know that for i N there exists a solution u i L (, T ; V θ 2 ) L 2 (, T ; V ε θ 2 ) to (4.2). Duality pairing (4.2) with Nu i an using elementary inequalities, we have t u i, Nu i + 2 A i u i, Nu i = 2 f, Nu i ε 1 f 2 θ θ 2 + ε u i 2 θ θ 2. (4.5) Choosing ε > small enough, then using (4.3), by Grönwall s inequality we have u i (t) 2 θ 2 e Ct. (4.6) Moreover, integrating (4.5), an taking into account (4.6) an (4.3), we infer A i u i 2 L 2 (,t;v ε θ 2 ) + u i 2 L 2 (,t;v θ θ 2 ) ψ(t), t [, ), (4.7) where ψ : [, ) (, ) is a continuous function. For any fixe T >, this gives u i L (, T ; V θ 2 ) L 2 (, T ; V θ θ 2 ) with uniformly (in i) boune norms. On the other han, we have u i γ A i u i γ + B(u i, u i ) γ + f γ. (4.8) By estimating the secon term in the right han sie as in the proof of Theorem 3.1, an taking into account (4.7), we conclue that u i is uniformly boune in L 2 (, T ; V γ ). By employing Theorem A.2, an passing to a subsequence, we infer the existence of u satisfying (4.4). Now taking into account the weak convergence of A i to A, the rest of the proof procees similarly to that of Theorem 3.1.

A GENERAL FAMILY OF REGULARIZED NAVIER-STOKES AND MHD MODELS 19 For example, setting ε = 1, with θ = an θ 2 = 1, an checking all the requirements i) v) of Theorem 4.1, the viscous solutions to the SBM converge to the invisci solution as the viscosity tens to zero. Recall that the global existence of weak solution to the invisci SBM (first establishe in [5]) is also establishe in Theorem 3.1. Similarly, setting ε =, with θ = an θ 2 = 1, the viscous solutions to the Leray-α moel converge to the invisci solution as the viscosity tens to zero. This result gives another proof of the global existence of a weak solution for the invisci Leray-α moel. On the other han, the convergence of viscous solutions of ML-α an NS-α to its invisci solutions, respectively, are not covere here since both moels fail conition i) of Theorem 4.1. Notice that the global existence of weak solution to the invisci ML-α an NS-α are not establishe in Theorem 3.1. Besies the invisci SBM, there are no global well-poseness results reporte previously in the literature for the other invisci α-moels. 4.2. Perturbations involving the nonlinear part. For i N, let A i : V s V s 2ε an N i : V s V s+2ε 2 be boune linear operators, satisfying an v 2 θ+θ 2 A i N 1 i v, v + v 2 θ 2, v V θ+θ 2, (4.9) v 2 θ 2 N 1 i v, v, v V θ 2, (4.1) where we also assume that N i is invertible. In this subsection, we continue with perturbations of (4.1) of the form u i + A i u i + B i (u i, u i ) = f, (i N), (4.11) where B i is some bilinear map. Again assuming that both problems (4.1) an (4.11) have the same initial conition u, an that A i A an B i B in some topology, we are concerne with the behavior of u i as i. For reference, efine the trilinear form b i (u, v, w) = B i (u, v), w. Theorem 4.2. Assume the above setting, an in aition let the following conitions hol. i) b i : V σ V σ V γ R is uniformly boune for some σ [ θ 2, θ + θ 2 2ε 2 ), an γ [θ + ε 2, ) (ε 2, ); ii) b i (v, v, N i v) = for any v V; iii) u V θ 2, an f L 2 (, T ; V θ θ 2 ), T > ; iv) A i : V θ θ 2 V γ is uniformly boune an converges weakly to A; v) N 1 i : V s+2θ 2 V s+2θ 2 2ε 2 is uniformly boune; vi) N 1 i N : V θ θ 2 V θ+θ 2 2ε 2 converges strongly to the ientity map; vii) For any v V θ θ 2, B i (v, v) converges weakly to B(v, v). Then, there exists a solution u L (, T ; V θ 2 ) L 2 (, T ; V θ θ 2 ) to (4.1) such that up to a subsequence, y i = N 1 N i u i satisfies y i u weak-star in L (, T ; V θ 2 ), y i u weakly in L 2 (, T ; V θ θ 2 ), y i u strongly in L 2 (, T ; V s ) for any s < θ θ 2, (4.12) as i.