论文标题
Navier-Stokes方程的全局弱解决方案，用于间歇性初始数据的半空间
Global weak solutions of the Navier-Stokes equations for intermittent initial data in half-space
论文作者
论文摘要
我们证明，在半空间$ \ mathbb {r}^3 _+$中存在不可压缩的Navier-Stokes方程的全球弱解，并在加权空间中具有初始数据，在间隔意义上可以在空间无限的情况下在空间无限内生长出非均匀的本地平方集成函数。初始数据的空间建立在立方体上，其侧面$ r $与原始距离成正比，并且允许数据的正方形积分作为$ r $的功率增长。 存在通过新的先验估计和稳定性在加权空间以及新的压力估计中获得。另外，我们证明了这种弱解决方案的最终规律性，直至边界，以$（x，t）$满足$ t> c_1 | x | x |^2 + c_2 $，其中$ c_1，c_1，c_2> 0 $，对于大型的初始数据$ u_0 $，$ c_1 $ thit $ c_1 $。作为存在定理的应用，我们构建了全局离散自相似的解决方案，从而将其在半空间上扩展到与整个空间相同的一般性。
We prove existence of global-in-time weak solutions of the incompressible Navier-Stokes equations in the half-space $\mathbb{R}^3_+$ with initial data in a weighted space that allow non-uniformly locally square integrable functions that grow at spatial infinity in an intermittent sense. The space for initial data is built on cubes whose sides $R$ are proportional to the distance to the origin and the square integral of the data is allowed to grow as a power of $R$. The existence is obtained via a new a priori estimate and stability result in the weighted space, as well as new pressure estimates. Also, we prove eventual regularity of such weak solutions, up to the boundary, for $(x,t)$ satisfying $t>c_1|x|^2 + c_2$, where $c_1,c_2>0$, for a large class of initial data $u_0$, with $c_1$ arbitrarily small. As an application of the existence theorem, we construct global discretely self-similar solutions, thus extending the theory on the half-space to the same generality as the whole space.
