Gallay Thierry, Wayne C Eugene
Université de Grenoble I, Institut Fourier, BP 74, 38402 Saint-Martin d'Hères, France.
Philos Trans A Math Phys Eng Sci. 2002 Oct 15;360(1799):2155-88. doi: 10.1098/rsta.2002.1068.
We use the vorticity formulation to study the long-time behaviour of solutions to the Navier-Stokes equation on R(3). We assume that the initial vorticity is small and decays algebraically at infinity. After introducing self-similar variables, we compute the long-time asymptotics of the rescaled vorticity equation up to second order. Each term in the asymptotics is a self-similar divergence-free vector field with Gaussian decay at infinity, and the coefficients in the expansion can be determined by solving a finite system of ordinary differential equations. As a consequence of our results, we are able to characterize the set of solutions for which the velocity field satisfies ||u(.,t)||(L(2)) = o(t(-5/4)) as t-->+ infinity. In particular, we show that these solutions lie on a smooth invariant submanifold of codimension 11 in our function space.
我们使用涡度公式来研究(\mathbb{R}^3)上纳维-斯托克斯方程解的长时间行为。我们假设初始涡度很小且在无穷远处呈代数衰减。引入自相似变量后，我们计算重标涡度方程直至二阶的长时间渐近性。渐近性中的每一项都是在无穷远处具有高斯衰减的自相似无散向量场，并且展开式中的系数可以通过求解一个有限的常微分方程组来确定。作为我们结果的一个推论，我们能够刻画当(t\to +\infty)时速度场满足(|u(.,t)|_{L^2}=o(t^{-\frac{5}{4}}))的解的集合。特别地，我们表明这些解位于我们函数空间中余维数为(11)的光滑不变子流形上。
