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Global existence and regularity for the dynamics of viscous oriented fluids

  • We prove global existence of weak solutions to regularized versions of balance equations representing the dynamics over a torus of complex fluids, with microstructure described by a vector field taking values in the unit ball. Regularization is offered by the presence of second-neighbor microstructural interactions and our choice of filtering the balance of macroscopic momentum by inverse Helmholtz operator with unit length scale.

    Citation: Luca Bisconti, Paolo Maria Mariano. Global existence and regularity for the dynamics of viscous oriented fluids[J]. AIMS Mathematics, 2020, 5(1): 79-95. doi: 10.3934/math.2020006

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  • We prove global existence of weak solutions to regularized versions of balance equations representing the dynamics over a torus of complex fluids, with microstructure described by a vector field taking values in the unit ball. Regularization is offered by the presence of second-neighbor microstructural interactions and our choice of filtering the balance of macroscopic momentum by inverse Helmholtz operator with unit length scale.


    For sufficiently differentiable maps ˜u:T2×[0,T]~R2 and ˜ν:T2×[0,T]S2, with T2 a torus and S2 the unit sphere, we have shown in reference [12] that the system

    ut+(u)uΔu+π=(νν)νΔνt,u=0,Δνt+Δ((u)ν)Δ2ν=νt+(u)ν+|ν|2νΔν,

    reasonably describes the dynamics over T2 of oriented (i.e., polarized or spin) fluids, a representation in which we account for second-neighbor director interactions in a minimalistic way, the one giving us sufficient amount of regularity to allow existence of a certain class of weak solutions.

    In the balance of microstructural actions governing the evolution of ν, an hyper-stress behaving like 2ν accounts for second-neighbor interactions; it enters the equation through its double divergence, which generates the term Δ2ν. A viscous-type contribution (namely νΔνt) affects Ericksen's stress in the balance of macroscopic momentum, an equation in which π is pressure, i.e., the reactive stress associated with the volume-preserving constraint u=0.

    We have explicitly underlined in reference [12] the terms neglected in the previous balance equations with respect to a complete representation of second-neighbor director interactions, and their contribution to the Ericksen stress.

    Also, to tackle the analysis of such balances, in reference [12] we considered transient states foreseeing |ν|1 (i.e., a polarized fluid not in saturation conditions) and replaced the nonlinear term |ν|2ν with its approximation 1ε2(1|ν|2)ν, ε a positive parameter. Eventually, we established just local existence of a certain class of weak solutions.

    The description of such fluids falls within the general model-building framework of the mechanics of complex materials (a format involving manifold-valued microstructural descriptors) in references [25] and [26] (see also [27,28]). By following that format, if we derive balance equations by requiring invariance of the sole external power of actions under isometric changes in observers even just for first-neighbor interactions, since the infinitesimal generator of SO(3) action over S2 is ν×, we find the possible existence of a conservative self-action proportional to ν, i.e., something like λν, with λ0.

    Consequently, we consider here a relaxed version of the balances above by accounting for |ν|1 and introducing the self-action λν. Then, we write

    ut+(u)uΔu+π=(νν)νΔνt, (1.1)
    u=0, (1.2)
    Δνt+Δ((u)ν)Δ2ν=νt+(u)νΔν+1ε2(1|ν|2)νλν,|ν|1, (1.3)

    with initial conditions

    u|t=0=u0,ν|t=0=ν0. (1.4)

    We tackle its analysis by filtering the balance of macroscopic momentum by (IΔ)1. In the process, we define the regularized velocity

    w:=(IΔ)1u,

    and approximate the filtered version of equation (1.1) by considering that (IΔ)1(uu)(ww). Then, we apply the inverse filter (IΔ) (and we write once again π and ν for pressure and director field, respectively). The resulting system reads

    wtΔwt+(w)wΔw+π=(νν)νΔνt, (1.5)
    w=0, (1.6)
    Δνt+Δ((w)ν)Δ2ν=νt+(w)νΔν+1ε2(1|ν|2)νλν,|ν|1. (1.7)

    For it, we prove global existence of weak solutions (defined as in reference [12]).

