what is the meaning of Well-posedness of a mathematical model of a physical phenomena for example stokes equation in fluid dynamics ?
what is the necessity to prove that a model is well-posed?
how we can prove that a mathematical model is well-posed?
I don't know too much about mathematical physics but here is what we mean by saying a problem is well-posed.
A well-posed problem satisfies 3 conditions:
$(1)$: the solution exists
$(2)$: The solution is unique
$(3)$: The solution continuous depends on data
For prove, it is really depends on the problem you are studying... I don't know there is a general way of it...