论文标题
不可压缩的Navier-Stokes系统的本地化和退化控件
Localized and degenerate controls for the incompressible Navier-Stokes system
论文作者
论文摘要
我们考虑了由物理局部和退化力驱动的二维不可压缩的Navier-Stokes系统的全球近似可控性。换句话说，流体是通过四个标量控件来调节的，这些标量控制仅取决于时间，并在给定子域支持的有效构造的驱动力中作为系数出现。我们的想法包括将低模式控制挤压到一个小区域，从本质上讲，通过跟踪其沿线性涡度方程的特征曲线的作用。通过这种方式，通过显式构造，并通过将Coron的返回方法与几何控制的最新概念联系起来，非线性Navier-Stokes系统的原始问题将减少到一个由全球力量引导的线性传输方程的一个。可以将本文视为试图解决由于Agrachev引起的众所周知的开放问题。
We consider the global approximate controllability of the two-dimensional incompressible Navier-Stokes system driven by a physically localized and degenerate force. In other words, the fluid is regulated via four scalar controls that depend only on time and appear as coefficients in an effectively constructed driving force supported in a given subdomain. Our idea consists of squeezing low mode controls into a small region, essentially by tracking their actions along the characteristic curves of a linearized vorticity equation. In this way, through explicit constructions and by connecting Coron's return method with recent concepts from geometric control, the original problem for the nonlinear Navier-Stokes system is reduced to one for a linear transport equation steered by a global force. This article can be viewed as an attempt to tackle a well-known open problem due to Agrachev.
