We kindly invite you to the Miklós Farkas Seminar.
17 October (Thursday) 10.15, BME, H607
László Székelyhidi (Universität Leipzig)
Renormalized and non-renormalized solutions of the transport equation
The linear transport equation is the possibly simplest of all PDE, describing the evolution of a density under the flow of a given vectorfield.
It is well-known that, as long as the vectorfield is Lipschitz continuous, solutions of the linear transport equation are closely related, via the method of characteristics, to the Lagrangian flow map generated by the vectorfield. However, this link breaks down if the vectorfield is merely Sobolev, since in this case the ODE does not make sense classically. Of course, formulation of the PDE poses no problems even without differentiability assumptions. On the other hand there are numerous applications in fluid mechanics and kinetic theory, where the transport equation appears with a vectorfield which is in some Sobolev space, possibly even continuous, but not Lipschitz. For such cases DiPerna and Lions developed in the late 1980s a theory of renormalization, leading to a well-posed solution concept for both the PDE and the ODE. In the talk we discuss the limits of this theory and present examples showing that unless certain additional integrability conditions are imposed, the theory of renormalization does not lead to a unique solution.
(István Faragó, János Karátson, Róbert Horváth, Miklós Mincsovics, Gabriella Svantnerné Sebestyén)
Visit the homepage of the seminar: http://math.bme.hu/AlkAnalSzemi