In this paper, we study the existence and uniqueness of global weak solution, the regularity of the solutions and the existence of global attractor for a fourth order parabolic equation modeling epitaxial thin film growth with Neumann boundary conditions in two space dimensions.

In this paper, we consider the problem for optimal control of the sixth-order convective Cahn-Hilliard type equation. The optimal control under boundary condition is given, the existence of an optimal solution to the equation is proved and the optimality system is established.

<abstract><p>In this paper, the small initial data global well-posedness and time decay estimates of strong solutions to the Cauchy problem of 3D incompressible liquid crystal system with general Leslie stress tensor are studied. First, assuming that $ \|u_0\|_{\dot{H}^{\frac12+\varepsilon}}+\|d_0-d_*\|_{\dot{H}^{\frac32+\varepsilon}} $ ($ \varepsilon &gt; 0) $ is sufficiently small, we obtain the global well-posedness of strong solutions. Moreover, the $ L^p $–$ L^2 $ ($ \frac32\leq p\leq2 $) type optimal decay rates of the higher-order spatial derivatives of solutions are also obtained. The $ \dot{H}^{-s} $ ($ 0\leq s &lt; \frac12 $) negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates.</p></abstract>

<abstract><p>In this paper, we consider the decay rates of the higher-order spatial derivatives of solution to the Cauchy problem of 3D compressible nematic liquid crystals system. The $ \dot{H}^s\; (\frac12 &lt; s &lt; \frac32) $ negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates.</p></abstract>