论文标题
本地Cahn-Hilliard-Navier-Stokes系统的粘度解决方案的存在和独特性
Existence and Uniqueness of viscosity solutions of Value function of Local Cahn-Hilliard-Navier-Stokes system
论文作者
论文摘要
在这项工作中，我们考虑了局部的Cahn-Hilliard-Navier-Stokes方程，并具有二维有限域的定期潜力。我们将分布式最佳控制问题提出，因为最小化适当的成本功能，可应对受控的本地Cahn-Hilliard-navier- stokes系统，并定义关联的值函数。我们证明了值函数满足的动态编程原理。由于缺乏值函数的平滑度特性，我们使用粘度解的方法来获得无限尺寸汉密尔顿 - 雅各布 - 雅各比 - 贝尔曼方程的相应溶液。我们表明该值函数是汉密尔顿 - 雅各比 - 贝尔曼方程的唯一粘度解。粘度解的独特性是通过比较原理确定的。
In this work, we consider the local Cahn-Hilliard-Navier-Stokes equation with regular potential in two dimensional bounded domain. We formulate distributed optimal control problem as the minimization of a suitable cost functional subject to the controlled local Cahn-Hilliard-Navier- Stokes system and define the associated value function. We prove the Dynamic Programming Principle satisfied by the value function. Due to the lack of smoothness properties for the value function, we use the method of viscosity solutions to obtain the corresponding solution of the infinite dimensional Hamilton-Jacobi-Bellman equation. We show that the value function is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation. The uniqueness of the viscosity solution is established via comparison principle.
