http://www.aimspress.com/journal/Math DOI:10.3934/Math.2018.x.xxx Received: 23 July 2019 Accepted: 11 October 2019 Published: xxxx Research article

### Global existence and regularity for the dynamics of viscous oriented fluids

Luca Bisconti1 _{and Paolo Maria Mariano}2, ∗
1 _{DIMAI, Universit`a di Firenze}

viale Morgagni 67/a, I-50134 Firenze, Italy

2 _{DICeA, Universit`a di Firenze}

via Santa Marta 3, I-50136 Firenze, Italy

* Correspondence: [email protected]; Tel:+39 0552758893.

Abstract: 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.

Keywords: Mechanics of complex materials; spin fluids; second-neighbor interactions; hyperbolic equations

Mathematics Subject Classification: 76D05, 76D03, 35L65, 74N15

1. Introduction

For sufficiently differentiable maps ˜u : T2× [0, T ] → fR2 and ˜ν : T2× [0, T ] → S2, with T2a torus
and S2_{the unit sphere, we have shown in reference [12] that the system}

ut+ (u · ∇)u − ∆u + ∇π = −∇ · (∇ν>∇ν) − ∇ν>∆νt,

∇ · u= 0,

∆νt + ∆((u · ∇)ν) − ∆2ν = νt+ (u · ∇)ν + |∇ν|2ν − ∆ν,

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∆2ν. A viscous-type contribution (namely ∇ν>∆νt) affects the Ericksen stress

in the balance of macroscopic momentum, an equation where π is the 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|2_{ν with its approximation} 1

ε2(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 [26] and [27] (see also [28], [29]). 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 S O(3) action over S2is −ν×, we find the possible existence of a conservative self-action proportional to ν, i.e., something like λν, with λ ≥ 0.

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 · ∇)u − ∆u + ∇π = −∇ · (∇ν>∇ν) − ∇ν>∆νt, (1.1) ∇ · u = 0, (1.2) ∆νt+ ∆((u · ∇)ν) − ∆2ν = νt + (u · ∇)ν − ∆ν + 1 ε2(1 − |ν| 2 )ν − λν, |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 (I −∆)−1. In the process, we define the regularized velocity

w:= (I − ∆)−1u,

and approximate the filtered version of equation (1.1) by considering that ∇·(I −∆)−1(u⊗u) ≈ ∇·(w⊗w). Then, we apply the inverse filter (I −∆) (and we write once again π and ν for pressure and director field, respectively). The resulting system reads

wt −∆wt + (w · ∇)w − ∆w + ∇π = −∇ · (∇ν>∇ν) − ∇ν>∆νt, (1.5)
∇ · w = 0, (1.6)
∆νt+ ∆((w · ∇)ν) − ∆2ν = νt + (w · ∇)ν − ∆ν +
1
ε2(1 − |ν|
2_{)ν − λν, |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.

2. Notation and preliminaries
For p ≥ 1, by Lp _{= L}p

(T2_{) we indicate the usual Lebesgue space with norm k · k}

p. When p = 2,

we use the notation k · k := k · kL2 and denote by ( · , · ) the related inner product. Moreover, with k

a nonnegative integer and p ≥ 1, we denote by Wk,p := Wk,p(T2_{) the usual Sobolev space with norm}

k · kk,p(using k · kkwhen p= 2 ). We write W−1,p

0

:= W−1,p0(T2), p0 = p/(p − 1), for the dual of W1,p(T2) with norm k · k−1,p0.

Let X be a real Banach space with norm k · kX. We will use the customary spaces Wk,p(0, T ; X),

with norm denoted by k · kWk,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), p ≥ 1, 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 W}k,p_{, while (H}s_{)}n_{is}

the space of vector-valued maps with components in Hs _{= W}s,2_{∩ {w : ∇ · w} _{= 0}. We also define the}

following spaces:

H := closure of C_{0}∞(T2, R2) ∩ {w | ∇ · w= 0} in (L2)2,
Hs := closure of C_{0}∞(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 H−s _{we indicate the space dual to H}s_{. We denote by h · , · i :}= h · , · i

H−1_{,H}1 the duality

pairing between H−1and H1. We will also assume that the vector fields u and w have null average on T2. In particular, under such an assumption, Poincar´e’s inequality holds true.

