Do Initial Conditions for PDEs Need to Satisfy Governing Equations?

[pavanakumar]

I would really like to know whether initial conditions given to a time evolution PDE has to satisfy the governing equations. For example, if I have to solve numerically an incompressible flow equation do I need to give initial solution for the velocity field which is divergence free so as to obey the governing equation.

thanks in advance

 

[Chris Hillman]

It sounds like you are asking about systems of PDEs which decompose into evolution equations plus constraint equations. If so, yes, the constraint equations place constraints on the initial values, and once you find a "legal" solution to those on some slice such as $t=0$, you use the evolution equations to obtain the complete solution. For example, you can consider Maxwell's source-free field equations to be a pair of evolution equations
$$E_t = \nabla \times \vec{B}, \; \; B_t = -\nabla \times \vec{E}$$
plus a pair of constraint equations
$$\nabla \cdot \vec{E} = 0, \; \; \nabla \cdot \vec{B} = 0$$
So initial data on the slice $t=0$ consists of two incompressible spatial vector fields $\vec{E}, \, \vec{B}$.

 

[pavanakumar]

Thanks chris,
So my initial conditions must be mathematically consistent with the governing eqn.

I have one more query. If I want to give some sort of forcing in say E = f(t) (t -> time) along some boundary curve C. Then for each time I have no guarantee that the fields are divergence free. In this case too I must make the function f(t) divergence free. But how ? as I don't have prior knowledge of the fields at every time t (beforehand).