    The obtained regularity could allow us to obtain a uniqueness result. Also, the granted global existence of weak solutions can be used for analyzing possible weak or strong attractors, which we may foresee in appropriate state spaces. All these aspects will be matter of a forthcoming work.

    For p1, by Lp=Lp(T2) we indicate the usual Lebesgue space with norm p. When p=2, we use the notation :=L2 and denote by (,) the related inner product. Moreover, with k a nonnegative integer and p1, we denote by Wk,p:=Wk,p(T2) the usual Sobolev space with norm k,p (using k when p=2). We write W1,p:=W1,p(T2), p=p/(p1), for the dual of W1,p(T2) with norm 1,p.

    Let X be a real Banach space with norm X. We will use the customary spaces Wk,p(0,T;X), with norm denoted by Wk,p(0,T;X). In particular, W0,p(0,T;X)=Lp(0,T;X) are the standard Bochner spaces.

    (Lp)n:=L2(T2,Rn), p1, is the function space of vector-valued L2-maps. Similarly, (Wk,p)n:=(Wk,p(T2))n is the usual Sobolev space of vector-valued maps with components in Wk,p, while (Hs)n is the space of vector-valued maps with components in Hs=Ws,2{w:w=0}. We also define the following spaces:

    H:=closure   of C0(T2,R2){w|w=0} in (L2)2 ,Hs:=closure   of  C0(T2,R2){w|w=0} in (Ws,2)2,Hs:={ν(Ws,2)3}.

    This last space is the usual Sobolev space of vector fields with components Ws,2-functions. Again H:=H0. By Hs we indicate the space dual to Hs. We denote by ,Hs,Hs the duality pairing between Hs and Hs. We will also assume that the vector fields u and w have null average on T2. In particular, under such an assumption, Poincaré's inequality holds true.

    Here and in the sequel, we denote by c (or ˉc) positive constants, which may assume different values.

    We'll make use of the following well-known inequalities (see, e.g., [1,2,15,18,21,22,32]): Ladyzhenskaya's,

    vL4Cv12v12,vH1, (2.1)

    Agmon's,

    vLCv12Δv12,vH2. (2.2)

    In the sequel (especially to get estimates in Hs, with s non-integer) we'll also make use of commutator-type estimates as the one in the following lemma concerning the operator Λs, sR+ (see, e.g., [19,20,30], see also [6,31]), with Λ:=(Δ)1/2.

    Lemma 2.1. For s>0 and 1<r, and for smooth enough u and v, we get

    Λs(uv)Lrc(uLp1ΛsvLq1+vLp2ΛsuLq2), (2.3)

    where 1/r=1/p1+1/q1=1/p2+1/q2 and c is a suitable positive constant.

    We also recall the following result about product-laws in Sobolev spaces ([16, Theorem 2.2], see also [29])

    Lemma 2.2. Let s0, s1, s2R. The product estimate

    fgHs0cfHs1gHs2 (2.4)

    holds, provided that

    s0+s1+s2n2,  where  n  is  the  space  dimension, (2.5)
    s0+s10, (2.6)
    s0+s20, (2.7)
    s1+s20, (2.8)
    If  in  (2.5)  the  equality  sign  holds,inequalities  (2.6)(2.8)  must  be  strict. (2.9)

    Set T2:=2πZ2/L. T2 is the torus defined by T2:=(R2/T2). We can expand wHs(T2) in Fourier series as

    w(x)=kT2ˆwkeikx,

    with k=(k1,k2)Z2 the wave-number, |k|=|k1|2+|k2|2. The Fourier coefficients for w are defined by ˆwk:=1(2π)2T2w(x)eikxdx. The norm in Hs is given by

    w2Hs=|k|2s|ˆwk|2,

    and the inner product (,)Hs=(Λs,Λs) is characterized by

    (w,v)Hs=|k|1|k|2sˆwk¯ˆvk,

    where the over-bar denotes, as usual, complex conjugation. Consider the inverse Helmholtz operator

    G:=(IΔ)1, (2.10)

    taking values

    Gw(x):=T2G(x,y)w(y))dy, (2.11)

    where G(x,y) is the associated Green function (see, e.g., [5,7,8,9,10]). For wHs, take the Fourier expansion w=kT2ˆwkeikx, so that, by inserting this expression in (2.11), we get

    Gw:=kT211+|k|2ˆwkeikx. (2.12)

    G is self-adjoint. It commutes with differential operators (see, e.g., [4,5,7]). We get also

    (Gv,w)=(G1/2v,G1/2w)L2=(v,w)H1and(G1/2v,G1/2v)L2=vH1. (2.13)

    (see also [5,18]).

