**NOTE:** I also asked this question [here in MSE][3].


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][1] 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][2] 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!


  [1]: https://en.wikipedia.org/wiki/Aubin%E2%80%93Lions_lemma
  [2]: https://link.springer.com/book/10.1007/978-3-0348-0513-1

  [3]: https://math.stackexchange.com/questions/4945846/bochner-sobolev-spaces-with-second-time-derivative-and-embeddings