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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..)

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  • $\begingroup$ Does spaces related with $\ell^2$ are good for you? $\endgroup$
    – Tomás
    Jul 1, 2013 at 20:28
  • $\begingroup$ @Tomás I guess so $\endgroup$ Jul 2, 2013 at 20:03
  • $\begingroup$ So I am gonna post a answer with those kind of spaces @michael_faber, what do you think? $\endgroup$
    – Tomás
    Jul 2, 2013 at 20:06
  • $\begingroup$ @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! $\endgroup$ Jul 2, 2013 at 20:09
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    $\begingroup$ @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. $\endgroup$
    – Tomás
    Jul 7, 2013 at 15:37

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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 $L^p$.

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

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  • $\begingroup$ 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 $H=L^2$. $\endgroup$
    – bartgol
    Jul 6, 2013 at 17:47

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