Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain

In this paper, we prove some logarithmically improved regularity 
 criteria for the 3D nematic liquid crystals models, Boussinesq 
 system, and MHD equations in a bounded domain.

1. Introduction. In this paper we improve in different ways criteria concerning continuation of strong solutions of several models linked with the Navier-Stokes equations. Since the proofs have several points in common, we collect results for different models in a single paper, instead of scattering them through the literature. In particular, we prove with full details especially results for the nematic liquid crystals model below, and in the rest of the paper we show the changes needed to adapt the same techniques to different equations.
A particular feature of this paper is that we consider the problem in a bounded domain, with Dirichlet conditions on the velocity. Several results in literature concern the Cauchy problem (or the space periodic case) where proofs of regularity criteria are considerably less technical. In this paper we mainly focus in obtaining results with realistic boundary conditions, dealing carefully with the various boundary terms. To this end, let Ω be a bounded domain in R 3 with smooth boundary ∂Ω, and ν is the unit outward normal vector to ∂Ω. First, we consider the regularity criterion for the density-dependent, incompressible, nematic liquid crystals model [8,10,17,19]: div u = 0 in Ω × (0, ∞), in Ω × (0, ∞) with initial and boundary conditions (ρ, ρu, d)(·, 0) = (ρ 0 , ρ 0 u 0 , d 0 ) in Ω,

LUIGI C. BERSELLI AND JISHAN FAN
where ρ denotes the density, u = (u 1 , u 2 , u 3 ) the velocity, π the pressure, and d = (d 1 , d 2 , d 3 ) represents the macroscopic molecular orientations. The for each component of d, the symbol ∇d ⊗ ∇d denotes a matrix whose (i, j)-th entry is ∂ i d ∂ j d. (We consider the viscosity and other physical parameters set equal to one, to avoid unessential complications) When d = 0, then (1)-(3) represent the well-known density-dependent Navier-Stokes system, which has been object of many studies [9,16,14]. In particular, Kim [16] proved in this case that the following assumption is enough to continue smooth solutions: u ∈ L 2q q−3 (0, T ; L q w (Ω)), with 3 < q ≤ ∞, here L q w (Ω) denotes the (Marcinkiewicz) weak-L q (Ω) space and L ∞ w (Ω) = L ∞ (Ω). We recall that for the Navier-Stokes system this type of results in weak space date back to Sohr [23], see also [6], concerning Marcinkiewicz regularity with respect to the time variable.
When ρ = 1, (that is the nematic model with constant density) Guillén-González, Rojas-Medar, and Rodriguez-Bellido [15] proved the following regularity criterion (which we are going to logarithmically improve): then strong solution can be continued beyond T > 0.
It is well-known that problem (1)-(6) has a unique local-in-time strong solution (see [18]) and thus we omit the details here, we only recall that in particular a strong solution, we mean a quadruplet (ρ, u, π, d) satisfying (1)-(4) almost everywhere with the initial-boundary conditions (5)- (6). (In particular the quantities u H 2 , d H 3 , and ρ W 1,p are bounded almost everywhere). The first aim of this paper is to prove the following regularity criterion.
Let also the following compatibility condition at time t = 0 holds true: ∃ (π 0 , g) ∈ H 1 (Ω) × L 2 (Ω) such that Let (ρ, u, d) be a local strong solution to the problem (1)- (6). If u satisfies one of the following two conditions: with 0 < T < ∞, then, the strong solution (ρ, u, d) can be extended beyond T > 0.
In the case of constant density ρ, for simplicity set equal to 1, we consider also the following simplified liquid crystals model: Clearly, for the initial boundary problem (9)-(13) the same regularity criterion (7) or (8) holds true. Furthermore, we will prove also the following result concerning a more classical scaling-invariant criterion in weak-spaces.
Observe that this condition is the same (since it does not involve the variable d) which holds true for the Navier-Stokes equations, see [23].
As a further application of related techniques, we next consider the 3D viscous/diffusive Boussinesq system: Here the scalar field θ is the temperature and e 3 := (0, 0, 1) t . As expected, the regularity criteria (7) and (8) are valid also for the Boussinesq initial boundary value problem (15)- (19). Extensions are proved in the following theorem.
. Let (u, θ) be a local strong solution to the problem (15)- (19). If u satisfies one of the following two conditions: for some with 0 < T < ∞. Then, the solution (u, θ) can be extended beyond T > 0.

