# Mathematisches Kolloquium

Am Donnerstag, 8. Mai 2014, um 16:30 Uhr spricht

## Josef Malek (Universität Prag)

im HS A027 über das Thema

## Implicitly constituted fluids and solids: modeling and PDE analysis

Zusammenfassung: Starting from the seminal work of Rajagopal (2003), it has been observed that the framework of implicitly constituted materials provides an appropriate theoretical basis for derivation/justi cation of many models that have been used, mostly successfully, in various areas (engineering, chemistry, food processing) but that have been proposed in an adhoc manner. It concerns both fluid and solid mechanics. This new framework generates models and their settings that are different from the classical ones bringing thus challenges to the mathematical theory of nonlinear partial differential equations (PDE). In the first part of the talk, after motivating the structure of the implicit constitutive relations and introducing the framework, I will provide several examples of implicitly constituted models. Then we focus on mathematical properties of the flows and deformation of such implictly constituted materials. This will bring us to the area of PDE analysis of relevant initial and boundary value problems. After mentioning several concepts of solutions and their importance (from my personal viewpoint) I will focus on two (or perhaps three) examples to show what questions one may ask and study and how they are related to or differ from the classical theories. The elastic solids with limiting linearized strain will serve as the first example. In these models the stress can increase unboundedly while the strain yet remains small. We shall discuss mathematical properties of the solution corresponding to the anti-plane stress weak solution to the anti-plane stress problem in V-notch (non-convex) domains. Although the equations governing the deformation that are being considered share certain similarities with the minimal surface problem, the boundary conditions and the presence of an additional model parameter that appears in the equation and its specific range makes the problem, as well as the result, different from those associated with the minimal surface problem. The second example will bring us to the analysis flows of incompressible fluids with activation criteria (such as Bingham and Herschel?Bulkley fluids) subject to stick-slip boundary conditions. We will underline the advantages of new settings.
Alle Interessierten sind hiermit herzlich eingeladen. Eine halbe Stunde vor dem Vortrag gibt es Kaffee und Tee im Sozialraum (Raum 448) im 4. Stock.
Treffpunkt zum Abendessen um 18.00 Uhr wird noch bekannt gegeben.