State-constrained optimal control of the three-dimensional stationary Navier-Stokes equations

In this paper, an optimal control problem for the stationary Navier-Stokes equations in the presence of state constraints is investigated. Existence of optimal solutions is proved and first order necessary conditions are derived. The regularity of the adjoint state and the state constraint multiplier is also studied. Lipschitz stability of the optimal control, state and adjoint variables with respect to perturbations is proved and a second order sufficient optimality condition for the case of pointwise state constraints is stated.     

Pressure representation and boundary regularity of the Navier-Stokes equations with slip boundary condition

We first represent the pressure in terms of the velocity in . Using this representation we prove that a solution to the Navier-Stokes equations is in under the critical assumption that , with r?3, while for r=3 the smallness is required. In [H.J. Choe, Boundary regularity of weak solutions of the Navier-Stokes equations, J. Differential Equations 149 (2) (1998) 211-247], a boundary L^∞ estimate for the solution is derived if the pressure on the boundary is bounded. In our work, we remove the boundedness assumption of the pressure. Here, our estimate is local. Indeed, employing Moser type iteration and the reverse Hölder inequality, we find an integral estimate for L^∞-norm of u.     

Regularity criteria for suitable weak solutions of the Navier-Stokes equations near the boundary

We present some new regularity criteria for “suitable weak solutions” of the Navier-Stokes equations near the boundary in dimension three. We prove that suitable weak solutions are Hölder continuous up to the boundary provided that the scaled mixed norm with 3/p+2/q?2, 2<q?∞, (p,q)≠(3/2,∞) is small near the boundary. Our methods yield new results in the interior case as well. Partial regularity of weak solutions is also analyzed under some additional integral conditions.     

一个新的边界层方程及其在形状控制中的应用

本文给出固壁边界上(即一个二维流形上) 的流体速度梯度和压力的二阶偏微分方程, 从而也给出边界上法向应力, 以及流体中运动物体所受的阻力和升力的计算公式. 本方法的创新在于边界上法向速度梯度不是通过在边界层内速度梯度的数值微分达到, 而是通过它与其他变量一起作为一组偏微分方程的解而得到, 证明边界层方程组的适定性问题, 并且给出解关于边界形状的Gâteaux 导数所满足的偏微分方程. 本文将本方法应用于飞机外形的形状最优控制, 给出阻力泛函关于形状第一变分的可计算形式. 数值例子表明, 用本方法得到的阻力精度比通用程序得到要高.     

Local regularity of the Navier-Stokes equations near the curved boundary

We present some regularity conditions for suitable weak solutions of the Navier-Stokes equations near the curved boundary of a sufficiently smooth domain. Our extend the work that was results established near a flat boundary by Gustafson, Kang and Tsai (2006) [6].     

The boundary integral method for the linearized rotating Navier-Stokes equations in exterior domain

In this paper, we apply the boundary integral method to the linearized rotating Navier-Stokes equations in exterior domain. Introducing some open ball which decomposes the exterior domain into a finite domain and an infinite domain, we obtain a coupled problem by the linearized rotating Navier-Stokes equations in finite domain and a boundary integral equation without using the artificial boundary condition. For the coupled problem, we show the existence and uniqueness of solution. Finally, we study the finite element approximation for the coupled problem and obtain the error estimate between the solution of the coupled problem and its approximation solution.     

Analysis of an optimal control problem for the three-dimensional coupled modified Navier-Stokes and Maxwell equations

The mathematical formulation and analysis of an optimal control problem associated with a viscous, incompressible, electrically conducting fluid in a bounded three-dimensional domain with fixed perfectly conducting boundaries is considered. The objective of control is the matching of the velocity and magnetic fields to given target fields; control is effected through distributed mechanical force and current controls. The existence of optimal solutions is shown, the Gâteaux differentiability for the magnetohydrodynamic system with respect to controls is proved, and the optimality system is obtained.     

A trace regularity result for thermoelastic equations with application to optimal boundary control

We consider a mixed problem for a Kirchoff thermoelastic plate model with clamped boundary conditions. We establish a sharp regularity result for the outer normal derivative of the thermal velocity on the boundary. The proof, based upon interpolation techniques, benefits from the exceptional regularity of traces of solutions to the elastic Kirchoff equation. This result, which complements recent results obtained by the second and third authors, is critical in the study of optimal control problems associated with the thermoelastic system when subject to thermal boundary control. Indeed, the present regularity estimate can be interpreted as a suitable control-theoretic property of the corresponding abstract dynamics, which is crucial to guarantee well-posedness for the associated differential Riccati equations.     

On the rotating Navier-Stokes equations with mixed boundary conditions

The stationary and nonstationary rotating Navier-Stokes equations with mixed boundary conditions are investigated in this paper. The existence and uniqueness of the solutions are obtained by the Galerkin approximation method. Next, θ-scheme of operator splitting algorithm is applied to rotating Navier-Stokes equations and two subproblems are derived. Finally, the computational algorithms for these subproblems are provided.     