Here and in the sequel, we denote by c (or ¯c) positive constants, which may assume different values. We’ll make use of the following well-known inequalities (see, e.g., [1, 2, 16, 19, 22, 23, 33]): Ladyzhenskaya’s, kvkL4 ≤ Ckvk 1 2k∇vk 1 2, v ∈ H1, (2.1) Agmon’s, kvkL∞ ≤ Ckvk 1 2k∆vk 1 2, v ∈ H2. (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, s ∈ R+
(see, e.g., [20, 21, 31], see also [32, 6]), withΛ := (−∆)1/2_{.}

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

kΛs(uv)kLr ≤ c(kuk_{L}p1kΛsvkLq1 + kvkLp2kΛsukLq2), (2.3)

where1/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 ([17, Theorem 2.2], see also [30])

Lemma 2.2. Let s0, s1, s2 ∈ R. The product estimate

k f gkH−s0 ≤ ck f kHs1kgkHs2 (2.4)

holds, provided that

s0+ s1+ s2≥

n

2, where n is the space dimension, (2.5)

s0+ s1≥ 0, (2.6)

s0+ s2≥ 0, (2.7)

s1+ s2≥ 0, (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 w ∈ Hs(T2) in Fourier

series as

w(x)= X

k∈T_{2}?

b wkeik·x,

with k= (k1, k2) ∈ Z2 the wave-number, |k| = p|k1|2+ |k2|2. The Fourier coefficients for w are defined

by_{b}wk := _{(2π)}12

R

T2w(x)e

−ik·x_{dx. The norm in H}s

is given by
kwk2_{H}s =
X
|k|2s|
b
wk|2,

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

(w, v)Hs =

X

|k|≥1

|k|2s_{b}wk·bvk,

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

G:= (I − ∆)−1, (2.10) taking values Gw(x) := Z T2 G(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 w ∈ Hs, take the Fourier expansion w= Pk∈T?

2 bwke

ik·x

, so that, by inserting this expression in (2.11), we get

Gw:= X
k∈T?
2
1
1+ |k|2wbke
ik·x_{.}
(2.12)

Gis 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)_{H}−1 and (G1/2v, G1/2v)_{L}2 = kvk_{H}−1. (2.13)

3. Existence and regularity result We set

fε(ν) := _{ε}1_{2}(1 − |ν|2)ν, ε > 0.

Then, we rewrite the filtered balances as

wt−∆wt+ (w · ∇)w − ∆w + ∇π = −∇ · (∇ν>∇ν) − ∇ν>∆νt, (3.1)

∇ · w= 0, (3.2)

∆νt+ ∆((w · ∇)ν) − ∆2ν = νt+ (w · ∇)ν − ∆ν + 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)–to–(3.3) if (w, ν) ∈ L∞(0, T ; H32 ×H
5
2_{), (∂}
tw, ∂tν) ∈ L2(0, T ; H1×H
3
2_{), and}
Z T
0
(wt(s), v(s))+ (∇wt(s), ∇v(s))+ ((w(s) · ∇)w(s), v) + (∇w(s), ∇v(s))
ds
=Z
T
0
(∇ν>∇ν(s), ∇v(s)) + (∇ν>∇νt(s), ∇v(s))
ds,
(3.5)

holds true for every v ∈ C_{0}∞ (0, T ) × T2_{, and}

Z T 0 (∇νt(s),∇h(s))+ ∇(w(s) · ∇)ν(s), ∇h(s) + (∆ν(s), ∆h(s)) ds = Z T 0 (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 φ ∈ H52,ψ ∈ C∞

0(0, T ), and |ν(x, t)| ≤ 1 a.e. in (0, T ) × T
2_{.}

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) ∈ H 3 2×H 5 2, with |ν

0(x)| ≤ 1 for a.e. x ∈ T2. Then, system(3.1)–to–(3.3),

supplied with(3.4), admits a weak solution (w, ν) which is defined for any fixed time T ≥ 0.

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 3.1. For the integralR
T2∇ (w · ∇)ν · ∇ω dx, with w ∈ H
1_{,}_{ω ∈ H}1
, andν ∈ H2, we get
Z
T2
∇ (w · ∇)ν · ∇ω dx =
Z
T2
∂j w
i_{∂}
iν
k∂
jω
k
dx
=Z
T2
∂jwi∂iνk∂jωk dx+
Z
T2
wi∂i jνk∂jωk dx.

