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We investigate regularity criteria for weak solutions to the Cauchy problem of the 3d co-rotational Beris-Edwards system for nematic liquid crystals, which couples the Navier–Stokes equations for the fluid velocity u with an evolution-diffusion equations for the Q -tenser. Our results yield that for any positive constant γ > 0 , if either the negative part of the associated pressure Π satisfies Π − [ ln ⁡ (1 + Π −) ] 1 + γ ∈ L ∞ (R + ; L 3 2 , ∞ (R 3)) , or the quantity 2 Π + | u | 2 + | ∇ Q | 2 satisfies (2 Π + + | u | 2 + | ∇ Q | 2) [ ln ⁡ (1 + 2 Π + + | u | 2 + | ∇ Q | 2) ] 1 + γ ∈ L ∞ (R + ; L 3 2 , ∞ (R 3)) , then the weak solution (u , Q) , to the 3d co-rotational Beris-Edwards system, is global-in-time smooth. Here, the subscript "−" and "+" denote the negative and the nonnegative part, respectively. L 3 2 , ∞ (R 3) denotes the standard weak Lebesgue space. If Q ≡ 0 , then our results extend some previous known results from the theory of the 3d Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

We study the partial regularity of suitable weak solutions to the 3d co-rotational Beris-Edwards system modeling the hydrodynamical motion of nematic liquid crystal flows with the Landau-De Gennes bulk potential, and present an improved version of the Caffarelli-Kohn-Nirenberg criterion obtained recently by Du-Hu-Wang (2020) [9] in terms of the velocity gradient only. The result yields that the velocity field plays a more important role than the Q -tensor field in the partial regularity theory of the 3d co-rotational Beris-Edwards system. [ABSTRACT FROM AUTHOR]

We study the partial regularity problem for a three dimensional simplified Ericksen–Leslie system, which consists of the Navier–Stokes equations for the fluid velocity coupled with a convective Ginzburg-Landau type equations for the molecule orientation, modelling the incompressible nematic liquid crystal flows. Base on the recent studies on the Navier–Stokes equations, we first prove some new local energy bounds and an εε-regularity criterion for suitable weak solutions to the simplified Ericksen-Leslie system, i.e., for σ ∈[0,1], there exists a ε > 0 such that if (u,d,P) is a suitable weak solution in Qr(z0) with 0 < r ≤ 1 and z0 = (x0,t0), and satisfies r −3/2−σ ∫t0t0−r2(∥|u|2∥2/2−σ/H−σ(Br(x0)) + ∥|∇d|2∥2/2−σ/H−σ(Br(x0)) + ∥P∥2/2−σ/H−σ(Br(x0))) dt ≤ ε, then (u,d) is regular at z0. Here, H−σ(Br(x)) is the dual space of Hσ0(Br(x)), the space of functions ƒ in the homogeneous Sobolev space Hσ(R3) such that supp ƒ ⊂ Br(x). Inspired by this ε-regularity criterion, we then improve the known upper Minkowski dimension of the possible interior singular points for suitable weak solutions from 95/63(≈1.50794) given by [24] (Nonlinear Anal. RWA, 44 (2018), 246–259.) to 835/613(≈1.36215). [ABSTRACT FROM AUTHOR]

In this paper, we study the possible singular set of suitable weak solutions ( u , d ) to the 3d simplified nematic liquid crystal flows, and prove that the parabolic upper Minkowski dimension of the possible singular set is bounded by 95 63 . Moreover, if the suitable weak solution ( u , d ) of the 3d simplified nematic liquid crystal flows satisfies ( u , ∇ d ) ∈ L w ( 0 , T ; L s ( R 3 ) ) with 3 < w , s < ∞ , then the parabolic upper Minkowski dimension of the singular set is no greater than max { w , s } ( 2 w + 3 s − 1 ) . If the suitable weak solution ( u , d ) satisfies ( u , ∇ d ) ∈ L ∞ ( 0 , T ; L 3 , ∞ ( R 3 ) ) , where L 3 , ∞ ( R 3 ) denotes the standard weak L 3 -Lebesgue space, then the parabolic upper Minkowski dimension of the singular set is bounded by 1. [ABSTRACT FROM AUTHOR]

