I am dealing with fluid-structure interaction problem and I have a pde subjected to a constraint .
I must use Lagrange multipliers but I don’t know how. Please, any one give a simple example for how to apply Lagrange multiplier to pde. So I can deal with my problem.
$\Omega $ is the fluid region and $B(t)$ is region occupied by a particle.$u$ is the velocity of the fluid ,$F$ is the external force and $\rho$ is the fluid density constant.p is the pressure and $\mu$ is the viscosity of the fluid $$\rho (\frac{\partial u}{\partial t} + (u \cdot \nabla ))= F + \nabla \cdot \sigma \qquad \text{in} \quad \Omega \backslash B(t)$$ $$\nabla \cdot u=0 \qquad \text{in} \quad \Omega $$ $$\nabla \cdot \sigma = \frac{1}{2} \mu \nabla \cdot(\nabla u+\nabla u^t) -\nabla p$$
We also have the constraint defined in $B(t)$ $$u(x,t)= V(t)+w(t) \times (x-G(t))$$ Where $V$ is the translation velocity of the particle, $G$ is its center of mass and $w$ is the angular velocity.
Note that I am dealing with fictitious domain so we will integrate over the entire domain $\Omega$ in $\textbf{2 dimensions}$
$$u(x,0)=u_0 \qquad \text{in} \quad \Omega \backslash B(t) $$ and $u=0 $ on the boundary