    We set

    fε(ν):=1ε2(1|ν|2)ν,ε>0.

    Then, we rewrite the filtered balances as

    wtΔwt+(w)wΔw+π=(νν)νΔνt, (3.1)
    w=0, (3.2)
    Δνt+Δ((w)ν)Δ2ν=νt+(w)νΔν+fε(ν)λν,|ν|1, (3.3)

    with initial conditions

    w|t=0=w0,ν|t=0=ν0. (3.4)

    To keep the notation compact, here and in the sequel we omit the dependence of w and ν on ε.

    Definition 3.1 (Regular weak solution). For a given T>0, a pair (w,ν) is a regular weak solution of (3.1)–(3.3) if (w,ν)L(0,T;H32×H52), (tw,tν)L2(0,T;H1×H32), and

    T0((wt(s),v(s))+(wt(s),v(s))+((w(s))w(s),v)+(w(s),v(s)))ds=T0((νν(s),v(s))+(ννt(s),v(s)))ds, (3.5)

    holds true for every vC0((0,T)×T2), and

    T0((νt(s),h(s))+([(w(s))ν(s)],h(s))+(Δν(s),Δh(s)))ds=T0((νt(s),h(s))+((w(s)))ν(s),h(s))+(fε(ν(s)),h(s))+(ν(s),h(s))λ(ν(s),h(s)))ds (3.6)

    for every h(t,x)=ψ(t)ϕ(x), with ϕH52, ψC0(0,T), and |ν(x,t)|1 a.e. in (0,T)×T2.

    In the following, we'll always refer to "regular weak solutions" simply as "weak solutions", for the sake of brevity.

    Theorem 3.1. Assume (w0,ν0)H32×H52, with |ν0(x)|1 for a.e. xT2. Then, systems (3.1)–(3.3), supplied with (3.4), admits a weak solution (w,ν) which is defined for any fixed time T0.

    The chosen regularity for the initial data allows the reader to compare easily the result here with what we got in reference [12], realizing our passage from local (short time) to global (large fixed time) existence. Also, by renouncing to a certain amount of solution regularity (i.e., considering a weaker class) we could accept data (w0,ν0)H1×H2, obtaining for them once again an existence result (see Remark 4.1 below).

    Remark 3.1. For the integral T2((w)ν)ω dx, with wH1, ωH1, and νH2, we get

    T2((w)ν)ω dx=T2j(wiiνk)jωk dx=T2jwiiνkjωk dx+T2wiijνkjωk dx.

    The first term on the right-hand side of the above identity is such that

    T2jwiiνkjωk dx=T2iνkjωkjwi dx=T2(νω)w dx,

    and for the second term we find

    T2wiijνkjωk dx=T2ijνkjωkwi dx=T2(ν)ωw dx.

    For the second term on the right-hand side of (3.1), we compute

    νΔνt=(ννt)(ν)νt. (3.7)

    We introduce Galerkin's approximating functions {(wn,νn)}, prove a maximum principle, by which the constraint |νn|1 is verified, and compute some a-priori estimates. The Aubin-Lions compactness theorem [24] allows us to get convergence of a subsequence. Actually, we apply Galerkin's procedure originally used for the standard Navier-Stokes equations, by adapting it to systems (3.1)–(3.3). (Further details about such a scheme appear in references [23, §2], [13, Appendix A], [7,11]).

    Note: In the sequel, for the sake of conciseness we often avoid writing explicitly the integration measure in some integrals, every time we find it appropriate.

