scholarly journals Second Order Pressure Estimates for the Crank--Nicolson Discretization of the Incompressible Navier--Stokes Equations

2020 ◽  
Vol 58 (1) ◽  
pp. 375-409 ◽  
Author(s):  
Florian Sonner ◽  
Thomas Richter
Keyword(s):  
Stokes Equations ◽  
Second Order ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Crank Nicolson

Pontryagin maximum principle and second order optimality conditions for optimal control problems governed by 2D nonlocal Cahn–Hilliard–Navier–Stokes equations

Analysis ◽  
2020 ◽  
Vol 40 (3) ◽  
pp. 127-150
Author(s):  
Tania Biswas ◽  
Sheetal Dharmatti ◽  
Manil T. Mohan
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Stokes Equations ◽  
Minimum Principle ◽  
Immiscible Fluids ◽  
Second Order ◽  
Navier Stokes ◽  
Distributed Optimal Control ◽  
Navier Stokes Equations

AbstractIn this paper, we formulate a distributed optimal control problem related to the evolution of two isothermal, incompressible, immiscible fluids in a two-dimensional bounded domain. The distributed optimal control problem is framed as the minimization of a suitable cost functional subject to the controlled nonlocal Cahn–Hilliard–Navier–Stokes equations. We describe the first order necessary conditions of optimality via the Pontryagin minimum principle and prove second order necessary and sufficient conditions of optimality for the problem.

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