In this paper, we investigate large time behavior of global‐in‐time strong solution to the three‐dimensional compressible flow of nematic liquid crystal with low regularity assumptions on initial datum. More precisely, we show that the negative Besov space B˙2,∞−s‐norms (s ≥ 0) of solution are preserved along time evolution; by using this fact together with the conventional energy estimates in Besov space framework and the interpolation inequalities, we establish that, for the initial perturbation ϱ0−1,u0,d0−d‾ just small in homogeneous Besov space B˜12,32×B˙12×B˜12,32, the global‐in‐time strong solution to the Cauchy problem of the compressible flow of nematic liquid crystal has the following optimal temporal decay rate: ‖ϱ(t)−1‖B˙ℓ+‖u(t)‖B˙ℓ+d(t)−d‾B˙ℓ+1≤C(1+t)−ℓ+s2,for−s<ℓ≤12,provided that we further assume that ϱ0−1,u0,d0−d‾ still belongs to B˙2,∞−s×B˙2,∞−s×B˙2,∞−s+1. Here, d‾∈S2 is a constant unite vector. To illustrate our methods clearly, we also revisit the optimal temporal decay of solutions to the heat equation in the framework of Besov space. [ABSTRACT FROM AUTHOR]

In this paper, we consider partial regularity of the Leray–Hopf type weak solutions to the Cauchy problem of the 3d co-rotational Beris–Edwards system for the nematic liquid crystal flows with Landau–De Gennes potential. The system under investigation consists of the Navier–Stokes equations for the velocity u coupled with an evolution equations for the Q -tensor. It is proved that a Leray–Hopf type weak solution (u , Q) ∈ L ∞ (0 , T ; L 2 (R 3 ; R 3)) ∩ L 2 (0 , T ; H 1 (R 3 ; R 3)) × L ∞ (0 , T ; H 1 (R 3 ; S 0 (3) )) ∩ L 2 (0 , T ; H 1 (R 3 ; S 0 (3) )) , to the 3d co-rotational Beris–Edwards system, is locally bounded if there exists a absolute constant ɛ so that the scaled local L p (1 ≤ p ≤ 5) norm of (u , ∇ Q) is less than ɛ. This implies that the one-dimensional parabolic Hausdorff measure for the possible singular point set is zero, which extends the corresponding result of Du et al. (2020) to the Leray–Hopf type weak solution. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global existence and uniqueness of solution to the 3D inhomogeneous incompressible nematic liquid crystal flows with variable density in the framework of Besov spaces. It is proved that there exists a global and unique solution to the nematic liquid crystal flows if the initial data ( ρ 0 − 1 , u 0 , n 0 − e 3 ) ∈ M ( B ˙ p , 1 3 p − 1 ( R 3 ) ) × B ˙ p , 1 3 p − 1 ( R 3 ) × B ˙ p , 1 3 p ( R 3 ) with 1 ≤ p < 6 , and satisfies ‖ ρ 0 − 1 ‖ M ( B ˙ p , 1 3 p − 1 ) + ‖ u 0 ‖ B ˙ p , 1 3 p − 1 + ‖ n 0 − e 3 ‖ B ˙ p , 1 3 p ≤ c for some small c > 0 depending only on p . Here M ( B ˙ p , 1 3 p − 1 ( R 3 ) ) is the multiplier space of Besov space B ˙ p , 1 3 p ( R 3 ) . Using the Lagrangian approach in Danchin and Mucha (2012, 2013) [7,8] is the key to our results. [ABSTRACT FROM AUTHOR]

In this paper, we concern the 3D nematic liquid crystal equations and prove three almost Serrin-type blow-up criteria for the breakdown of local in time smooth solutions in terms of pressure and gradient of the orientation field. More precisely, let T ∗ be the maximal time of the local smooth solution, then T ∗ < + ∞ if and only if ∫ 0 T ∗ ‖ ‖ ‖ P ( ⋅ , t ) ‖ L x 1 p ‖ L x 2 q ‖ L x 3 r β + ‖∇d(⋅,t)‖ L 4 8 d t = ∞ , with 2 β + 1 p + 1 q + 1 r = 2 and 2 ≤ p , q , r ≤ ∞ , 1 − ( 1 p + 1 q + 1 r ) ≥ 0 , and ∫ 0 T ∗ ‖ ‖ ‖ ∇ P ( ⋅ , t ) ‖ L x 1 p ‖ L x 2 q ‖ L x 3 r β + ‖ ∇ d ( ⋅ , t ) ‖ L 4 8 d t = ∞ , with 2 β + 1 p + 1 q + 1 r = 3 and 1 ≤ p , q , r ≤ ∞ , 1 − ( 1 2 p + 1 2 q + 1 2 r ) ≥ 0 , and ∫ 0 T ∗ ‖ ‖ ∂ 3 P ( ⋅ , t ) ‖ L x 3 γ ‖ L x 1 x 2 α β + ‖ ∇ d ( ⋅ , t ) ‖ L 4 8 d t = ∞ , with 2 β + 1 γ + 2 α = k ∈ [ 2 , 3 ) and 3 k ≤ γ ≤ α < 1 k − 2 . [ABSTRACT FROM AUTHOR]

