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NOTE: I also asked this question here in MSE.

In the weak theory of evolution PDEs, the Bochner-Sobolev spaces are frequently used. For $a,b \in \mathbb{R}$ and $X,Y$ banach spaces, we define these spaces as $$W^{1,p,q}(a,b,X,Y) = \left\{v, \ \text{ such that } \ \ v \in L^p(a,b;X) \ \text{ and } \ \frac{d}{dt} v \in L^q(a,b;Y) \right\}.$$

The Aubin-Lions lemma then gives compact embedding results given certain conditions are met for $X, Y$ and $p,q$.

My question is what about such spaces with higher time derivative order? For example we can define $$W^{2,p,q,l}(a,b,X,Y,Z) = \left\{v, \ \text{ such that } \ \ v \in L^p(a,b;X) \ \text{ and } \ \frac{d}{dt} v \in L^q(a,b;Y) \ \text{ and } \ \frac{d^2}{dt^2} v \in L^l(a,b;Z) \right\}.$$

This definition is given in Roubíček's Nonlinear PDEs With Applications for $p= \infty$ (formula (7.4)) but I cannot find any embedding results mentioned there.

Even more general, what about space for time derivatives of order $k$?

Any references would be appreciated!

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    $\begingroup$ It is a bit unclear to me what kind of result you expect or whether you want to pose different assumptions on the relation of $X,Y$ and $Z$. (Obviously anything that applies to the first-order space also aplies to the second-order space.) $\endgroup$
    – Hannes
    Commented Jul 15 at 7:08
  • $\begingroup$ @Hannes What I had in mind is probably a better regularity in time for the space we embed in. For example, the first order space embed compactly in $C([0,T], W)$ where $W$ is a space between $X$ and $Y$ with given assumptions. For the second order space and since we have control over the second time derivative, would an embedding in $C^1([0,T], W)$ be valid? $\endgroup$
    – MathsGoose
    Commented Jul 15 at 8:38
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    $\begingroup$ That depends on the constellation with the Banach spaces, but you can of course apply the classical Aubin-Lions type results to the time derivative in the second order space setting to go for a continuously differentiable function space in time. $\endgroup$
    – Hannes
    Commented Jul 15 at 9:48
  • $\begingroup$ @Hannes Yes, you are right! Now I wonder whether $W$ should be between $X$ and $Y$ or $Y$ and $Z$ and what kind of embeddings does it have with its surrounding spaces? Because if the embeddings are compact, some special cases such as $W=Y$ will not be true. $\endgroup$
    – MathsGoose
    Commented Jul 15 at 10:11
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    $\begingroup$ Well a priori only between $Y$ and $Z$ in your notation. There is a very comprehensive literature on Aubin-Lions type results so I recommend to start digging there for possible setups. For example in the seminal paper on Compact sets in the space $L^p(B)$ by Simon. $\endgroup$
    – Hannes
    Commented Jul 16 at 10:03

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