PARTIAL DIFFERENTIAL INCLUSIONS OF TRANSPORT TYPE WITH STATE CONSTRAINTS

摘要

The focus is on the existence of weak solutions to the quasilinear first-order partial differential inclusionpartial derivative t f is an element of -div(x) (g (t, f) f) + u(t, f) . f + W(t, f)with values in L-P(R-N) for p is an element of (1, infinity). The solution is to satisfy state constraints in addition, i.e., all its values belong to a given set nu subset of L-P(R-N) of constraints. We specify sufficient conditions such that every function in nu initializes at least one weak solution with all its values in nu (so-called weak invariance a.k.a. viability of nu). Due to the regularity assumptions about the set-valued coefficient mappings, these solutions prove to be renormalized (in the sense of DI PERNA and LIONS).