In this paper, we consider the initial and boundary value problem of a simplified compressible nematic liquid crystal flow in Ω ⊂ R 3 . We establish the existence of global weak solutions, provided the initial orientational director field d 0 lies in the hemisphere S 2 + . IntroductionThe continuum theory of liquid crystals was developed by Ericksen [3] and Leslie [7] during the period of 1958 through 1968, see also the book by De Gennes [2]. Since then there have been remarkable research developments in liquid crystals from both theoretical and applied aspects. When the fluid containing nematic liquid crystal materials is at rest, we have the well-known Oseen-Frank theory for static nematic liquid crystals, see Hardt-Lin-Kinderlehrer [8] on the analysis of energy minimal configurations of nematic liquid crystals. In general, the motion of fluid always takes place. The so-called Ericksen-Leslie system is a macroscopic continuum description of the time evolution of the material under influence of both the flow velocity field u and the macroscopic description of the microscopic orientation configurations d of rod-like liquid crystals.When the fluid is an incompressible, viscous fluid, Lin [10] first derived a simplified Ericksen-Leslie system (i.e. ρ = 1 and divu = 0 in the equation (1.1) below) modeling liquid crystal flows in 1989. Subsequently, Lin and Liu [11,12] have made some important analytic studies, such as the global existence of weak and strong solutions and the partial regularity of suitable weak solutions, of the simplified Ericksen-Leslie system, under the assumption that the liquid crystal director field is of varying length by Leslie's terminology or variable degree of orientation by Ericksen's terminology. When dealing with the system (1.1) with ρ = 1 and divu = 0, in dimension two Lin-Lin-Wang [13] and Lin-Wang [14] have established the existence of a unique global weak solution, that has at most finitely many possible singular time, for the initial-boundary value problem in bounded domains (see also Hong [9], Xu-Zhang [36], and Lei-Li-Zhang [15] for some related works); and in dimension three Lin-Wang [18] have obtained the existence of global weak solutions very recently when the initial director field d 0 maps to the hemisphere S 2 + . When the fluid is compressible, the simplified Ericksen-Leslie system (1.1) becomes more complicate, which is a strongly coupling system between the compressible Navier-Stokes equation and the transported harmonic map heat flow to S 2 . It seems worthwhile to be explored for the mathematical analysis of (1.1). We would like to mention that there have been both modeling study, see Morro [24], and numerical study, see , on the hydrodynamics of compressible nematic liquid crystals under the influence of temperature gradient or electromagnetic forces. Now let's introduce the simplified Ericksen-Leslie system for compressible nematic liquid crystal flow. Let Ω ⊂ R 3 be a bounded, smooth domain, S 2 ⊂ R 3 be the unit sphere, and 0 < T ≤ +∞. We will consider a simplifi...

Let B ⊂ R N B \subset \mathbb {R}^N be the unit ball. We study the structure of solutions to the semilinear biharmonic problem \[ { Δ 2 u = λ ( 1 − u ) − p a m p ; in  B , 0 > u > 1 a m p ; in  B , u = ∂ ν = 0 ( resp.~ u = Δ u = 0 ) a m p ; on  ∂ B , \begin {cases} \Delta ^2 u=\lambda (1-u)^{-p} & \text {in $B$},\\ 0>u>1 & \text {in $B$},\\ u=\partial _\nu =0\; (\text {resp.~$u = \Delta u = 0$}) & \text {on $\partial B$}, \end {cases} \] where p , λ > 0 p, \lambda >0 , which arises in the study of the deflection of charged plates in electrostatic actuators. We study in particular the structure of solutions for N = 2 N=2 or 3 3 and show the existence of mountain-pass solutions under suitable conditions on p p . Our results contribute to completing the picture of solutions in previous works. Moreover, we also analyze the asymptotic behavior of the constructed mountain-pass solutions as λ → 0 \lambda \to 0 .

We study forward self-similar solutions to the 3-D Navier-Stokes equations with the fractional diffusion (−∆) α . First, we construct a global-time forward self-similar solutions to the fractional Navier-Stokes equations with 5/6 < α ≤ 1 for arbitrarily large self-similar initial data by making use of the so called blow-up argument. Moreover, we prove that this solution is smooth in R 3 × (0, +∞). In particular, when α = 1, we prove that the solution constructed by Korobkov-Tsai [23, Anal. PDE 9 (2016), 1811-1827 satisfies the decay estimate by establishing regularity of solution for the corresponding elliptic system, which implies this solution has the same properties as a solution which was constructed in [17, Jia andŠverák, Invent. Math. 196 (2014), 233-265]. n 2 , γ(r) = +∞ 0 s r−1 e −s ds, r > 0.

We are interested in entire solutions for the semilinear biharmonic equation ∆ 2 u = f (u) in R N , where f (u) = e u or −u −p (p > 0). For the exponential case, we prove that for the polyharmonic problem ∆ 2m u = e u with positive integer m, any classical entire solution verifies ∆ 2m−1 u < 0, this completes the results in [6,14]; we obtain also a refined asymptotic expansion of radial separatrix solution to ∆ 2 u = e u in R 3 , which answers a question in [2]. For the negative power case, we show the nonexistence of the classical entire solution for any 0 < p ≤ 1. (2000): 35J91, 35B08, 35B53, 35B40. Mathematics Subject Classification

We establish the global well-posedness of the Landau-Lifshitz-Gilbert equation in R n for any initial data m 0 ∈ H 1 * (R n , S 2 ) whose gradient belongs to the Morrey space M 2,2 (R n ) with small norm ∇m 0 M 2,2 (R n ) . The method is based on priori estimates of a dissipative Schrödinger equation of Ginzburg-Landau types obtained from the Landau-Lifshitz-Gilbert equation by the moving frame technique.