Associated Project 6 • 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

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 \begin{align*}\partial_t v + v \cdot \nabla v - \nabla p & = 0, \\ \text{div}(v) & = 0,\end{align*} 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

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. \begin{align*}\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.\end{align*} 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

"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.

Publications

1. X. Liao and C. Zillinger. On variable viscosity and enhanced dissipation. Nonlinearity, 36(11):6071–6103, November 2023. URL https://doi.org/10.1088/1361-6544/acfec0. [preprint] [bibtex]

2. C. Zillinger. On echo chains in the linearized Boussinesq equations around traveling waves. SIAM J. Math. Anal., 55(5):5127–5188, September 2023. URL https://doi.org/10.1137/21M1458053. [preprint] [bibtex]

3. N. Knobel and C. Zillinger. On echoes in magnetohydrodynamics with magnetic dissipation. J. Differential Equations, 367:625–688, September 2023. URL https://doi.org/10.1016/j.jde.2023.05.020. [preprint] [bibtex]

4. C. Zillinger. On stability estimates for the inviscid Boussinesq equations. J. Nonlinear Sci., 33(6):106, 38, September 2023. URL https://doi.org/10.1007/s00332-023-09958-2. [preprint] [bibtex]

5. C. Zillinger. Linear inviscid damping in Sobolev and Gevrey spaces. Nonlinear Anal., 213:112492, December 2021. URL https://doi.org/10.1016/j.na.2021.112492. [preprint] [bibtex]

6. Y. Deng and C. Zillinger. Echo chains as a linear mechanism: norm inflation, modified exponents and asymptotics. Arch. Rational Mech. Anal., 242(1):643–700, October 2021. URL https://doi.org/10.1007/s00205-021-01697-6. [preprint] [bibtex]

7. F. Della Porta, A. Rüland, J. Taylor, and C. Zillinger. On a probabilistic model for martensitic avalanches incorporating mechanical compatibility. Nonlinearity, 34(7):4844–4896, June 2021. URL https://doi.org/10.1088/1361-6544/abfca9. [preprint] [bibtex]

8. C. Zillinger. On enhanced dissipation for the Boussinesq equations. J. Differential Equations, 282:407–445, May 2021. URL https://doi.org/10.1016/j.jde.2021.02.029. [preprint] [bibtex]

9. C. Zillinger. On the Boussinesq equations with non-monotone temperature profiles. J. Nonlinear Sci., 31(4):64, 38, May 2021. URL https://doi.org/10.1007/s00332-021-09723-3. [preprint] [bibtex]

10. C. Zillinger. On echo chains in Landau damping: traveling wave-like solutions and Gevrey 3 as a linear stability threshold. Ann. PDE, 7(1):1–29, January 2021. URL https://doi.org/10.1007/s40818-020-00090-y. [preprint] [bibtex]

11. Y. Deng and C. Zillinger. On the smallness condition in linear inviscid damping: monotonicity and resonance chains. Nonlinearity, 33(11):6176–6194, October 2020. URL https://doi.org/10.1088/1361-6544/aba236. [preprint] [bibtex]

12. P. Cesana, F. Della Porta, A. Rüland, C. Zillinger, and B. Zwicknagl. Exact constructions in the (non-linear) planar theory of elasticity: from elastic crystals to nematic elastomers. Arch. Ration. Mech. Anal., 237(1):383–445, July 2020. URL https://doi.org/10.1007/s00205-020-01511-9. [preprint] [bibtex]

Preprints

13. N. Knobel and C. Zillinger. On the Sobolev stability threshold for the 2D MHD equations with horizontal magnetic dissipation. CRC 1173 Preprint 2023/21, Karlsruhe Institute of Technology, September 2023. [bibtex]

14. C. Zillinger. On nonlinear Landau damping and Gevrey regularity. CRC 1173 Preprint 2023/20, Karlsruhe Institute of Technology, September 2023. [bibtex]