Obviously solutions in $C^k$ are nicer but it seems people are happy with obtaining weak solutions in some Sobolev space that only satisfy the weak formulation. Why should this, in the real world, be seen as a "solution"? After all (unless you are lucky enough to be in the right dimension to apply an embedding theorem), you can't say that when $u \in L^2(0,T;H^1)$ that
$u(x_1,t_1)$ is the temperature of a fuel tank at $x_1$ at time $t_1$
So it is useless right?