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!

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!