    We apply directly Galerkin's method only to the velocity field w (this scheme is also known as "semi-Galerkin formulation"; see, e.g, [13]).

    For any positive integer i, let us denote by (ωi,πi)H2×W1,2 the unique solution of the following Stokes problem:

    Δωi+πi=λiωi,in  T2,ωi=0,in  T2, (4.1)

    with T2πidx=0, for i=1,2, and 0<λ1λ2λn with λn+, as n. Functions {ωi}+i=1 determine an orthonormal basis in H made of the eigenfunctions pertaining to (4.1).

    Let Pn:H3/2Hn:=H3/2span{ω1,ω2,,ωn} be the orthonormal projection of H3/2 on its finite dimensional subspace Hn. Take T>0. For every positive integer n, we look for an approximate solution (wn,νn)C1(0,T;Hn)×L(0,T,H32)L2(0,T,H52) to systems (3.1)–(3.3) with

    wn(t,x)=ni=1ϕni(t)ωi(x),ϕni to  be  determined. (4.2)

    Consider the following problem defined a.e. in (0,T)×T2:

    (wnt(t)Δwnt(t),vn)H12+((wn(t))wn(t),vn)H12+(wn(t),vn)H12=(((νn)νn)(t),vn)H12+(((νn)Δνt)(t),vn)H12,vnHn, (4.3)
    (IΔ)[tνn(t)Δνn(t)+((wn(t))νn(t))]=fε(νn(t))+λνn, (4.4)
    |νn|1, (4.5)
    wn(x,0)=wn0(x):=Pn(w0)(x),νn(0,x)=ν0(x), for  xT2, (4.6)

    where w0H32 and ν0H52, with |ν0(x)|1 a.e. in T2.

    Instead of exploiting test functions in L2, we take directly the formulation in H1/2, for it provides the needed regularity, The pertinent analysis develops in two steps:

    Step A: Let ¯wnC1(0,T;Hn) be a given velocity field of the form ¯wn(t,x)=ni=1¯ϕni(t)ωi(x), with ¯ϕni assigned. For

    (IΔ)[νnt(t)Δνn(t)+(¯wn(t))νn(t)]=fε(νn(t))+λνn(t),a.e.in$(0,T)×T2$,

    with νn(0,x)=ν0(x),for$xT2$, we actually look for a vector field

    νnL(0,T;H32)L2(0,T;H52),νtL2(0,T;H1)

    solving a.e. on (0,T)×T2 the following system:

    νnt(t)Δνn(t)+(¯wn(t))νn(t)=G(fε(νn(t)))+λG(νn(t)), (4.7)
    νn(0,x)=ν0(x),for  xT2, (4.8)

    where G is once again the inverse Helmholtz operator G=(IΔ)1 introduced in (4.1). Since G has Fourier symbol corresponding to the inverse of two spatial derivatives, the right-hand side part of (4.7) results to be regularized (i.e., the terms Gfε(νn) gains two additional spatial derivatives with respect to fε(νn); the same occurs for Gνn(t)). Thus, this new expression can be rewritten equivalently as a semilinear parabolic equation in the unknown νn. The existence of such νn is guaranteed by the classical theory of parabolic equations (see, e.g., [17]), which also provides higher regularity results (see [17, Theorem 6, Ch. 7.1]). They allow us to use the regularity of initial data ν0H52 to get νnL(0,T;H32)L2(0,T;H52) and νntL2(0,T;H1) (by interpolation we also have that νtC(0,T;H1)). The following lemma (see [12, Lemma 4.1] and also [14, Lemma 2.1]) guarantees the constraint |ν|1.

    Lemma 4.1 (Weak maximum principle). Let ν0H52 be such that |ν0(x)|1 for a.e. xT2. Take ˉwnC(0,T;Hn). Then, there exists a weak solution νnL(0,T;H32)L2(0,T;H52) to the problems (4.7)–(4.8). Moreover, fixed ϵ>0 large enough in the definition of fϵ, every such weak solution verifies |νn(x,t)|1 a.e. on T2×[0,T].

