多尺度流体方程及其耦合模型的数学理论

本项目拟研究多尺度流体方程及其耦合模型的数学理论，包括Boltzmann型方程、Navier-Stokes方程、及描述流-场耦合的液晶模型与液-粒耦合的两相流模型等，围绕诸多前沿核心问题，如高维可压Navier-Stokes方程含真空强解的适定性、三维不可压Navier-Stokes方程（轴对称）光滑解的整体适定性、高维Prandtl边界层的适定性、三维Ericksen-Leslie方程弱解的整体存在性、高维Vlasov-Maxwell-Landau方程弱解的整体适定性等富有挑战性的公开问题，重点研究解的存在性、唯一性、正则性、稳定性、时空渐近行为、及多尺度极限等。这些科学前沿问题的解决既需要借鉴源于流体力学、等离子物理、液晶材料等的规律和想法，同时也要挖掘这些问题本身所蕴含的数学结构，并结合诸多现代数学研究领域的不同研究思想和技术方法，加以创新发展，有重要的科学理论意义和应用价值。

The mathematical study on the well-posedness theory and behaviors of the fluid-dynamical equations and the related coupling models, which consist of the Boltzmann equation and its variants at mesoscopic scale, Navier-Stokes system and Euler equations at macroscopic scale, microscopic-macroscopic coupled Doi-Onsager liquid crystals models, and fluid-particle coupled two-phase models, is one of the most important and challenging issues in the fields of applied analysis and nonlinear partial differential equations. This project is devoted to the study on some important aspects of the fluid-dynamical equations and the related coupling models, such as the existence, uniqueness and regularity of weak solutions, and their nonlinear stability and multi-scaling asymptotical behaviors. In particular, it shall be taken into granted the important issues including the well-posedness and behaviors of the compressible Navier-Stokes system in energy space for general initial data containing vacuum, the global well-posedness of (axisymmetric) smooth solution to three dimensional incompressible Navier-Stokes equations for general smooth initial data with finite energy, the well-posedness (or ill-posedness) and stability analysis of multi-dimensional Prandtl equations, and the global existence and behaviors of weak solutions to multi-dimensional Vlasov-Maxwell-Landau system. These are fundamental and frontier issues in the research fields of nonlinear partial differential equations, and we expect that the study about these open problems can make important breakthrough on these basic issues of great mathematical challenges.