Liouville theorems for the Navier-Stokes equations and applications

• 成果类型: Article
• 署名作者: Koch, Gabriel; Nadirashvili, Nikolai; Seregin, Gregory A.; Sverak, Vladimir
• 署名单位: University of Chicago; Aix-Marseille Universite; Centre National de la Recherche Scientifique (CNRS); University of Oxford; University of Minnesota System; University of Minnesota Twin Cities
• 刊物名称: ACTA MATHEMATICA
• ISSN/ISSBN: 0001-5962
• DOI: 10.1007/s11511-009-0039-6
• 发表日期: 2009
• 页码: 83-105
• 关键词: self-similar solutions; blow-up; superlinear problems; elliptic-equations; well-posedness; WEAK SOLUTIONS; REGULARITY; decay; singularity; FLOWS

摘要

We study bounded ancient solutions of the Navier-Stokes equations. These are solutions with bounded velocity defined in R (n) x (-1, 0). In two space dimensions we prove that such solutions are either constant or of the form u(x, t) = b(t), depending on the exact definition of admissible solutions. The general 3-dimensional problem seems to be out of reach of existing techniques, but partial results can be obtained in the case of axisymmetric solutions. We apply these results to some scenarios of potential singularity formation for axi-symmetric solutions, and obtain extensions of results in a recent paper by Chen, Strain, Tsai and Yau [4].