    In performing the next calculations, we could relax hypotheses by assuming that ν0H1 and is such that |ν0(x)|1 a.e. xT2, with ˉwnC(0,T;Hn). Then, there would exist a weak solution νL(0,T;H1)×L2(0,T;H2), with |ν(x,t)|1 a.e. in T2×(0,T). However, for the sake of simplicity, we still use the same regularity assumptions previously introduced, and we denote by ν and w the quantities νn and ˉwn, respectively for the sake of conciseness.

    Proof. Existence of the solution νnL(0,T;H32)L2(0,T;H52) to (4.7)–(4.8) has been already mentioned above.

    Define φ(x,t)=(|ν(x,t)|21)+, where z+=max{z,0} for each zR. Assume there exists a measurable subset BT2 with positive measure |B|>0 such that |ν(x,t)|>1 a.e. in B×(t1,t2], 0t1<t2T, and |ν(x,t)|=1 a.e. in B×(t1,t2]. By taking φν as a test function against (4.7), we get

    12T2t(|ν|2)φ+T2(w)|ν|2φ+T2ν(φν)1ϵ2T2G(|ν|21)νφν+λB(Gν)(φν)=0,

    which is equivalent to

    12Bt(|ν|2)φ+B(w)|ν|2φ+Bν(φν)1ϵ2BG1/2(φν)G1/2(φν)+λB(G1/2ν)(G1/2ν)=0. (4.9)

    With indicating L2(B), we can also write

    12Bt(|ν|2)φ=12Bt(|ν|21)φ=14ddtφ2,B(w)|ν|2φ=B(w)(|ν|21)φ=B(w)φφ=0,Bν(φν)=12B(|ν|2)φ+B|ν|2φ=12B(|ν|21)φ+B|ν|2φ=12φ2+B|ν|2φ12φ20.

    Then, Eq. (4.9) becomes

    ddtφ2+2φ2+4B|ν|2φ4ϵ2BG1/2(φν)G1/2(φν)+4λB(G1/2ν)G1/2(φν)=0. (4.10)

    Since φ(t2)φ(t1) (here, φ(t1)=0), by integrating in time over (t1,t2], we get

    2t2t1φ2+4t2t1(B|ν|2φ4ϵ2B(G1/2ν)G1/2(φν)+4λB(G1/2ν)G1/2(φν))0.

    In principle, B may have more than one connected component with positive measure. However, these components are finite in number for ¯B is compact. Thus, previous inequality can be rewritten as

    i(2t2t1Bi|φ|2+4t2t1(Bi|ν|2φ4ϵ2BiG1/2(φν)G1/2(φν)+4λBi(G1/2ν)G1/2(φν)))0.

    Then, there exists at least one connected component Bj, with |Bj|>0, on which

    2t2t1Bj|φ|2+4t2t1(Bj|ν|2φ4ϵ2BjG1/2(φν)G1/2(φν)+4λBjG1/2(ν)G1/2(φν))0,

    and hence, by(2.13), we have

    2t2t1φ2L2(Bj)+4t2t1Bj|ν|2φ4ϵ2t2t1BjG1/2(φν)G1/2(φν)4λt2t1BjG1/2(ν)G1/2(φν))cϵ2t2t1φν2H1(Bj)+4λ|t2t1BjG1/2(ν)G1/2(φν)|. (4.11)

    Since |ν(x,t)|=1 a.e. on B×(t1,t2), we get φ(x,t)=0 a.e. on B×(t1,t2) and, in particular, φ(x,t)=0 a.e. on Bj×(t1,t2). Assume that Bj is the closure of an open set. By using the Poincaré inequality on left-hand side first term of (4.11), along with the control (2.4) (see also [29]), we obtain

    φν2H1φ2H1(Bj)ν2L2(Bj)cφ2H1(Bj)ν2L(0,T;L2(T2))=˜cφ2H1(Bj),

    and

    4λ|BG1/2(ν)G1/2(φν)|4λνφνH14λν2φH1cλ(Bj(ν21)dx+1)φH1cλ(Bj(ν21)2dx)12φH1+cλφH1cλφφH1+ˉcλφ2H1ˆcλφ2H1. (4.12)

    Hence, the inequality

    Ct2t1φ2L2(Bj)+t2t1φ2L2(Bj)+4t2t1Bj|ν|2φ(˜cϵ2+ˆcλ)t2t1φ2H1(Bj),

    where C is the constant involved in the Poincaré inequality, holds true. Then, we find

    ct2t1φ2H1(Bj)+4t2t1Bj|ν|2φ(˜cϵ2+ˆcλ)t2t1φ2H1(Bj), (4.13)

    which gives an absurd by assuming that ϵ is sufficiently large as λ is small.

