I'm studying the PDE $$ \frac{\partial u}{\partial t} + A_x\frac{\partial u}{\partial x} + A_y\frac{\partial u}{\partial y} + A_z\frac{\partial u}{\partial z} = 0 $$ with $A_x, A_y, A_z$ being constant matrices, that arise either from the acoustic or linear elasticity system. The equation is solved in some complicated 2- or 3D domain (for example, with cavities or sharp edges), with boundary conditions of the type $Bu = f$. These boundary conditions may represent the condition of given normal stress on the boundary for the linear elasticity or given normal velocity on the boundary for acoustics.
I would like to know, how to check if the problem is well-posed. In particular, I need to know
- The properties of the domain, i.e. if it is simpy connected, or if the boundary is piecewise smooth, etc. The papers considering the well-posedness for this system that I've found so far consider $u(t)$ to be defined either on the set with $C^\infty$ boundary (for example, this one). Maybe there is a generalisation to the case of piecewise smooth boundary?
- The number of boundary conditions or rank of B matrices. My intuition tells me that this sholud depend on the postiton of the characteristic surfaces on the boundary, but I haven't found any useful information yet.
