Research Summary

• Current research project progress: Firstly, we studied the 3D compressible Navier-Stokes-Korteweg (NSK) system, which describes the motions of the compressible isothermal viscous capillary fluids. In [1], by using a general energy method, we obtain the optimal time decay rates of the solutions to the NSK system, when the initial datas are small perturbations of a given constant state (ρ̄, θ̄). In particular, the optimal decay rates of the higher-order spatial derivatives of solutions are obtained. In our proof, the negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates. Instead of using the traditional linear decay analysis, we apply a family of scaled energy estimates with minimum derivative counts and interpolations among them. Secondly, we studied the multi-dimensional hydrodynamic equations of active nematic liquid crystals, described in the Landau-de Gennes’ Q-tensor framework, both in incompressible and compressible cases. The system governs fluids with active nematic suspensions. And one of the distinctions of this active nematic model from its well-studied passive counterparts is that: it is the energy that the active suspensions consume and dissipate that drives the system out of equilibrium, rather than the external force applied at the boundary of the passive system. On account of such internal drive in the active system, the system’s dynamics would be dramatically changed. Moreover, many novel effects, like the occurrence of giant density fluctuations, the spontaneous laminar flow, unconventional rheological properties, low Reynolds number turbulence, and different spatial and temporal patterns in passive systems, are disclosed from the interaction of the orientational order and the