Renormalized solutions for a class of nonlinear evolution problems

Nonlinear parabolic problem with lower order terms in Musielak-Orlicz spaces

“…As a consequence, each term in the right hand side of (59) is bounded either in W −1,x L M (Q T ) or in L 1 (Q T ) which shows that (57) holds true. Arguing again as in [22] estimates (56), (57) and the following remark (1), we can show (53) and (54).…”

“…Démonstration: : Proof of (53) and (54): Proceeding as in [22], we have for any S ∈ W 2,∞ (IR), such that S , has a compact support…”

Renormalized Solutions for Nonlinear Degenerate Elliptic Problems with L¹ Data

“…As already mentioned in the introduction Problem (1), (2) does not admit a weak solution under assumptions (5)- (9). Indeed, as the growth of a(x, u, ∇u) is not controlled with respect to u, the field a(x, u, ∇u) is not, in general, defined as a distribution.…”

“…This notion was introduced by P.-L. Lions and Di Perna [12] for the study of the Boltzmann equation (see also P.-L. Lions [15] for a few applications to fluid mechanics models). This notion was then adapted to the elliptic version of (1) and (2) in Boccardo, J.-L. Diaz, D. Giachetti, F. Murat [11], in P.-L. Lions and F. Murat [16], and F. Murat [16,17] (see also [8,9] for nonlinear parabolic problems). At the same time the equivalent notion of entropy solutions have been developed independently by Bénilan and al.…”

Renormalized solutions for a non-uniformly parabolic equation

“…The notion of renormalized solutions was first introduced by DiPerna and Lions [20] for the study of Boltamann equation. It was then adapted to the study of some nonlinear elliptic and parabolic problems and evolution problems in fluid mechanics ( [3,4,5,9,15,26]). We hope that the renormalized solution is still existent and unique, and it is equivalent to the entropy solution of problem (1.1).…”