The next seminar “Modelling of materials – theory, model reduction and efficient numerical methods” will take place next Wednesday (March 10, 2020) from 9:00 till 10:00. The talk will be given by Michal Bathory. Please note that the seminar Modelling of materials takes place on Wednesday in this semester. Please see the details below.
Speaker: Michal Bathory
Title: A variational approach to incompressible fluids
Abstract: It is believed since the 19th century that the motion of a Newtonian fluid is truthfully described by the Navier-Stokes equations. Despite the apparent mathematical simplicity of the system, there are still many unsolved problems, including the Millennium problem of existence of smooth solutions and uniqueness of suitable weak solutions. Another famous problem is finding a “natural” boundary condition on an abstract interface, such as the outlet. One useful approach to both these problems is by looking for additional selection principles for weak solutions, such as the energy conservation. We construct Leray-Hopf weak solutions which, in addition, arise in the limit of minima of certain (WIDE) functionals, even in the presence of inflows and outflows. We also explicitly identify the (outflow) boundary conditions that are automatically satisfied by such solutions; for instance we are able to derive the boundary condition -(1/2) (v.v) n – p n + (D v) n = 0 that is of importance in numerical computations. Our conclusions remain valid for fluids with more complicated rheology, such as shear thickening or thinning fluids.