论文标题
Navier-Stokes-Korteweg型号的局部确切可控性
Local exact controllability to constant trajectories for Navier-Stokes-Korteweg model
论文作者
论文摘要
在本文中，我们研究了dimension $ d \ in \ {1,2,3 \} $的圆环上的可压缩Navier-Stokes-korteweg系统的局部精确可控性，当该控件在开放子集上作用时。更确切地说，我们获得了恒定状态$（ρ_{\ star}，0）$的局部确切可控性，以进行任意的小积极时间，并且在控制区域没有任何几何条件。为此，我们分析了线性方程的控制特性，并详细研究了伴随方程的可观察性。特别是，我们将展示这些伴随方程的抛物线（可能也是分散）结构。基于此，我们将能够通过Carleman估计来恢复伴随系统的可观察性。
In this article, we study the local exact controllability to a constant trajectory for a compressible Navier-Stokes-Korteweg system on the torus in dimension $ d\in\{1,2,3\}$ when the control acts on an open subset. To be more precise, we obtain the local exact controllability to the constant state $(ρ_{\star},0)$ for arbitrary small positive times and without any geometric condition on the control region. In order to do so, we analyze the control properties of the linearized equation, and present a detailed study of the observability of the adjoint equations. In particular, we shall exhibit the parabolic (possibly also dispersive) structure of these adjoint equations. Based on that, we will be able to recover observability of the adjoint system through Carleman estimates.