The first term on the right-hand side of the above identity is such that Z T2 ∂jwi∂iνk∂jωk dx= Z T2 ∂iνk∂jωk∂jwi dx= Z T2 ∇ν>∇ω · ∇w dx, and for the second term we find

Z T2 wi∂i jνk∂jωk dx= Z T2 ∂i jνk∂jωkwi dx= Z T2 ∇(∇ν)>∇ω · w dx. For the second term on the right-hand side of (3.1), we compute

∇ν>_{∆ν}
t = ∇ · ∇ν
>_{∇ν}
t − ∇(∇ν)
>_{∇ν}
t. (3.7)
4. Proofs

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 [25] 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 system (3.1)–to–(3.3). (Further details about such a scheme appear in references [24, §2], [13, Appendix A], [7] [11].)

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

4.1. Approximate Galerkin solutions

Galerkin’s method is applied directly 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:

∆ωi+ ∇πi = −λiωi, in T2,

∇ ·ωi = 0, in T2,

(4.1) withR

T2πidx = 0, for i = 1, 2, . . . and 0 < λ1 ≤ λ2 ≤ . . . λn. . . with λn → +∞, as n → ∞. Functions

{ωi}+∞_{i}_{=1} determine an orthonormal basis in H made of the eigenfunctions pertaining to (4.1).

Let Pn: H3/2 → Hn := H3/2 ∩ span{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_{) ∈ C}1_{(0, T ; H}

n) × L∞(0, T, H

3

2) ∩ L2(0, T, H 5

2) to system (3.1)–to–(3.3) with

wn(t, x)= n X i=1 φn i(t)ωi(x), φni to be determined. (4.2)

Consider the following problem defined a.e. in (0, T ) × T2:
(wn_{t}(t) −∆wn_{t}(t), vn)
H12 + (w
n
(t) · ∇)wn(t), vn
H12 + (∇w
n
(t), ∇vn)
H12
= (∇(νn
)>∇νn)(t), ∇vn
H12 + (∇(ν
n
)>∆νt)(t), vn
H12, ∀v
n_{∈ H}
n,
(4.3)
(I −∆)∂tνn(t) −∆νn(t)+ (wn(t)·∇)νn(t)=− fε(νn(t))+ λνn, (4.4)
|νn
| ≤ 1, (4.5)
wn(x, 0)= wn_{0}(x) := Pn(w0)(x), νn(0, x)= ν0(x), for x ∈ T2, (4.6)
where w0 ∈ H
3
2 and ν_{0}∈ H
5
2, with |ν
0(x)| ≤ 1 a.e. in T2.

Instead of exploiting test functions in L2_{, we take directly the formulation in H}1/2_{, for it provides}

the needed regularity, The pertinent analysis develops in two steps:
–Step A: Let wn ∈ C1_{(0, T ; H}

n) be a given velocity field of the form w
n
(t, x)= Pn_{i}_{=1}φn_{i}(t)ωi(x), with φ
n
i
assigned. For
(I −∆)νn_{t}(t) −∆νn(t)+ wn(t) · ∇νn
(t)= − f
ε(νn(t))+ λνn(t), a.e. in (0, T ) × T2,
with νn_{(0, x)}= ν

0(x), for x ∈ T2, we actually look for a vector field

νn

∈ L∞(0, T ; H32) ∩ L2(0, T ; H 5 2), ν

t ∈ L2(0, T ; H1)

solving a.e. on (0, T ) × T2the following system: νn t(t) −∆ν n (t)+ wn(t) · ∇νn (t)= −G fε(νn(t))+ λG(νn(t)), (4.7) νn (0, x) = ν0(x), for x ∈ T2, (4.8)

where G is once again the inverse Helmholtz operator G = (I − ∆)−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 −G fε(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., [18]), which also provides higher regularity results (see [18, Theorem 6, Ch. 7.1]). They allow us to use the regularity of initial data ν0 ∈ H

5
2 _{to get ν}n ∈
L∞(0, T ; H32)∩ L2(0, T ; H
5
2) and νn
t ∈ L2(0, T ; H

1_{) (by interpolation we also have that ν}

t ∈ C(0, T ; H1)).

