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.

In the balance of microstructural actions governing the evolution of ν, an hyper-stress behaving like ∇ 2 ν accounts for second-neighbor interactions; it enters the equation through its double divergence, which generates the term ∆ 2 ν. A viscous-type contribution (namely ∇ν ∆ν t ) 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 ε 2 (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 S 2 is −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 We tackle its analysis by filtering the balance of macroscopic momentum by (I − ∆) −1 . In the process, we define the regularized velocity w := (I − ∆) −1 u, 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 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.

Notation and preliminaries
For p ≥ 1, by L p = L p (T 2 ) we indicate the usual Lebesgue space with norm · p . When p = 2, we use the notation · := · L 2 and denote by ( · , · ) the related inner product. Moreover, with k a nonnegative integer and p ≥ 1, we denote by W k,p := W k,p (T 2 ) the usual Sobolev space with norm · k,p (using · k when p = 2 ). We write W −1,p := W −1,p (T 2 ), p = p/(p − 1), for the dual of W 1,p (T 2 ) with norm · −1,p .
Let X be a real Banach space with norm · X . We will use the customary spaces W k,p (0, T ; X), with norm denoted by · W k,p (0,T ;X) . In particular, W 0,p (0, T ; X) = L p (0, T ; X) are the standard Bochner spaces.
(L p ) n := L 2 (T 2 , R n ), p ≥ 1, is the function space of vector-valued L 2 -maps. Similarly, (W k,p ) n := (W k,p (T 2 )) 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 H s = W s,2 ∩ {w : ∇ · w = 0}. We also define the following spaces: This last space is the usual Sobolev space of vector fields with components W s,2 -functions. Again H := H 0 . By H −s we indicate the space dual to H s . We denote by · , · := · , · H −1 ,H 1 the duality pairing between H −1 and H 1 . We will also assume that the vector fields u and w have null average on T 2 . In particular, under such an assumption, Poincaré's inequality holds true.
Here and in the sequel, we denote by c (orc) positive constants, which may assume different values.
The norm in H s is given by where the over-bar denotes, as usual, complex conjugation. Consider the inverse Helmholtz operator where G(x, y) is the associated Green function (see, e.g. [5,7,8,9,10]). For w ∈ H s , take the Fourier expansion w = k∈T 2 w k e ik·x , so that, by inserting this expression in (2.11), we get (2.12) G is self-adjoint. It commutes with differential operators (see, e.g., [4,5,7]). We get also (see also [5,19]).

Existence and regularity result
We set Then, we rewrite the filtered balances as To keep the notation compact, here and in the sequel we omit the dependence of w and ν on ε.
In the following, we'll always refer to "regular weak solutions" simply as "weak solutions", for the sake of brevity,.
, 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 (w 0 , ν 0 ) ∈ H 1 × H 2 , obtaining for them once again an existence result (see Remark 4.1 below).
The first term on the right-hand side of the above identity is such that and for the second term we find For the second term on the right-hand side of (3.1), we compute

Proofs
We introduce Galerkin's approximating functions {(w n , ν n )}, prove a maximum principle, by which the constraint |ν n | ≤ 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.)

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 (ω i , π i ) ∈ H 2 × W 1,2 the unique solution of the following Stokes problem: with T 2 π i dx = 0, for i = 1, 2, . . . and 0 < λ 1 ≤ λ 2 ≤ . . . λ n . . . with λ n → +∞, as n → ∞. Functions {ω i } +∞ i=1 determine an orthonormal basis in H made of the eigenfunctions pertaining to (4.1). Let P n : H 3/2 → H n := H 3/2 ∩ span{ω 1 , ω 2 , . . . , ω n } be the orthonormal projection of H 3/2 on its finite dimensional subspace H n . Take T > 0. For every positive integer n, we look for an approximate solution (w n , ν n ) ∈ C 1 (0, T ; H n ) × L ∞ (0, T, H Consider the following problem defined a.e. in (0, T ) × T 2 : where w 0 ∈ H 3 2 and ν 0 ∈ H 5 2 , with |ν 0 (x)| ≤ 1 a.e. in T 2 . Instead of exploiting test functions in L 2 , 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 w n ∈ C 1 (0, T ; H n ) be a given velocity field of the form w n (t, with ν n (0, x) = ν 0 (x), for x ∈ T 2 , we actually look for a vector field solving a.e. on (0, T ) × T 2 the following system: 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 f ε (ν n ) gains two additional spatial derivatives with respect to f ε (ν n ); the same occurs for Gν n (t)). Thus, this new expression can be rewritten equivalently as a semilinear parabolic equation in the unknown ν n . The existence of such ν n 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 ν 0 ∈ H 5 2 to get ν n ∈ L ∞ (0, T ; H 3 2 )∩ L 2 (0, T ; H 5 2 ) and ν n t ∈ L 2 (0, T ; H 1 ) (by interpolation we also have that ν t ∈ C(0, T ; H 1 )). The following lemma (see [12,Lemma 4.1] and also [15,Lemma 2.1]) guarantees the constraint |ν| ≤ 1. In performing the next calculations, we could relax hypotheses by assuming that ν 0 ∈ H 1 and is such that |ν 0 (x)| ≤ 1 a.e. x ∈ T 2 , withw n ∈ C(0, T ; H n ). Then, there would exist a weak solution ν ∈ L ∞ (0, T ; H 1 ) × L 2 (0, T ; H 2 ), with |ν(x, t)| ≤ 1 a.e. in T 2 × (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 ν n andw n , respectively for the sake of conciseness.
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 ⊂ T 2 with positive measure |B| > 0 such that |ν(x, t)| > 1 a.e. in B × (t 1 , t 2 ], 0 ≤ t 1 < t 2 ≤ T , and |ν(x, t)| = 1 a.e. in ∂B × (t 1 , t 2 ]. By taking ϕν as a test function against (4.7), we get which is equivalent to With · indicating · L 2 (B) , we can also write Then, equation (4.9) becomes d dt (4.10) Since ϕ(t 2 ) ≥ ϕ(t 1 ) (here, ϕ(t 1 ) = 0), by integrating in time over (t 1 , t 2 ], we get 2 t 2 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 Then, there exists at least one connected component B j , with |B j | > 0, on which and hence, by(2.13), we have (4.11) Since |ν(x, t)| = 1 a.e. on ∂B×(t 1 , t 2 ), we get ϕ(x, t) = 0 a.e. on ∂B×(t 1 , t 2 ) and, in particular, ϕ(x, t) = 0 a.e. on ∂B j × (t 1 , t 2 ). Assume that B j 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 [30]), we obtain and 4λ (4.12) Hence, the inequality where C is the constant involved in the Poincaré inequality, holds true. Then, we find which gives an absurd by assuming that is sufficiently large as λ is small. The general case, when B j is not the closure of an open set, follows the same line of the argument in reference [12].
2 ) be the vector field just determined in the previous step. We search the approximating velocity field w n ∈ C 1 (0, T ; H n ) satisfying the equation where both ν n and w n are given. Thanks to the Cauchy-Lipschitz theorem, we can prove existence of a unique maximal solution w n of the above problem.

Global existence
In the sequel, for the sake of compactness, we use the same symbol · L p (0,T ;L k ) for the norm in L p (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 L p (0, T ; (W s,k ) n ) (also L p (0, T ; H s ) and L p (0, T ; (H s ) n )).
As a final step in our argument, to extract a convergent subsequence from {(w n , 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.
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].