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I encountered a physics problem, which requires solving two (coupled) PDEs:

The dynamic variables are the vector field $v_i$ and the tensor field $\varepsilon_{ij}$, which is defined by a vector field $u_i$: $\varepsilon_{ij}=\frac{1}{2}(\partial_iu_j+\partial_ju_i)$. I assume that all fields are just functions of one spatial coordinates and one for time.

The dynamic equation for $v_i$ is:

$\dot{v}_i+\partial_{j}\sigma_{ij}=0.$

The dynamic equation for $\varepsilon_{ij}$ is

$\dot{\varepsilon}_{ij}-A_{ij}+\varepsilon_{ij}/\tau-\partial_k\partial_k\varepsilon_{ij}=0,$

with $\tau$ a positive constant.

The tensor field $\sigma_{ij}$ is a function (in general a nonlinear) of $\varepsilon_{ij}$ and $A_{ij}=\frac{1}{2}(\partial_iv_j+\partial_jv_i).$

For the variable $v_i$ the boundary condition reads: $v_i=0$ at the boundary. For the variable $\varepsilon_{ij}$ I do not know what kind of a boundary condition to use. I have a clearly defined boundary condition of $u_i$ through which $\varepsilon_{ij}$ is defined. I want $u_i=0$ at $z=0$ and $u_i=u_0 \delta_{ix}$ at $z=d$ (d is the length of the system).

Say that my initial condition is $v_i(z,t=0)=0$ and $u_i(z,t=0)=u_0/d~\delta_{ix} z$. Then I can calculate $\varepsilon_{ij}$ and perform one time step with a numerical method. Then I will have $\varepsilon_{ij}(t=\Delta t)$. My question is how to keep the boundary conditions for $u_i$ fixed (dynamically). Apart from determining the initial $\varepsilon_{ij}$, the field $u_i$ seems redundant, which should not be.

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