The following lemma (see [12, Lemma 4.1] and also [15, Lemma 2.1]) guarantees the constraint |ν| ≤ 1. Lemma 4.1 (Weak maximum principle). Let ν0 ∈ H

5

2 be such that |ν

0(x)| ≤ 1 for a.e. x ∈ T2. Take

¯

wn _{∈ C(0, T ; H}

n). Then, there exists a weak solution νn ∈ L∞(0, T ; H

3

2) ∩ L2(0, T ; H 5

2) to the problem

(4.7)–(4.8). Moreover, fixed > 0 large enough in the definition of f, 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 ν0 ∈ H1 and is

such that |ν0(x)| ≤ 1 a.e. x ∈ T2, with ¯wn ∈ C(0, T ; Hn). Then, there would exist a weak solution

ν ∈ L∞

(0, T ; H1) × L2_{(0, T ; H}2

), 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
wthe quantities νn_{and ¯}_{w}n_{, respectively for the sake of conciseness.}

Proof. Existence of the solution νn _{∈ L}∞

(0, T ; H32) ∩ L2(0, T ; H 5

2) to (4.7)–(4.8) has been already

mentioned above.

Define ϕ(x, t) = (|ν(x, t)|2 _{− 1)}

+, where z+ = max{z, 0} for each z ∈ R. Assume there exists a

measurable subset B ⊂ T2 with positive measure |B| > 0 such that |ν(x, t)| > 1 a.e. in B × (t1, t2],

0 ≤ t1 < t2 ≤ T , and |ν(x, t)|= 1 a.e. in ∂B × (t1, t2]. By taking ϕν as a test function against (4.7), we

get 1 2 Z T2 ∂t(|ν|2)ϕ+ Z T2 (w · ∇)|ν|2ϕ + Z T2 ∇ν · (ϕν) − 1 2 Z T2 G(|ν|2− 1)ν · ϕν+ λ Z B (Gν) · (ϕν)= 0, which is equivalent to 1 2 Z B ∂t(|ν|2)ϕ+ Z B (w · ∇)|ν|2ϕ + Z B ∇ν · ∇(ϕν) − 1 2 Z B G1/2 ϕν · G1/2(ϕν)+ λ Z B (G1/2ν) · (G1/2ν) = 0. (4.9)

With k · k indicating k · kL2_{(B)}, we can also write

1
2
Z
B
∂t(|ν|2)ϕ =
1
2
Z
B
∂t(|ν|2− 1)ϕ=
1
4
d
dtkϕk
2_{,}
Z
B
(w · ∇)|ν|2ϕ =
Z
B
(w · ∇)(|ν|2− 1)ϕ=
Z
B
(w · ∇)ϕ · ϕ= 0,
Z
B
∇ν · ∇(ϕν) = 1
2
Z
B
∇(|ν|2) · ∇ϕ+
Z
B
|∇ν|2_{ϕ}
= 1
2
Z
B
∇(|ν|2− 1)∇ϕ+
Z
B
|∇ν|2ϕ
= 1
2k∇ϕk
2_{+}
Z
B
|∇ν|2_{ϕ ≥} 1
2k∇ϕk
2
≥ 0.
Then, equation (4.9) becomes

d
dtkϕk
2_{+ 2k∇ϕk}2_{+ 4}
Z
B
|∇ν|2ϕ − 4
2
Z
B
G1/2 ϕν · G1/2(ϕν)
+ 4λ
Z
B
(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

2 Z t2 t1 k∇ϕk2+ 4 Z t2 t1 Z B |∇ν|2ϕ − 4 2 Z B (G1/2ν) · G1/2(ϕν) + 4λ Z B (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

X
i
2
Z t2
t1
Z
Bi
|∇ϕ|2_{+ 4}
Z t2
t1
Z
Bi
|∇ν|2_{ϕ −} 4
2
Z
Bi
G1/2 ϕν · G1/2(ϕν)
+4λZ
Bi
(G1/2ν) · G1/2(ϕν)
!
≤ 0.
Then, there exists at least one connected component Bj, with |Bj|> 0, on which

2
Z t2
t1
Z
Bj
|∇ϕ|2_{+ 4}
Z t2
t1
Z
Bj
|∇ν|2_{ϕ −} 4
2
Z
Bj
G1/2 ϕν · G1/2(ϕν)
+4λZ
Bj
G1/2 ν · G1/2(ϕν)
!
≤ 0,
and hence, by(2.13), we have