Remark 2.
The results of the previous theorem improve those collected in refs. [4,5,7]. Moreover, results extend to the bounded domain previous results from refs. [25,13,26,11] 640 LUIGI C. BERSELLI AND JISHAN FAN Next, we consider the regularity criterion for the 3D MHD system: where the vector field b = (b 1 , b 2 , b 3 ) is the magnetic field. For the MHD initial boundary value problem (22)-(26) one can prove scaling invariant regularity criteria involving just the velocity field, as those with conditions (7) and (8). Furthermore, we will prove the following result.
be a local strong solution to the problem (22)- (26). If (14) holds true, then the solution (u, b) can be extended beyond T > 0.
1.1. Notation and some preliminary lemmas. In the following we will use customary Lebesgue L p (Ω) and Sobolev spaces W k,p (Ω) and H s (Ω) = W s,2 (Ω), without distinction between scalar, vector, and tensor fields. We will denote by C generic constants changing from line to line, but independent on relevant quantities and for simplicity we will also write · dx := Ω · dx. In our proofs, we will use the following basic lemmas, which are at the basis of the derivation of most of the estimates. To handle boundary terms arising when looking for L p estimates we will use the following integration by parts and trace inequality.
Let Ω ⊂ R 3 be a smooth bounded domain, let b : Ω → R 3 be a smooth vector field, and let 1 < p < ∞. Then and Let Ω be a smooth and bounded open set and let 1 < p < ∞. Then the following trace estimate holds for any b ∈ W 1,p (Ω).
When b satisfies b · ν = 0 on ∂Ω, we will also use the identity which is valid for any sufficiently vector field b. 16]). Let f ∈ H 1 (Ω) and Ω ⊆ R 3 . Then there holds We will also use the following generalization of Hölder and Sobolev inequality. 24]). There holds the generalized Hölder inequality: 22]). There holds the following logarithmic Sobolev inequality: for any f ∈ W s,p (Ω) and Ω ⊆ R 3 .

LUIGI C. BERSELLI AND JISHAN FAN
Then it is obvious that and we have, see [1, p. 71], Thus (34) is proved.

2.
On density-dependent, incompressible, nematic liquid crystals. Due to the aforementioned existence and uniqueness theorem for local strong-solutions, by using a standard continuation argument, we only need to establish a priori estimates on smooth enough local solutions. Hence, by using a standard argument we assume existence of strong solutions in the maximal interval [0, T [, and all the calculations we are going to perform are completely justified. By proving uniform bounds for all t ∈ [0, T [, for appropriate norms of the unknowns, we will show that the solution can be continued beyond T , contradicting its maximality. In particular, in this case it will be enough to show boundedness for the H 2 (Ω)-norm of the velocity and for the H 3 (Ω)-norm of d.
Proof of Theorem 1.1. First, observe that, thanks to the maximum principle, it follows from (1) and (2) that By testing (3) by u, by suitable integration by parts (with boundary terms vanishing due to the boundary conditions (6)), and, by using (1) and (2), we see that Next, we test (4) by −∆d + |d| 2 d − d and by using (1), we easily obtain the equality d dt Summing up these two estimates we get the well-known energy inequality (37) Next, we prove the following estimate on d: Without loss of generality, we can assume that 1 ≤ d 0 L ∞ . Multiplying (4) by 2d, we get . Then, estimate (38) follows from (39) by the using the standard maximum principle.
for any 0 < δ < 1 and for some C > 0 depending on δ.

LUIGI C. BERSELLI AND JISHAN FAN
We use (35) to bound I 3 , I 4 , I 5 and I 6 as follow, for any 0 < δ < 1 and with a constant C depending on δ : Applying ∂ t to (4), testing by −∆d t and using (1) and (38), we get for any 0 < δ < 1. Plugging the above estimates for I 3 , I 4 , I 5 and I 6 into (51) and using (52) and taking δ small enough, and integrating over [t 0 , t] and using (50), we have Similarly to (46) and (48), we have and (55) Combining (54) and (55) and using (50) and (53), we conclude that and thus, if C 0 < 1 we can absorb terms from the right hand side to get Having proved the above estimate, it is now standard proving that ending the proof of statement i).
Concerning Part II, the proof is very close. In the second case, let (8) hold true. Similarly to (41), we take q = ∞ and using (34), we still get (41) provided that The rest calculations are similar if we take q = ∞ and using (59), and thus we arrive at (56), (57), and (58) ending the proof.
3. On a simplified liquid crystals model. This section is devoted to the proof of Theorem 1.2. As usual by a standard continuation argument, we only need to establish a priori estimates.

4.
On the viscous/diffusive Boussinesq system. Also in this case we show just a-priori estimates for smooth solutions.

5.
On a magneto-hydrodynamics system. We follow the same path as before.
6. On a simplified Ericksen-Leslie model. Also the results of this section are based on the same machinery, hence we need just a priori estimates to prove Theorem 1.5.

7.
On a simplified Ericksen-Leslie model with constant density. The proof of this last result is very similar to the previous one.