# Examples of truly abstract evolution PDEs?

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.

• Does spaces related with $\ell^2$ are good for you? Jul 1, 2013 at 20:28
• @Tomás I guess so Jul 2, 2013 at 20:03
• So I am gonna post a answer with those kind of spaces @michael_faber, what do you think? Jul 2, 2013 at 20:06
• @Tomás Yes I would be interested in that. I don't think this question will receive an another answer to be honest so you should post it! Jul 2, 2013 at 20:09
• @michael_faber, I dont think my answer can be of any help. It does not have any pratical application. In favt, my idea was to take Brezis example of a Hilbert triple and then consider any differential equations in this space. You can find Brezis's example in his book of PDE in the chapter of Hilbert spaces. Jul 7, 2013 at 15:37

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 $L^p$.

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 $$u \in L^2(0,T;V) \cap L^{\infty}(0,T;H) \\ u \in L^8(0,T; L^4(\Omega))$$

• For NS equations H is still $L^2$ though. I haven't read that book, but I think the fact that you can prove that the solution lies also in some other crazy $L^p$ space is a consequence of the settings $V=H^1$ and $H=L^2$. Jul 6, 2013 at 17:47