2
Z t2
t1
k∇ϕk2
L2_{(B}
j)+ 4
Z t2
t1
Z
Bj
|∇ν|2_{ϕ ≤} 4
2
Z t2
t1
Z
Bj
G1/2 ϕν · G1/2(ϕν)
− 4λ
Z t2
t1
Z
Bj
G1/2 ν · G1/2(ϕν)
≤ c
2
Z t2
t1
kϕνk2
H−1_{(B}
j)
+ 4λ
Z t2
t1
Z
Bj
G1/2 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 Bjis the closure of an open set. By using the Poincar´e inequality on

left-hand side first term of (4.11), along with the control (2.4) (see also [30]), we obtain
kϕνk2
H−1 ≤ kϕk
2
H1_{(B}
j)kνk
2
L2_{(B}
j)≤ ckϕk
2
H1_{(B}
j)kνk
2
L∞_{(0,T ;L}2_{(T}2_{))} = ˜ckϕk
2
H1_{(B}
j),
and
4λ
Z
B
G1/2 ν · G1/2(ϕν)
≤ 4λkνkkϕνkH−1
≤ 4λkνk2kϕkH1
≤ cλZ
Bj
(kνk2− 1) dx+ 1kϕkH1
≤ cλ
Z
Bj
(kνk2− 1)2 dx
!1_{2}
kϕk_{H}1 + cλkϕk_{H}1
≤ cλkϕkkϕkH1 + ¯cλkϕk2
H1
≤ ˆcλkϕk2_{H}1.
(4.12)

Hence, the inequality
C
Z t2
t1
kϕk2
L2_{(B}
j)+
Z t2
t1
k∇ϕk2
L2_{(B}
j)+ 4
Z t2
t1
Z
Bj
|∇ν|2ϕ ≤ ˜c
2 + ˆcλ
Z t2
t1
kϕk2
H1_{(B}
j),

where C is the constant involved in the Poincar´e inequality, holds true. Then, we find

c
Z t2
t1
kϕk2
H1_{(B}
j)+ 4
Z t2
t1
Z
Bj
|∇ν|2_{ϕ ≤} ˜c
2 + ˆcλ
Z t2
t1
kϕk2
H1_{(B}
j), (4.13)

which gives an absurd by assuming that 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 νn ∈ L∞(0, T ; H32) ∩ L2(0, T ; H 5

2) be the vector field just determined in the previous step.

We search the approximating velocity field wn _{∈ C}1_{(0, T ; H}

n) satisfying the equation

(wnt(t), v
n
)
H12+(∇w
n
t(t), ∇v
n
)
H12 + (∇w
n
(t), ∇vn)
H12 + (w
n
(t) · ∇)wn(t), vn
H12,
= (∇(νn
)>∇νn), ∇vn
H12+ ∇(ν
n
)>∆νt, vn
H12, ∀v
n _{∈ H}
n,
with
wn(x, 0) = wn_{0}(x)= Pn(w0)(x), for x ∈ T2,

where both νn_{and w}n

are given. Thanks to the Cauchy-Lipschitz theorem, we can prove existence of a unique maximal solution wnof the above problem.

4.2. Global existence

In the sequel, for the sake of compactness, we use the same symbol k · kLp_{(0,T ;L}k_{)} for the norm

in Lp_{(0, T ; L}k_{) and L}p_{(0, T ; (L}k_{)}n_{).} _{We employ the same convention also for L}p_{(0, T ; W}s,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 equation (3.3), to which we apply the operator G = (I −∆)−1, and take the L2-product with test ν, obtaining

1
2
d
dtkνk
2_{+ k∇νk}2 _{≤}1
ε2
Z
T2
|G (1 − |ν|2)ν||ν| dx + λ
Z
T2
|G12ν|2dx
≤ c
ε2k(1 − |ν|
2
)νkH−2kνk + cλkνk2
H−1
≤ c
ε2k(1 − |ν|
2_{)νkkνk}_{+ cλkνk}2
≤ c
ε2 + cλ
kνk2_{,}
(4.14)

where the constraint |ν|2≤ 1 plays a role. Given T > 0, the Gronwall Lemma implies ν ∈ L∞(0, T ; L2)∩
L2_{(0, T ; H}1_{).}

By taking the L2-product of (3.1) with w, we compute
1
2
d
dt kwk
2_{+ k∇wk}2+ k∇wk2 _{=}
Z
T2
∇ν>_{∇ν · ∇w dx −}
Z
T2
∇ν>_{∆ν}
t· w dx
=Z
T2
∇ν>_{∇}_{ν · ∇w dx +}
Z
T2
∇ν>_{∇ν}
t · ∇w dx
+
Z
T2
∇(∇ν)>∇νt· w dx,
(4.15)

where, to get the second equality, we have used relation (3.7), integrating by parts the first term obtained.