    The general case, when Bj is not the closure of an open set, follows the same line of the argument in reference [12].

    Step B: Let νnL(0,T;H32)L2(0,T;H52) be the vector field just determined in the previous step. We search the approximating velocity field wnC1(0,T;Hn) satisfying the equation

    (wnt(t),vn)H12+(wnt(t),vn)H12+(wn(t),vn)H12+((¯wn(t))wn(t),vn)H12,=(((νn)νn),vn)H12+((νn)Δνt,vn)H12,vnHn,

    with

    wn(x,0)=wn0(x)=Pn(w0)(x), for  xT2,

    where both νn and ¯wn are given. Thanks to the Cauchy-Lipschitz theorem, we can prove existence of a unique maximal solution wn of the above problem.

    In the sequel, as short-hand notation, we use the same symbol Lp(0,T;Lk) for both the norm in Lp(0,T;Lk) and the one in Lp(0,T;(Lk)n). We employ the same convention also for Lp(0,T;Ws,k) and Lp(0,T;(Ws,k)n) (also Lp(0,T;Hs) and Lp(0,T;(Hs)n)).

    Proof of Theorem 3.1. First, we deduce a priori estimates. Then, we apply a compactness criterion proving that the limiting pair (ˆw,ˆν) is actually a weak solution to (3.1)–(3.3), supplemented by (1.4). Only for the sake of conciseness we use (w,ν) instead of (wn,νn).

    Step 1: Energy a priori estimates. Consider Eq. (3.3), to which we apply the operator G=(IΔ)1, and take the L2-product with test ν, obtaining

    12ddtν2+ν21ε2T2|G((1|ν|2)ν)||ν|dx+λT2|G12ν|2dxcε2(1|ν|2)νH2ν+cλν2H1cε2(1|ν|2)νν+cλν2(cε2+cλ)ν2, (4.14)

    where the constraint |ν|21 plays a role. Given T>0, the Gronwall Lemma implies νL(0,T;L2)L2(0,T;H1).

    By taking the L2-product of (3.1) with w, we compute

    12ddt(w2+w2)+w2=T2(νν)wdxT2νΔνtwdx=T2(νν)wdx+T2(ννt)wdx+T2(ν)νtwdx, (4.15)

    where, to get the second equality, we have used relation (3.6), integrating by parts the first term obtained.

    By multiplying (3.3) by νt and integrating over T2, we obtain

    12ddt(ν2+Δν2)+νt2+νt2=T2((w)ν)νtdxT2(w)ν)νtdxT2fε(ν)νtdx+λT2ννtdx. (4.16)

    Remark 1 implies that the first term in the right-hand side of (4.16) can be rewritten as

    T2((w)ν)νt dx=T2(ννt)wdx+T2(ν)νtwdx.

    Then, by summing up (4.15) and (4.16), we infer

    12ddt(w2+w2+ν2+Δν2)+w2+νt2+νt2T2|w||ν|2dx+T2|w||ν||νt|dx+T2|fϵ(ν)||νt|dx+λT2|ν||νt|dx=:4i=1Ii. (4.17)

    For the terms Ii, i=1,2,3, we have the following bounds

    I1wν2L4cwνΔνcεw2ν2+CεΔν2, (4.18)
    I2wL4νL4νtcw12|w12ν12Δν12νtcδwwνΔν+δνt2ˉcδwνΔν+δνt2¯cϵδ2w2ν2+cϵΔν2+δνt2, (4.19)
    I32ε2ννtcε4δν2+δνt2, (4.20)

    and

    I4λννtcλδν2+λδνt2. (4.21)

    Estimates above, together with inequalities (4.14) and (4.17), allow us to write

    ddt(w2+w2+ν2+Δν2)+w2+(1(λ+2)δ)νt2+νt2cε,δ,λν2+cεΔν2+cε,δw2ν2. (4.22)

    In the present case the penalisation parameter ε>0 is constant, so we omit such a term along with δ and λ in the next calculations. From (4.22) we obtain

    ddt(w2+w2+ν2+Δν2)c(ν2+Δν2+w2ν2)cν2+c(w2+Δν2)(1+ν2).

