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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, 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]

In Guo and Wang (2012) [10] , Y. Guo and Y. Wang developed a general new energy method for proving the optimal time decay rates of the solutions to dissipative equations. In this paper, we generalize this method in the framework of homogeneous Besov spaces. Moreover, we apply this method to a model arising from electro-hydrodynamics, which is a strongly coupled system of the Navier–Stokes equations and the Poisson–Nernst–Planck equations through charge transport and external forcing terms. We show that some weighted negative Besov norms of solutions are preserved along time evolution, and obtain the optimal time decay rates of the higher-order spatial derivatives of solutions by the Fourier splitting approach and the interpolation techniques. [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]

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]

In this paper, we develop the energy argument in homogeneous Besov space framework to study the large time behavior of global-in-time strong solutions to the Cauchy problem of the three-dimensional incompressible nematic liquid crystal flows with low regularity assumptions on initial data. More precisely, if the small initial data [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]

We prove that weak-strong uniqueness holds for the $$\beta $$ β -generalized surface quasi-geostrophic equation in the regular class $$\nabla \theta \in L^{q}(0,T; L^{p}(\mathbb{R }^{2}))$$ ∇ θ ∈ L q ( 0 , T ; L p ( R 2 ) ) with $$\frac{\alpha }{q}+\frac{2}{p}=\alpha +\beta -1$$ α q + 2 p = α + β − 1 , where $$\alpha \in (0,1], \beta \in [1,2)$$ α ∈ ( 0 , 1 ] , β ∈ [ 1 , 2 ) and $$\frac{2}{\alpha +\beta -1}

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]