Talks

The presentations aims to give a rather short non-technical introduction in the general concept of a functional calculus including several examples and applications.

The presentation will (mostlikely) make use of the tool 'Zoom'. The audience will not have to do much but simply follow a link the speaker provides everybody with who asks in advance (jan.meichsner@tuhh.de). Members of the institute will get the link via the common email list.

Quite often, we wonder about the people around us but we are too shy to actually ask. On Thursday 27.02.2020, the Institute für Mathematik has organised a presentation about the one that is – up to the current date and not for very long – its latest “outstanding” acquisition.

During the presentation, you will finally be able to respond to the following questions:

- Why is he so fascinating?

- What was his last piece of work?

- What has he done during his first month?

These and any other question will be discussed during the meeting and who knows, maybe one day, in your closest cinema.

Moving-boundary flow simulations are an important design and analysis tool in many areas, including civil and biomedical engineering, as well as production engineering. Interface-capturing offers flexibility for complex free-surface motion, while interface-tracking is very attractive due to its mass conservation properties at low resolution. We focus on these alternatives in the context of flow simulations based on stabilized finite element discretizations of Navier-Stokes equations, including space-time formulations that allow extra flexibility concerning grid design at the interface.

Space-time approaches offer some not-yet-fully-exploited advantages; among them, the potential to allow some degree of unstructured space-time meshing. A method for generating simplex space-time meshes has been developed, allowing arbitrary temporal refinement in selected portions of space-time slabs. The method increases the flexibility of space-time discretizations, even in the absence of dedicated space-time mesh generation tools. The resulting tetrahedral and pentatope meshes are being used in the context of cavity filling flow simulations, such as those necessary to design injection molding processes.

We show the existence of solutions of nonlinear non-autonomous Cauchy problems
\[
\partial_t u(t,x) - \nabla_x \cdot (a(t,x)\nabla_xu(t,x))= f(t,x,u(t,x),\nabla u(t,x)),
\qquad u(0,\cdot)=u_0
\]
for a bounded open set $\Omega\subseteq \mathbb R^n$.
The coefficient matrix $a$ is supposed to be symmetric, uniformly elliptic,
Lipschitz continuous w.r.t.\ $t\in(0,\tau)$ and measurable w.r.t.\ $x\in\Omega$;
the nonlinearity $f$ is required to satisfy a linear growth condition.
We show that, given $u_0\in H_0^1(\Omega)$, there exists $u\in L_2(0,\tau;H_0^1(\Omega))
\cap H^1(0,\tau;L_2(\Omega))$ solving the problem mentioned above.

The proof relies on Schaefer's fixed point theorem. In the
course of the proof one uses maximal regularity properties of solutions of
inhomogeneous linear problems and compact embeddings of vector-valued Sobolev spaces.

The result partly generalises [ArCh10].

The talk is based on joint work with Wolfgang Arendt and Jürgen Voigt.


[ArCh10] W. Arendt, R. Chill: Global existence for quasilinear
diffusion equations in isotropic nondivergence form. Ann. Sc. Norm. Super. Pisa
Cl. Sci. (5) IX, 523-539 (2010).