    Set y=(w2+w2+ν2+Δν2). The differential inequality

    ycν2+y(1+ν2)

    implies

    (w2+w2+ν2+Δν2)(t)(w0,w0,ν0,Δν02)et0(1+ν(s)2)ds+νL(0,T;L2)t0ets(1+ν()2)dds. (4.23)

    Since (w,ν) stands for (wn,νn), as a consequence of the above estimates, for any fixed T>0, it follows that wn,wn2L(0,T;H)+νn,Δνn2L(0,T;H) is uniformly bounded with respect to nN. The control (4.22) implies νtL2(0,T;H1).

    Step 2: Further a priori estimates. We take the H1/2-inner product of (3.1) and (3.2) with w and νt, respectively, as in the case of Eqs (4.15) and (4.16). After integration by parts, we obtain

    12ddt(w2H12+w2H12)+w2H12=((w)w),w)H12+((νν),w)H12+((ννt),w)H12+((ν)νt,w)H12, (4.24)

    and

    12ddt(ν2H12+Δν2H12)+νt2H12+νt2H12=((ννt),w)H12((ν)νt,w)H12((w)ν),νt)H12(fε(ν),νt)H12+λ(ν,νt)H12. (4.25)

    From Eqs (4.24) and (4.25), we get

    12ddt(w2H12+w2H12+ν2H12+Δν2H12)+w2H12+νt2H12+νt2H12|((w)w),w)H12|+|((νν),w)H12|+|((w)ν),νt)H12|+|(fε(ν),νt)H12|+λ|(ν,νt)H12|=:5i=1Li. (4.26)

    For the terms Li, i=1,,5 we actually use the norm induced by (,)˙H12=(Λ12(),Λ12()) instead of the full norm H12, although we still keep the same norm notation H12. Previous evaluation of the lower-order terms in the steps already described motivates our notational choice. Also, for the velocity vector filed w, the norm w˙H12 is equivalent to the full norm wH12.

    Consider L1. Since T2(w)Λ12wΛ12wds=0, we get

    L1T2|(Λ12w)w||Λ12w|dsΛ12w2L4wΛ12wΛ12wwcwH12wwH12cϵw2H12w2+ϵw2H12. (4.27)

    Then, by exploiting (2.3), with s=1/2, r=2 and p1=p2=q1=q2=4, we find

    L2ννH12wH12cΛ12νL4νL4wH12c(Λ12ν12Λ12Δν12ν12Δν12)wH12cϵ(ν2H12ν2+Δν2H12Δν2)+ϵw2H12, (4.28)
    L3(w)νH12νtH12(Λ12wL4νL4+wL4Λ12νL4)νtH12cϵ(w2ν2H12+w2H12Δν2)+ϵνt2H12, (4.29)
    L4=|(Λ12fε(ν),Λ12νt)|cfε(ν)H1νtH12cϵε4(ν2+ν2)+ϵνt2H12, (4.30)

    and

    L5=λ|(Λ12ν,Λ12νt)|cλνH12νtH12cλϵν2H12+ϵνt2H12, (4.31)

    after using Hölder's, Ladyzhenskaya's, and Young's inequalities as well as the continuous embedding W1/2,2(T2)L4(T2).

