Examples of truly abstract evolution PDEs?

Question

Let $V \subset H \subset V^*$. Consider the parabolic PDE $$y' = A(t)y + f$$ which is found in many books. Usually under some assumptions on $A(t)$ and $f$, there is a solution $y \in L^2(0,T;V)$ with $y' \in L^2(0,T;V^*)$.

In every example I've seen, $V$ and $H$ are chosen to be $H^k$ and $L^2$ respectively. Can someone give me an example of a different Hilbert triple that does not involve $L^2$ and a parabolic equation in that setting? Something completely wild would be interesting.

(I recently read this thread https://mathoverflow.net/questions/115825/abstract-evolution-equations and saw that the answers did not answer the question explicitly, so I ask it here. I read the answers and was not satisfied..)

Answer 1

You may want to look at Navier-Stokes Equations: Theory and Numerical Analysis by Temam chapter 3.

He defines the problem (3D NSE) to be to find $u$ such that

$$ u \in L^2(0,T;V), \, u^{\prime} \in L^1(0,T; V^*) $$

and then shows, in Thm. 3.3, that

$$ u \in L^{8/3}(0,T; L^4(\Omega)) \\ u^{\prime} \in L^{4/3}(0,T;V^*) $$ and in Thm. 3.4 that there is at most one solution in $$ u \in L^2(0,T;V) \cap L^{\infty}(0,T;H) \\ u \in L^8(0,T; L^4(\Omega)) $$

Answer 2

I think the fact that you can use various Sobolev spaces with various boundary conditions is already a good reason to call this approach abstract. But surely there are some examples beyond usual Sobolev spaces. Let me mention some that I know.

One can include various constraints into $H$ and $V$. For example, the Stokes' equations can be solved with $H$ and $V$ being the completions of the divergence free vector fields in the $L^2$ and $H^1$ norms, respectively.

If you consider the Hille-Yosida theorem as part of this approach, one can, for example, solve the heat equation in $C$, $C^\alpha$, or $L^p$.