$Webrelaunch 2020$

Department of Mathematics

Home Page

Institutes

Institute for Analysis

Junior Research Group Stability and Instability in Fluids and Materials

In our research we consider physically motivated problems in partial differential equations, which combine physical effects and interesting mathematical phenomena. Here a particular emphasis is on mixing, resonances and asymptotic stability in fluid systems with and without dissipation as well as rigidity and flexibility of convex integration in shape-memory alloys.

Mixing as a (de)stabilizing effect

$Illustration of mixing by a circular flow$

One of the major questions of fluid dynamics lies in the description of stability of fluids and their asymptotic behavior.
This analysis is of great interest due to the equations' mathematically challenging and intricate properties and also in view of their physical applications.

Here a central model is given by the incompressible Euler equations

$\partial_t v + v \cdot \nabla v - \nabla p = 0,$
$\text{div}(v)=0,$

where $v$ denotes the velocity of the fluid and $p$ is the pressure. Unlike the Navier-Stokes or Boltzmann equations these equations do not include common damping mechanisms such as dissipation or entropy increase. On the contrary the equations possess infinitely many conservation laws and are even time reversible.
Yet considering perturbations are shear flows $v=(U(y),0)$ one observes that small perturbations converge back to a shear flow as time tends to infinity, a phenomenon known as inviscid damping.
Mathematically this behavior can be described in terms of weak convergence and results in beneficial damping but also in deteriorating regularity due to the appearance of smaller and smaller mixing scales.
Moreover, the interaction and "un-mixing" of wave-like perturbations can lead to non-linear instability of the dynamics.

Resonance cascades in coupled fluid systems

$Resonances of two traveling waves$ $Crosssection for fixed y$

New effects arise in coupled fluid systems such as the Boussinesq equations with partial dissipation, magnetohydrodynamics or the Navier-Stokes equations with variable viscosity. In all these systems there is an interaction of different, competing physical effects. A prototypical example is the presence of Rayleigh-Benard instabilities in the Boussinesq equations which couple thermal diffusion to the evolution of a fluid.
$\partial_t v + v \cdot \nabla v - \nabla p = \theta e_2 + \nu \Delta v,$
$\partial_t \theta + v \cdot \nabla \theta = \kappa \Delta \theta,$
$\text{div}(v)=0.$
Here there is a competition between the destabilizing effects of buoyancy on the one hand, and mixing-enhanced dissipation on the other hand.

As one of the main results of our recent research we show that non-linear instability mechanisms can be accurately captured in terms of the interaction of special non-linear, global in time solutions called traveling waves. In particular, this descriptions allows us to study the competition of these (de)stabilizing effects and identify associated time scales, norm inflation mechanisms and optimal spaces to prove regularity. This complex interaction of physical effects in coupled systems is also at the core of ongoing PhD work on resonances in magnetohydrodynamics within this research group.

Convex integration in materials

$Rigidity and flexibility$

Shape-memory alloys are metal alloys which undergo a first-order diffusionless solid-solid phase transformation, passing from a highly symmetric high temperature phase (austenite) to a low temperature phase (martensite) with a less symmetric unit cell structure. This loss of symmetry gives rise to multiple variants of martensite. Hence, at low temperature deformations are energetically comparatively inexpensive, as different variants of martensite can accommodate macroscopic changes in shape. However, upon heating the material, all these variants are forced back into the austenite phase and thus the material recovers its shape. The material has a memory.

Mathematically, we describe these materials in terms of a variational model due to Ball-James. As this problem is highly non-convex the existence and description of minimizers is a very challenging problem.
Focusing on exactly stress-free states, we search for deformations
$u: \Omega \rightarrow \mathbb{R}^n$
which solve the differential inclusion problem
$\nabla u \in K.$
Here we observe that at relatively high regularity the equations are rigid: all solutions can only exhibit certain patterns.
In contrast, at lower regularity the equations are flexible and allow for infinitely many wild solutions with highly complex microstructures.
A main aim of our research is to develop a better understanding of this dichotomy between rigidity and flexibility and to construct wild solutions at higher Sobolev regularity.

Staff in the Junior Research Group
Name	Tel.	E-Mail
M.Sc. Niklas Knobel	0049 721 608 46197	niklas.knobel@kit.edu
PD Dr. Christian Zillinger	+49 721 608 43035	christian.zillinger@kit.edu

Current Offering of Courses
Semester	Titel	Typ
Winter Semester 2024/25	Klassische Methoden für partiellen Differentialgleichungen	Lecture
Winter Semester 2023/24	Introduction to convex integration	Lecture
Winter Semester 2022/23	Introduction to convex integration	Lecture
Summer Semester 2022	Selected methods in fluids and kinetic equations	Lecture
Winter Semester 2021/22	Introduction to Kinetic Equations	Lecture
Winter Semester 2020/21	Seminar: Microstructure in materials and fluid dynamics	Seminar