论文标题
Navier-Stokes方程的平滑可控性具有Navier条件。应用于拉格朗日可控性
Smooth controllability of the Navier-Stokes equation with Navier conditions. Application to Lagrangian controllability
论文作者
论文摘要
我们在平滑的简单连接的有限域中处理3D Navier-Stokes方程，并在边界的非空开放部分上进行控件，并在边界的其余部分，未受控制的一部分上进行了navier slip-lip-nith-niver-navier slip-niver-slip-niver-slip-slip-them-stokes。我们扩展了小型全局确切可控性结果[J。 M. Coron，F。Marbach和F. Sueur，带有Navier-Stokes方程的小型全球精确可控性，带有Navier Slip-with-th-n-with-n-t-th-n-e-eur条件。数学。 Soc。，22（2020），1625-1673。]从Leray弱解决方案到平滑溶液的情况。我们的策略依赖于粘性边界层的精心制作耗散方法的改进，该方法出现在边界的不受控制部分附近，该方法允许处理更精细的拓扑中的多尺度特征。 作为我们分析的副产品，我们还获得了一个小型的全球近似拉格朗日可控性结果，并扩展到Navier-Stokes方程的情况下，最新结果[O.玻璃和T. Horsin，2-D Euler方程的Lagrangian可控性。应用于控制漩涡贴片形状的控制，J。Math。 Pures Appl。 （9），93（2010），61-90]，[O。玻璃和T. Horsin，规定了三维完美流体中一组颗粒的运动，Siam J. Control Optim。，50（2012），2726-2742]，[T。 Horsin和O. Kavian，无粘性不可压缩流体的拉格朗日可控性：一种建设性方法，Esaim Control Optim。计算。在Euler方程和结果[O.玻璃和T. Horsin，低雷诺数的拉格朗日可控性，Esaim Control Optim。计算。在稳定的Stokes方程式的情况下，Var。，22（2016），1040-1053]。
We deal with the 3D Navier-Stokes equation in a smooth simply connected bounded domain, with controls on a non-empty open part of the boundary and a Navier slip-with-friction boundary condition on the remaining, uncontrolled, part of the boundary. We extend the small-time global exact controllability result in [J. M. Coron, F. Marbach and F. Sueur, Small-time global exact controllability of the Navier-Stokes equation with Navier slip-with-friction boundary conditions, J. Eur. Math. Soc., 22 (2020),1625-1673.] from Leray weak solutions to the case of smooth solutions. Our strategy relies on a refinement of the method of well-prepared dissipation of the viscous boundary layers which appear near the uncontrolled part of the boundary, which allows to handle the multi-scale features in a finer topology. As a byproduct of our analysis we also obtain a small-time global approximate Lagrangian controllability result, extending to the case of the Navier-Stokes equations the recent results [O. Glass and T. Horsin, Approximate Lagrangian controllability for the 2-D Euler equation. Application to the control of the shape of vortex patches, J. Math. Pures Appl. (9), 93 (2010), 61-90], [O. Glass and T. Horsin, Prescribing the motion of a set of particles in a three-dimensional perfect fluid, SIAM J. Control Optim., 50 (2012), 2726-2742], [T. Horsin and O. Kavian, Lagrangian controllability of inviscid incompressible fluids: a constructive approach, ESAIM Control Optim. Calc. Var., 23 (2017), 1179-1200] in the case of the Euler equations and the result [O. Glass and T. Horsin, Lagrangian controllability at low Reynolds number, ESAIM Control Optim. Calc. Var., 22 (2016), 1040-1053] in the case of the steady Stokes equations.
