The tempered disk and the tempered cohomology

Consider a non-archimedean valuation ring V (K its fraction field, in mixed characteristic): inspired by some views presented by Scholze, we introduce a new point of view on the non-archimedean analytic setting in terms of derived analytic geometry (then associating a "spectrum" to each ind-Banach algebra). We want to look at the behaviour of this spectrum from a differential point of view. In such a spectrum, for example, there exist open subsets having functions with log-growth as sections for the structural sheaf. In this framework, a transfer theorem for the log-growth of solutions of p-adic differential equations can be interpreted as a continuity theorem (analogue to the transfer theorem for their radii of convergence in the Berkovich spaces). As a dividend of such a theory, we define a new cohomology theory in terms of the Hodge-completed derived de Rham cohomology of the ind-Banach derived analytic space associated to a smooth k-scheme, X_k (k residual field of V), via the use of "tempered tubes". We finally compare our tempered de Rham cohomology with crystalline cohomology.