In this paper, we investigate the global existence and long time behavior of strong solutions for compressible nematic liquid crystal flows in three-dimensional whole space. The global existence of strong solutions is obtained by the standard energy method under the condition that the initial data are close to the constant equilibrium state in $H^2$-framework. If the initial data in $L^1$-norm are finite additionally, the optimal time decay rates of strong solutions are established. With the help of Fourier splitting method, one also establishes optimal time decay rates for the higher order spatial derivatives of director.

In this paper, we consider a hydrodynamic flow of nematic liquid crystal system. We prove the local well-posedness for the system in the critical Lebesgue space, and study the space-time regularity of the local solution.

Based on the range space property (RSP), the equivalent conditions between nonnegative solutions to the partial sparse and the corresponding weighted $l_1$-norm minimization problem are studied in this paper. Different from other conditions based on the spark property, the mutual coherence, the null space property (NSP) and the restricted isometry property (RIP), the RSP-based conditions are easier to be verified. Moreover, the proposed conditions guarantee not only the strong equivalence, but also the equivalence between the two problems. First, according to the foundation of the strict complementarity theorem of linear programming, a sufficient and necessary condition, satisfying the RSP of the sensing matrix and the full column rank property of the corresponding sub-matrix, is presented for the unique nonnegative solution to the weighted $l_1$-norm minimization problem. Then, based on this condition, the equivalence conditions between the two problems are proposed. Finally, this paper shows that the matrix with the RSP of order $k$ can guarantee the strong equivalence of the two problems.

Consider the Cauchy problems for an $n$-dimensional nonlinear system of fluid dynamics equations. The main purpose of this paper is to improve the Fourier splitting method to accomplish the decay estimates with sharp rates of the global weak solutions of the Cauchy problems. We will couple together the elementary uniform energy estimates of the global weak solutions and a well known Gronwall’s inequality to improve the Fourier splitting method. This method was initiated by Maria Schonbek in the 1980’s to study the optimal long time asymptotic behaviours of the global weak solutions of the nonlinear system of fluid dynamics equations. As applications, the decay estimates with sharp rates of the global weak solutions of the Cauchy problems for $n$-dimensional incompressible Navier-Stokes equations, for the $n$-dimensional magnetohydrodynamics equations and for many other very interesting nonlinear evolution equations with dissipations can be established.

In this paper, by using the Guo-Krasnoselskii’s fixed-point theorem, we establish the existence and multiplicity of positive solutions for a fourth-order nonlinear eigenvalue problem. The corresponding examples are also included to demonstrate the results we obtained.

We present some conditions for the existence and uniqueness of almost automorphic solutions of third order neutral delay-differential equations with piecewise constant of the form $$(x(t) + px(t − 1))′′′ = a_0x([t]) + a_1x([t − 1]) + f(t),$$ where [·] is the greatest integer function, $p,$ $a_0$ and $a_1$ are nonzero constants, and $f(t)$ is almost automorphic.