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.


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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