In this paper, the Cauchy problem for the 3D incompressible nematic liquid crystal flows is investigated. A new logarithmically improved blow-up criterion for the local in time smooth solution to the 3D incompressible nematic liquid crystal flows in an appropriate homogeneous Besov space is established. [ABSTRACT FROM AUTHOR]

We use a general energy method recently developed by [Guo Y, Wang Y. Decay of dissipative equations and negative sobolev spaces. Commun. Partial Differ. Equ. 2012;37:2165–2208.] to prove the global existence and temporal decay rates of solutions to the three-dimensional compressible nematic liquid crystal flow in the whole space. In particular, the negative Sobolev norms of solutions are shown to be preserved along time evolution, and then the optimal decay rates of the higher order spatial derivatives of solutions are obtained by energy estimates and the interpolation inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we investigate blow up criteria for the local smooth solutions to the 3D incompressible nematic liquid crystal flows via the components of the gradient velocity field $\nabla u$ and the gradient orientation field $\nabla d$ . More precisely, we show that $0< T_{ \ast}<+\infty$ is the maximal time interval if and only if or where $i,j,k\in\{1,2,3\}$ , $i\neq j$ , $i\neq k$ , and $j\neq k$ . [ABSTRACT FROM AUTHOR]

This paper concerns the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible nematic liquid crystal flows on the whole space R 2 with vacuum as far field density. It is proved that the 2D nonhomogeneous incompressible nematic liquid crystal flows admit a unique global strong solution provided that the initial data density and the gradient of orientation decay not too slow at infinity, and the initial orientation satisfies a geometric condition (see (1.3) ). In particular, the initial data can be arbitrarily large and the initial density may contain vacuum states and even have compact support. Furthermore, the large time behavior of the solution is also obtained. [ABSTRACT FROM AUTHOR]

We show that the solution to the Cauchy problem of the 3D nematic liquid crystal flows, with initial data belonging to a critical Besov space, belongs to a Gevrey class. More precisely, it is proved that for any ( u 0 , d 0 − d ¯ 0 ) ∈ B ̇ p , 1 3 p − 1 ( R 3 ) × B ̇ q , 1 3 q ( R 3 ) with some suitable conditions imposed on p , q ∈ ( 1 , ∞ ) , there exists T ∗ > 0 depending only on initial data, such that the nematic liquid crystal flows admit a unique solution ( u , d ) on R 3 × ( 0 , T ∗ ) , and satisfies ‖ e t Λ 1 u ( t ) ‖ L ˜ T ∗ ∞ ( B ̇ p , 1 3 p − 1 ) ∩ L ˜ T ∗ 1 ( B ̇ p , 1 3 p + 1 ) + ‖ e t Λ 1 ( d ( t ) − d ¯ 0 ) ‖ L ˜ T ∗ ∞ ( B ̇ q , 1 3 q ) ∩ L ˜ T ∗ 1 ( B ̇ q , 1 3 q + 2 ) < ∞ . Here, d ¯ 0 ∈ S 2 is a constant unit vector, and Λ 1 is the Fourier multiplier whose symbol is given by | ξ | 1 = | ξ 1 | + | ξ 2 | + | ξ 3 | . Moreover, if the initial data is sufficiently small, then T ∗ = ∞ . As a consequence of the results, decay estimates of higher-order derivatives of solutions in Besov spaces are deduced. [ABSTRACT FROM AUTHOR]

We consider the temporal decay estimates for weak solutions to the two-dimensional nematic liquid crystal flows, and we show that the energy norm of a global weak solution has non-uniform decay [ABSTRACT FROM AUTHOR]

