On a nonlinear compactness lemma in L(0, T ;B)

where A is elliptic and B monotone (not strictly). It is the case, for example, in porous medium, semiconductor equations, ... In our application, we considered the injection moulding of a thermoplastic, with a mold of small thickness with respect to its other dimensions. By averaging Navier-Stokes equations across the thickness of the mold, and under an assumption (of Hele-Shaw) stating that the velocity eld is proportional to the pressure gradient, the pressure equation can be written as a doubly nonlinear equation [6]. Note that in this context, the equation can degenerate to an elliptic one. In order to get existence of a solution, one usually perform a time discretization, use some result on elliptic operator and pass to the limit as the time step goes to zero. In nonlinear problems compactness in time and space is then required. The compactness in space is easily obtained for u from a coerciveness assumption on the elliptic part A, but we have no estimate on ∂u ∂t since B could degenerate. Theorem 1 uses the space compactness of u and some time regularity on B(u) to derive a compactness for B(u), which in turn can be useful to pass to the limit in nonlinear terms of A (provided A has a an appropriate structure, e.g. B−pseudomonotone [5]).