关于不可压Navier-Stokes方程的若干问题

三维不可压Navier-Stokes方程的研究无论对于偏微分方程理论还是对于实际应用，都有着重要的意义. 我们将从两方面着手：一方面，从Liouville定理的视角去研究稳态Navier-Stokes方程的Liouville公开问题，Leray公开问题，第一奇性下Navier-Stokes方程的Liouville定理；另一方面，我们将从正则性入手去考察，包括单个速度分量的临界空间正则性，超临界空间的正则性等. 申请人自从2012年开始发表有关Navier-Stokes方程的研究论文，迄今已在《JDE》，《 SIAM J. Math. Anal.》，《 J. Anal. Math.》，《Sci. China Math.》，《 St. Petersburg Math. J.》，《Nonlinearity》等权威杂志发表数篇，受到很多数学家引用，这为进一步研究积累了诸多经验.

The study of three-dimensional incompressible Navier Stokes equations is of great significance both for the theory of partial differential equations and for practical applications. We will start from two aspects: on the one hand, from the perspective of Liouville theorem, we will study the Liouville open problem of steady Navier Stokes equation, Leray open problem, and Liouville theorem of Navier Stokes equation under the Type I singularity; on the other hand, we will start from the regularity, including the regularity of critical space of single velocity component, the regularity of supercritical space, etc. Since 2012, the applicant has published research papers on Navier Stokes equation in 《JDE》，《 SIAM J. Math. Anal.》，《 J. Anal. Math.》，《Sci. China Math.》，《 St. Petersburg Math. J.》，《Nonlinearity》 etc., which has been cited by many mathematicians and accumulated a lot of experience for further research.