论文标题
对Navier-Stokes方程的一维类似物的有限时间爆破解决方案的重新归一化和存在
Renormalization and existence of the finite-time blow up solutions for a one-dimensional analogue of the Navier-Stokes equations
论文作者
论文摘要
一维的准地藻方程是著名的Navier-Stokes方程的一维傅立叶空间类似物。在他们的工作中，李和西奈提出了一种重新规范化的方法，以解决该方程式的有限时间爆破解决方案的问题。在这种情况下，有限的时间爆炸的存在是某个重新归一化操作员在适当的功能空间上存在固定点的结果。他们提供了复杂价值有限的时间的存在证明，炸毁了准地藻方程的解决方案。 在本文中，我们重新审查了准地斑型爆炸的重新规则化问题，证明存在重新归一化的固定点家族，并推断出真正的$ c^\ infty（[0，t），c^\ infty（c^\ infty（\ mathbb {r}）在有限的时期内，肠胃植物变得无限，与李和西奈先前的作品不同。
The one-dimensional quasi-geostrophic equation is the one-dimensional Fourier-space analogue of the famous Navier-Stokes equations. In their work Li and Sinai have proposed a renormalization approach to the problem of existence of finite-time blow up solutions of this equation. In this setting, existence of finite time blow ups is a consequence of existence of a fixed point for a certain renormalization operator on an appropriate functional space. They have provided a proof of existence of complex-valued finite time blow up solutions of the quasi-geostrophic equation. In this paper we revisit the renormalization problem for the quasi-geostrophic blow ups, prove existence of a family of renormalization fixed points, and deduce existence of real $C^\infty([0,T),C^\infty(\mathbb{R}) \cap L^2(\mathbb{R}))$ solutions to the quasi-geostrophic equation whose energy and enstrophy become unbounded in finite time, different from those found in the previous work of Li and Sinai.