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

1
2
d
dt k∇νk
2_{+ k∆νk}2+ kν
tk2+ k∇νtk2 = −
Z
T2
∇ (w · ∇)ν · ∇νtdx
−
Z
T2
(w · ∇)ν ·νtdx
−
Z
T2
fε(ν) · νtdx+ λ
Z
T2
ν · νtdx.
(4.16)

Remark 3.1 implies that the first term in the right-hand side of (4.16) can be rewritten as
Z
T2
∇ (w · ∇)ν · ∇νt dx=
Z
T2
∇ν>_{∇ν}
t · ∇w dx+
Z
T2
∇(∇ν)>∇νt· w dx.

Then, by summing up (4.15) and (4.16), we infer 1 2 d dt kwk2+ k∇wk2+k∇νk2+ k∆νk2 + k∇wk2+ kνtk2+ k∇νtk2 ≤ Z T2 |∇w||∇ν|2 dx+ Z T2 |w||∇ν||νt| dx +Z T2 | f(ν)||νt| dx+ λ Z T2 |ν||νt| dx=: 4 X i=1 Ii. (4.17)

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

I1 ≤k∇wkk∇νk2_{L}4
≤ck∇wkk∇νkk∆νk
≤cεk∇wk2_{k∇νk}2_{+ C}
εk∆νk2,
(4.18)
I2 ≤kwkL4k∇νk_{L}4kν_{t}k
≤ckwk12|∇wk
1
2k∇νk
1
2k∆νk
1
2kν
tk
≤c
δkwkk∇wkk∇νkk∆νk + δkνtk2
≤¯c
δk∇wkk∇νkk∆νk + δkνtk2
≤c¯
δ2k∇wk
2_{k∇νk}2_{+ c}
k∆νk2+ δkνtk2,
(4.19)

I3 ≤
2
ε2kνkkνtk
≤ c
ε4δkνk
2_{+ δkν}
tk2,
(4.20)
and
I4 ≤λkνkkνtk
≤ cλ
δ kνk2+ λδkνtk2.
(4.21)

Estimates above, together with inequalities (4.14) and (4.17), allow us to write
d
dt kwk
2_{+ k∇wk}2_{+ k∇νk}2_{+ k∆νk}2+ k∇wk2_{+ 1 − (λ + 2)δkν}
tk2+ k∇νtk2
≤ cε,δ,λkνk2+ cεk∆νk2+ cε,δk∇wk2k∇νk2.
(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

d dt kwk

2_{+ k∇wk}2_{+ k∇νk}2_{+ k∆νk}2_{ ≤ c kνk}2_{+ k∆νk}2_{+ k∇wk}2_{k∇νk}2_{)}

≤ ckνk2_{+ c k∇wk}2_{+ k∆νk}2_{(1}_{+ k∇νk}2_{).}

Set y= kwk2+ k∇wk2+ k∇νk2+ k∆νk2_{. The di}fferential inequality

y0 ≤ ckνk2+ y(1 + k∇νk2)
implies
kwk2+ k∇wk2+ k∇νk2+ k∆νk2(t)
≤ kw0, ∇w0, ∇ν0, ∆ν0k2e
Rt
0(1+k∇ν(s)k
2_{)ds}
+ kνkL∞_{(0,T ;L}2_{)}
Z t
0
e
Rt
s(1+k∇ν(`)k
2_{)d`}
ds.
(4.23)

