Conference Agenda

Thursday, 18/Mar/2021:
4:30pm - 5:30pm

Session Chair: Helmut Abels
Location: W003

External Resource:

Beyond the Navier-Stokes equations

J. Málek

In the first half of the 19th century Navier and Stokes
formulated the equations that describe the flow of water and many other
incompressible liquids under standard conditions. These equations now bear
the names of their inventors. A century later Leray developed the
mathematical foundations of the modern theory of the Navier-Stokes
equations both for planar and three-dimensional flows. He introduced the
concept of generalized solution to the Cauchy problem and proved its
existence for arbitrary (sufficiently regular) data and for an arbitrary
time interval. This concept not only reflects the physical assumptions
used when deriving the equations but it also forms the basis for the
construction of powerful numerical methods.
Despite the undeniable success of the Navier-Stokes equations, there are
many fluid-like incompressible materials that exhibit phenomena that can
not be described by the Navier-Stokes equations. In order to describe
these effects a number of macroscopic models, which are more complicated
than the Navier-Stokes equations, have been designed, developed, and used
in relevant applications. The aim of this lecture is to survey recent
developments, both in the area of theoretical continuum thermodynamics as
well as in the field of PDE analysis, which have led to the development of
Leray's programme beyond the Navier-Stokes equations. In particular,
results concerning viscous fluids in the presence of activation,
incorporated into the framework of implicitly constituted fluids,
and results concerning the existence of large-data weak solutions to
viscoelastic rate-type fluid flow models, without or with stress diffusion,
will be highlighted.