Regularity for the Navier-Stokes equations with slip boundary condition

For the Navier-Stokes equations with slip boundary conditions, we obtain the pressure in terms of the velocity. Based on the representation, we consider the relationship in the sense of regularity between the Navier-Stokes equations in the whole space and those in the half space with slip boundary data.

     

A two-grid algorithm based on Newton iteration for the stream function form of the Navier-Stokes equations

In this paper, we propose a two-grid algorithm for solving the stream function formulation of the stationary Navier-Stokes equations. The algorithm is constructed by reducing the original system to one small, nonlinear system on the coarse mesh space and two similar linear systems (with same stiffness matrix but different right-hand side) on the fine mesh space. The convergence analysis and error estimation of the algorithm are given for the case of conforming elements. Furthermore, the algorithm produces a numerical solution with the optimal asymptotic H ²-error. Finally, we give a numerical illustration to demonstrate the effectiveness of the two-grid algorithm for solving the Navier-Stokes equations.     

Stability of boundary layer and rarefaction wave to an outflow problem for compressible Navier-Stokes equations under large perturbation

In this paper, we investigate the large-time behavior of solutions to an outflow problem for compressible Navier-Stokes equations. In 2003, Kawashima, Nishibata and Zhu [S. Kawashima, S. Nishibata, P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys. 240 (2003) 483-500] showed there exists a boundary layer (i.e., stationary solution) to the outflow problem and the boundary layer is nonlinearly stable under small initial perturbation. In the present paper, we show that not only the boundary layer above but also the superposition of a boundary layer and a rarefaction wave are stable under large initial perturbation. The proofs are given by an elementary energy method.     

A regularization method for the numerical solution of elliptic boundary control problems with pointwise state constraints

A Lavrentiev type regularization technique for solving elliptic boundary control problems with pointwise state constraints is considered. The main concept behind this regularization is to look for controls in the range of the adjoint control-to-state mapping. After investigating the analysis of the method, a semismooth Newton method based on the optimality conditions is presented. The theoretical results are confirmed by numerical tests. Moreover, they are validated by comparing the regularization technique with standard numerical codes based on the discretize-then-optimize concept. The authors acknowledge support through DFG Research Center “Mathematics for Key Technologies” (FZT 86) in Berlin.     

Boundary Shape Control of the Navier-Stokes Equations and Applications

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.     

Multigrid Solution of a Lavrentiev-Regularized State-Constrained Parabolic Control Problem

A mesh-independent, robust, and accurate multigrid scheme to solve a linear state-constrained parabolic optimal control problem is presented. We first consider a Lavrentiev regularization of the state-constrained optimization problem. Then, a multigrid scheme is designed for the numerical solution of the regularized optimality system. Central to this scheme is the construction of an iterative pointwise smoother which can be formulated as a local semismooth Newton iteration. Results of numerical experiments and theoretical two-grid local Fourier analysis estimates demonstrate that the proposed scheme is able to solve parabolic state-constrained optimality systems with textbook multigrid efficiency.     

Local existence of solution to free boundary value problem for compressible Navier-Stokes equations

This paper is concerned with the free boundary value problem for multi-dimensional Navier-Stokes equations with density-dependent viscosity where the flow density vanishes continuously across the free boundary. Local (in time) existence of a weak solution is established; in particular, the density is positive and the solution is regular away from the free boundary.     

Ill-posedness of the Navier-Stokes equations in a critical space in 3D

We prove that the Cauchy problem for the three-dimensional Navier-Stokes equations is ill-posed in in the sense that a “norm inflation” happens in finite time. More precisely, we show that initial data in the Schwartz class S that are arbitrarily small in can produce solutions arbitrarily large in after an arbitrarily short time. Such a result implies that the solution map itself is discontinuous in at the origin.     

Robin-type boundary control problems for the nonlinear Boussinesq type equations

In this paper, we study a linear and a nonlinear boundary control problems arising from viscous flows. The equations are of nonlinear Navier-Stokes type for the velocity and pressure, of transport-diffusion type for the temperature and the salinity. The essential difficulties are due to the nonlinear nature of a part of the boundary conditions and to the nature of the equations: time-dependent, coupled and nonlinear. The existence and the conditions of the uniqueness of the solution, for the variational problem, are studied. The control is of linear or nonlinear Robin-type and acts on a part of the boundary during a time T. The cost function measures the distance between the observed and the computed vorticity. The existence of an optimal control in the admissible set of states and controls is proved. A first order necessary conditions of optimality are obtained.     

Free boundary value problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity

We study the free boundary value problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient in this paper. Under certain assumptions imposed on the initial data, we show that there exists a unique global strong solution, the interface separating the flow and vacuum state propagates along particle path and expands outwards at an algebraic time-rate, the flow density is strictly positive from blow for any finite time and decays pointwise to zero also at an algebraic time-rate as the time tends to infinity.     

On the regularity criteria of the 3D Navier-Stokes equations in critical spaces

Regularity criteria of Leray-Hopf weak solutions to the three-dimensional Navier-Stokes equations in some critical spaces such as Lorentz space, Morrey space and multiplier space are derived in terms of two partial derivatives, 1 u 1 , 2 u 2 , of velocity fields.