Since (w, ν) stands for (wn, νn_{), as a consequence of the above estimates, for any fixed T > 0, it follows}

that kwn, ∇wnk2

L∞_{(0,T ;H )}+ k∇ν

n_{, ∆ν}n_{k}2

L∞_{(0,T ;H)} is uniformly bounded with respect to n ∈ N. The control

(4.22) implies νt ∈ L2(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 equations (4.15) and (4.16). After integration by parts, we obtain
1
2
d
dt kwk
2
H12 + k∇wk
2
H12
+k∇wk2
H12
= − (w · ∇)w), w_{H}1
2 + ∇ν
>_{∇ν, ∇w}
H12
+ ∇ν>_{∇ν}
t, ∇w
H12 + ∇(∇ν)
>_{∇ν}
t, w
H12,
(4.24)

and
1
2
d
dt k∇νk
2
H12 + k∆νk
2
H12
+ kν
tk2
H12 + k∇νt
k2
H12
= − ∇ν>_{∇ν}
t, ∇w
H12 − ∇(∇ν)
>_{∇ν}
t, w
H12
− (w · ∇)ν, ν
t
H12 − fε(ν), νt
H12 + λ ν, νt
H12.
(4.25)

From equations (4.24) and (4.25), we get
1
2
d
dt kwk
2
H12 + k∇wk
2
H12 + k∇νk
2
H12 + k∆νk
2
H12
+ k∇wk2
H12 + kνt
k2
H12 + k∇νt
k2
H12
≤
(w · ∇)w), w
H12
+
∇ν
>_{∇ν, ∇w}
H12
+
(w · ∇)ν
, ν
t
H12
+
fε(ν), νt
H12
+ λ
ν, νt
H12
=:
5
X
i=1
Li.
(4.26)

For the terms Li, i= 1, . . . , 5 we actually use the norm induced by ( · , · )_{˙}

H12 = (Λ 1 2(·),Λ

1

2(·)) instead

of the full norm H12, although we still keep the same norm notation k · k

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 kwk_{˙}

H12 is equivalent to the full norm kwkH12.

Consider L1. Since
R
T2(w · ∇)Λ
1
2w ·Λ
1
2w ds = 0, we get
L1 ≤
Z
T2
|(Λ12_{w · ∇)w| |}Λ
1
2_{w| ds}
≤ kΛ12_{wk}2
L4k∇wk
≤ kΛ12wkkΛ
1
2∇wkk∇wk ≤ ckwk
H12k∇wkk∇wkH12
≤ c
kwk2H12
k∇wk2+ k∇wk2
H12
.
(4.27)

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

L2 ≤ k∇ν
>_{∇νk}
H12k∇wkH12 ≤ ckΛ
1
2∇νk
L4k∇νk_{L}4k∇wk
H12
≤ c kΛ12∇νk
1
2kΛ
1
2∆νk
1
2k∇νk
1
2k∆νk
1
2k∇wk
H12
≤ c
k∇νk2H12
k∇νk2_{+ k∆νk}2
H12
k∆νk2+k∇wk2
H12
,
(4.28)
L3 ≤ k(w · ∇)νk_{H}1
2kνtkH12 ≤ kΛ
1
2wk
L4k∇νk_{L}4 + kwk_{L}4kΛ
1
2∇νk
L4kν_{t}k
H12
≤ c
k∇wk2k∇νk2H12 + kwk
2
H12
k∆νk2+ kν
tk2
H12
, (4.29)
L4 =
Λ
1
2 _{f}_{ε}(ν),Λ
1
2ν
t
≤ ck fε(ν)kH1kνtkH12
≤ c
ε4(kνk
2_{+ k∇νk}2
)+ kνtk2
H12,
(4.30)

and L5= λ Λ 1 2ν, Λ 1 2ν t ≤ cλkνkH12kνtkH12 ≤ cλ kνk2H12 + kνt k2 H12 , (4.31)

after using H¨older’s, Ladyzhenskaya’s, and Young’s inequalities as well as the continuous embedding
W1/2,2(T2_{) ⊂ L}4_{(T}2_{).}

By using the estimates (4.27)–to–(4.30) along with (4.26), and absorbing the parameter ε−4 in a generic constant c, we obtain