    By using the estimates (4.27)–(4.30) along with (4.26), and absorbing the parameter ε4 in a generic constant c, we obtain

    12ddt(w2H12+w2H12+ν2H12+Δν2H12)+(1cϵ)w2H12+νt2H12+νt2H12cw2H12(1+w2+ν+Δν2)+cν2H12(1+w2+ν2+Δν2)+cΔν2H12Δν2

    with ϵ>0 small enough in a way that the coefficient ˉc:=(1cϵ) is positive. Fix T>0. By Grönwall's lemma, we get

    w(t)2H12+w(t)2H12+ν(t)2H12+Δν(t)2H12+ˉct0(w2H12+νt2H12+νt2H12)dsβexp{ct0[(1+w2+w2+ν2+Δν2)]ds}

    for any 0<tT, with

    β=c(w02H12+w02H12+ν02H12+Δν02H12),

    and the quantity on the right-hand side of the above inequality is bounded, for 0<tT, thanks to Eq. (4.23) and the hypotheses on initial data.

    Until here, we mainly used the notation (w,ν) in place of (wn,νn) but, in view of extracting a convergent subsequence, in the last part of the proof we'll employ the (wn,νn) notation. Step 3: Estimate for wnt. In order to extract a convergent subsequence of {(un,νn)}, we exploit the classical Aubin-Lions lemma; to this end we have first to provide a suitable control on wnt. The next calculations also fixes a minor issue present in the analogous control in reference [12], where we estimate acceleration in L1(0,T;H1).

    Consider Eq. (3.1). For φH1, T2φdx=0, with φ=1. Then, we get

    wntΔwnt,φH1,H1|((wn)φ,wn)|+|(wn,φ)|+T2|νn|2|φ|dx+cT2|νn||νnt||φ|dx+c|((νn)νnt,φ)|wn2L4φ+wnφ+νn2L4φ+cνnL4νntL4φ+c(νn)νntH1φc(wnwn+cwn+νnΔνn+νnH12νntH12+ΔνnH12νntH12), (4.32)

    after using the estimates performed in previous steps along with Hölder's, Ladyzhenskaya's, and Poincaré's inequalities. In the last inequality above, we have also exploited the continuous embedding W1/2,2(T2)L4(T2) and the Sobolev product laws (see, e.g., [?,3,29]) to get the estimate

    (νn)νntH1ΔνnνntH12ΔνnH12νntH12.

    Hence, we find

    T0wnt2H1dsc[(1+wn2L(0,T;L2))wn2L2(0,T;L2)+Δνn2L2(0,T;L2)+(νn2L(0,T;H12)+Δνn2L(0,T;H12))νnt2L2(0,T;H12)].

    As a final step in our argument, to extract a convergent subsequence from {(wn,νn)}, we can use the Aubin-Lions lemma following the same line as in the proof of [12, Theorem 3.1, Step 3]. Also, passage to the limit in weak formulation follows the same path exploited in reference [12]. So, we can conclude stating existence.

    Remark 4.1. By assuming initial data (w0,ν0)H1×H2, we can still reproduce the same calculations of Step 1, while Step 2 would require higher-order estimates, which are not available in the present setting. However, by using an approach similar to the one in Step 3, we could obtain a weaker control on wt by using Eq. (3.1) and providing a uniform estimate on

    ΔwntH2=supφ˙H2=1|Δwnt,φH2,H2|

    Indeed, also in this case, the worst term to be controlled is |((νn)νnt,φ)|. For it, we get

    T0|((νn)νnt,φ)|2dsT0((νn)νnt2H2φ2H2dsT0Δνn2νnt2dsΔνn2L(0,T;L2)T0νnt2ds,

    on the basis of inequalities (4.22), (4.23), and the product law (2.4). Then, to conclude about the existence of weak solutions, we can use again the same idea behind limiting and convergence procedures in [12,Theorem 3.1,Step 3].

    We thank Tommaso Ruggeri for a discussion on this matter at the end of a talk delivered in 2018 by one of us (PMM) in the Accademia Peloritana dei Pericolanti at Messina (Italy). This work is part of activities in the research group in "Theoretical Mechanics" of the "Centro di Ricerca Matematica Ennio De Giorgi" of the Scuola Normale Superiore in Pisa. We acknowledge the support of INDAM groups GNAMPA and GNFM.

    The authors declare no conflicts of interest in this paper.



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