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| Boundary Shape Control of the Navier-Stokes Equations and Applications |
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Citation： |
Kaitai LI,Jian SU,Aixiang HUANG.Boundary Shape Control of the Navier-Stokes Equations and Applications[J].Chinese Annals of Mathematics B,2010,31(6):879~920 |
| Page view： 1735
Net amount： 1148 |
Authors： |
Kaitai LI; Jian SU; Aixiang HUANG; |
Foundation： |
the National High-Tech Research and Development Program of China (No.
2009AA01A135), the National Natural Science Foundation of China (Nos. 10926080, 10971165, 10871156)
and Xian Jiaotong University (No. XJJ2008033). |
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| Abstract： |
In this paper, the geometrical design for the blade’s surface ? in an impeller
or for the profile of an aircraft, is modeled from the mathematical point of view by a
boundary shape control problem for the Navier-Stokes equations. The objective function
is the sum of a global dissipative function and the power of the fluid. The control variables
are the geometry of the boundary and the state equations are the Navier-Stokes equations.
The Euler-Lagrange equations of the optimal control problem are derived, which are an
elliptic boundary value system of fourth order, coupled with the Navier-Stokes equations.
The authors also prove the existence of the solution of the optimal control problem, the
existence of the solution of the Navier-Stokes equations with mixed boundary conditions,
the weak continuity of the solution of the Navier-Stokes equations with respect to the
geometry shape of the blade’s surface and the existence of solutions of the equations for
the G?ateaux derivative of the solution of the Navier-Stokes equations with respect to the
geometry of the boundary. |
Keywords： |
Blade, Boundary shape control, General minimal surface, Navier-Stokes
equations, Euler-Lagrange equations |
Classification： |
65N30, 76U05, 76M05 |
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