1 2 d dt kwk 2 H12 + k∇wk 2 H12 + k∇νk 2 H12 + k∆νk 2 H12 + (1 − c)k∇wk2 H12 + kνt k2 H12 + k∇νt k2 H12 ≤ckwk2 H12 (1+ k∇wk2+ k∇νk + k∆νk2)+ ck∇νk2 H12 (1+ kwk2+ k∇νk2+ k∆νk2) + ck∆νk2 H12 k∆νk2

with > 0 small enough in a way that the coefficient ¯c := (1−c) is positive. Fix T > 0. By Gr¨onwall’s
lemma, we get
kw(t)k2
H12+k∇w(t)k
2
H12 + k∇ν(t)k
2
H12 + k∆ν(t)k
2
H12
+ ¯cZ t
0
k∇wk2
H12 + kνt
k2
H12 + k∇νt
k2
H12
ds
≤β exp
(
c
Z t
0
(1+ kwk2_{+ k∇wk}2_{+ k∇νk}2_{+ k∆νk}2_{) ds}
)

for any 0 < t ≤ T , with

β = c kw0k2 H12 + k∇w0 k2 H12 + ∇ν0 k2 H12 + k∆ν0 k2 H12 ,

and the quantity on the right-hand side of the above inequality is bounded, for 0 < t ≤ T , thanks to equation (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 {(u

n_{, ν}n

)}, we exploit the classical Aubin-Lions lemma; to this end we have first to provide a suitable control on wn

t. The next

calculations also fixes a minor issue present in the analogous control in reference [12], where we estimate acceleration in L1(0, T ; H−1).

Consider equation (3.1). For ϕ ∈ H1,R

T2ϕ dx = 0, with k∇ϕk = 1. Then, we get

hwn_{t} −∆wn_{t}, ϕiH−1_{,H}1 ≤| (wn· ∇)ϕ, wn|+ |(∇wn, ∇ϕ)|
+Z
T2
|∇νn_{|}2_{|∇}_{ϕ| dx + c}
Z
T2
|∇νn_{||∇ν}n
t||∇ϕ| dx
+ c
(∇(∇ν
n
)>∇νnt, ϕ)
≤kwnk2
L4k∇ϕk + k∇w
n_{kk∇}_{ϕk}
+ k∇νn_{k}2
L4k∇ϕk + ck∇ν
n_{k}
L4k∇νn_{t}k_{L}4k∇ϕk
+ ck∇(∇νn
)>∇νn_{t}k_{H}−1k∇ϕk
≤c kwnkk∇wnk+ ck∇wnk+ k∇νnkk∆νnk
+ k∇νn_{k}
H12k∇ν
n
tk_{H}1_{2} + k∆ν
n_{k}
H12k∇ν
n
tk_{H}1_{2}
,
(4.32)

after using the estimates performed in previous steps along with H¨older’s, Ladyzhenskaya’s, and Poincar´e’s inequalities. In the last inequality above, we have also exploited the continuous embedding

W1/2,2(T2_{) ⊂ L}4_{(T}2_{) and the Sobolev product laws (see, e.g., [3, 14, 30]) to get the estimate}

k∇(∇νn)>∇νn_{t}k_{H}−1 ≤ k∆νnkk∇νn_{t}k
H12
≤ k∆νnk
H12k∇ν
n
tk_{H}1
2.
Hence, we find
Z T
0
kwntk
2
H1ds ≤c
h
(1+ kwnk2_{L}∞_{(0,T ;L}2_{)})k∇w
n
k2_{L}2_{(0,T ;L}2_{)}+ k∆ν
n
k2_{L}2_{(0,T ;L}2_{)}
+ (k∇νn_{k}2
L∞_{(0,T ;H}1_{2}_{)}+ k∆ν
n_{k}2
L∞_{(0,T ;H}1_{2}_{)})k∇ν
n
tk2
L2_{(0,T ;H}1_{2}_{)}i.

As a final step in our argument, to extract a convergent subsequence from {(wn_{, ν}n_{)}, 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 equation(3.1) and providing a uniform estimate on

k∆wn_{t}k_{H}−2 = sup

kϕkH2˙ =1

|h∆wn_{t}, ϕi_{H}−2_{,H}2|

Indeed, also in this case, the worst term to be controlled is (∇(∇ν

n_{)}>_{∇ν}n
t, ϕ)

. For it, we get
Z T
0
(∇(∇ν
n
)>∇νn_{t}, ϕ)
2
ds ≤
Z T
0
k(∇(∇νn)>∇νn_{t}k2_{H}−2kϕk
2
H2 ds
≤
Z T
0
k∆νnk2k∇νn_{t}k2 ds
≤ k∆νnk2
L∞_{(0,T ;L}2_{)}
Z T
0
k∇νn
tk
2_{ds,}

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].

Acknowledgments

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.

Conflict of Interest

The authors declare no conflicts of interest in this paper. References

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