AI 中文总结
研究不可压缩纳维-斯托克斯方程的位似自相似解，在三维中用刘维尔定理排除非平凡解，二维中确定唯一衰减解是奥森涡旋，还证明了围绕奥森涡旋线性化的欧拉算子的稳定性及发现一特殊位似解。
AI 中文摘要
我们研究不可压缩纳维-斯托克斯方程的位似向前自相似解：即存在非平凡的β，使得解\(\overline{U}\)和\(\beta \overline{U}\)均为自相似剖面的解。位似解还是唯一能用奇异极限论证沿Jia-Šverák程序证明勒雷-霍普夫解非唯一性的解。在三维中，对于充分正则的初始数据，我们证明一个刘维尔定理排除了非平凡位似解的存在。在二维中，我们的刘维尔定理证明唯一的衰减位似解是奥森涡旋。此外，我们证明围绕奥森涡旋线性化的欧拉算子是稳定的。另一方面，我们也发现一个存在线性化欧拉算子不稳定近似特征值的位似解。
英文摘要
We investigate homothetic forward self-similar solutions of the incompressible Navier-Stokes equations: the solutions $\overline{U}$ for which both $\overline{U}$ and $β\overline{U}$ are self-similar profiles for some nontrivial $β$. Homothetic solutions are, in addition, the only solutions for which a singular limit argument can be used to prove non-uniqueness of Leray-Hopf solutions along the lines of the Jia-Šverák program. In three dimensions, and for sufficiently regular initial data, we prove a Liouville theorem that rules out the existence of non-trivial homothetic solutions. In two dimensions, our Liouville theorem proves that the only decaying homothetic solution is the Oseen vortex. In addition, we prove that the Euler operator linearized around the Oseen vortex is stable. On the other hand, we also discover a homothetic solution for which an unstable approximate eigenvalue of the linearized Euler operator exists.
Comments49 pages