In recent paper [7] , Y. Du and K. Wang (2013) proved that the global-in-time Koch–Tataru type solution ( u , d ) to the n -dimensional incompressible nematic liquid crystal flow with small initial data ( u 0 , d 0 ) in BMO − 1 × BMO has arbitrary space–time derivative estimates in the so-called Koch–Tataru space norms. The purpose of this paper is to show that the Koch–Tataru type solution satisfies the decay estimates for any space–time derivative involving some borderline Besov space norms. More precisely, for the global-in-time Koch–Tataru type solution ( u , d ) to the nematic liquid crystal flow with initial data ( u 0 , d 0 ) ∈ BMO − 1 × BMO and ‖ u 0 ‖ BMO − 1 + [ d 0 ] BMO ≤ ε for some small enough ε > 0 , and for any positive integers k and m , one has ‖ t k + m 2 ( ∂ t k ∇ m u , ∂ t k ∇ m ∇ d ) ‖ L ˜ ∞ ( R + , B ˙ ∞ , ∞ − 1 ) ∩ L ˜ 1 ( R + ; B ˙ ∞ , ∞ 1 ) ≤ ε . Furthermore, we shall give that the solution admits a unique trajectory which is Hölder continuous with respect to space variables. [ABSTRACT FROM AUTHOR]

Abstract: We investigate blow-up criteria for the local in time classical solution of the nematic liquid crystal flows in dimensions two and three. More precisely, we prove that is the maximal time interval if and only if (i) for , or and (ii) for , [Copyright &y& Elsevier]

Abstract: We provide a sufficient condition for the regularity of solutions to the 3D nematic liquid crystal flow in the Morrey–Campanato space. More precisely, we prove that if the velocity satisfies or the gradient of the velocity satisfies then the solution remains smooth on . [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the regularity criterion for the solution of nematic liquid crystal flows in three dimensions ( ) and in two dimensions ( ). We prove that the solution is smooth up to time provided that there exists a positive constant such that (i) for , and (ii) for , [Copyright &y& Elsevier]

Abstract: In this work, we study the regularity criterion of the three-dimensional nematic liquid crystal flows. It is proved that if the vorticity satisfies where denotes the critical Besov space, then the solution becomes a regular solution on . This result extends the recent regularity criterion obtained by Fan and Ozawa (2012) [11]. [Copyright &y& Elsevier]

Abstract: This paper proves a logarithmically improved Beale–Kato–Majda’s criterion for the 3D nematic liquid crystal flows, which says that if then the solution is smooth up to the time . [Copyright &y& Elsevier]

In this article, we establish the Serrin-type regularity criteria for the 3D nematic liquid crystal flows in the terms of the multiplier space . Our criterion may be regarded as an extension of the recent result of Liu–Cui [Regularity of solutions to 3-D nematic liquid crystal flows, Electron. J. Diff. Equ. 2010(173) (2010), pp. 1–5]. [ABSTRACT FROM AUTHOR]

This paper proves some logarithmically improved regularity criteria in terms of the multiplier space for the 3D nematic liquid crystal flows. [ABSTRACT FROM AUTHOR]

We study the initial boundary value problem of a simplified Ericksen–Leslie system modeling the incompressible nematic liquid crystal flows in two dimensions of space, where the equations of the velocity field are characterized by a time-dependent external force g (t) and a no-slip boundary condition, and the equations for the molecular orientation are subjected to a time-dependent Dirichlet boundary condition h (t). Based on the recently addressed well-posedness and regularity results of the system, we present a rigorous proof to show the existence of optimal distributed controls, the control-to-state operator is Fréchet differentiable and first-order necessary optimality conditions for an associated optimal control problem. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the regularity of weak solutions to the Cauchy problem of the 3D simplified nematic liquid crystal flows with lower bound on pressure, and show that if the negative part of the pressure Π is controlled, or if the positive part of quantity | u | 2 + | ∇ d | 2 + 2 Π is controlled, then the weak solution (u , d) is global-in-time smooth. We also study the singular points of weak solutions, and prove that if a weak solution (u , d) on R 3 × (0 , T) satisfies ‖ u (⋅ , t) ‖ L p (R 3) + ‖ ∇ d (⋅ , t) ‖ L p (R 3) ≤ c (T − t) p − 3 2 p for some 3 < p < ∞ , where c is a positive constant, then the number of singular points is finite. [ABSTRACT FROM AUTHOR]