Talk by Josef Málek

The next seminar “Modelling of materials – theory, model reduction and efficient numerical methods” will take place next Wednesday (March 17, 2021) from 9:00 till 10:00 via Zoom (the same link as always). The talk will be given by Josef Málek. Please see the details below.

Speaker: Josef Málek

Title: Beyond the Navier-Stokes equations

Abstract: 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.