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Renormalized solutions of some transport equations with partially \(W^{1,1}\) velocities and applications. (English) Zbl 1170.35364
Summary: We prove existence and uniqueness of renormalized solutions of some transport equations with a vector field that is not \(W^{1,1}\) with respect to all variables but is of a particular form. Two specific applications of this new result are then treated, based upon the equivalence between transport equations and ordinary differential equations. The first one consists of a result about the dependance upon initial conditions for solutions of ODEs. The second one is related to some stochastic differential equations arising in the modelling of polymeric fluid flows.

MSC:
35F10 Initial value problems for linear first-order PDEs
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
76A05 Non-Newtonian fluids
76M35 Stochastic analysis applied to problems in fluid mechanics
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