论文标题
与表面张力的不可压缩的两阶段Navier-Stokes方程的最佳控制的不同性结果和灵敏度计算
Differentiability results and sensitivity calculation for optimal control of incompressible two-phase Navier-Stokes equations with surface tension
论文作者
论文摘要
我们分析具有表面张力的两阶段Navier-Stokes方程的最佳控制问题。基于$ l_p $ - 基础线性问题的最大规律性以及针对足够小的数据的最新问题结果，我们显示了解决方案与适当空间的初始和分布式控件相对于适当的空间的可不同性，从而形成了$ l_p $ -maximal的规律性设置。我们首先考虑将界面转换为超平面的公式。然后，我们推断物理坐标中解决方案的可怜性结果。最后，我们陈述了等效的流体类型公式，并使用所获得的可不同性结果来得出严格的流体量类型公式的相应灵敏度方程。对于涉及速度场或不连续压力或相位辅助场场的目标功能，我们在衍生物的对照和状态公式方面得出了可不同的结果。本文的结果构成了陈述最佳条件的分析基础，证明了基于衍生化的优化方法的应用并研究了基于流体量的离散量的离散灵敏度方案的收敛性，以最佳控制两阶段的纳维尔 - 斯托克斯方程。
We analyze optimal control problems for two-phase Navier-Stokes equations with surface tension. Based on $L_p$-maximal regularity of the underlying linear problem and recent well-posedness results of the problem for sufficiently small data we show the differentiability of the solution with respect to initial and distributed controls for appropriate spaces resulting form the $L_p$-maximal regularity setting. We consider first a formulation where the interface is transformed to a hyperplane. Then we deduce differentiability results for the solution in the physical coordinates. Finally, we state an equivalent Volume-of-Fluid type formulation and use the obtained differentiability results to derive rigorosly the corresponding sensitivity equations of the Volume-of-Fluid type formulation. For objective functionals involving the velocity field or the discontinuous pressure or phase indciator field we derive differentiability results with respect to controls and state formulas for the derivative. The results of the paper form an analytical foundation for stating optimality conditions, justifying the application of derivative based optimization methods and for studying the convergence of discrete sensitivity schemes based on Volume-of-Fluid discretizations for optimal control of two-phase Navier-Stokes equations.
