PD Dr. Christian Zillinger
- by appointment
- Kollegiengebäude Mathematik (20.30)
- 2.024
- +49 721 608 43035
- christian.zillinger@kit.edu
-
Englerstrasse 2
76131 Karlsruhe
Germany
You can find information about my project "Stability and Instability in Fluids and Materials" (AP6) in the CRC 1173 on its project page and here.
Semester | Titel | Typ |
---|---|---|
Winter Semester 2024/25 | Klassische Methoden für partiellen Differentialgleichungen | Lecture |
Winter Semester 2023/24 | Introduction to convex integration | Lecture |
Winter Semester 2021/22 | Introduction to Kinetic Equations | Lecture |
Winter Semester 2020/21 | Seminar: Microstructure in materials and fluid dynamics | Seminar |
Research
My research interests are in partial differential equations motivated by physical problems in particular fluid dynamics and material sciences and include:
- Mixing as a (de)stabilizing mechanism in fluids and inviscid damping.
- Cascades of resonances and instabilities in fluids and plasmas.
- Convex integration and microstructures in materials. Rigidity and flexibility.
- Magnetic fluids.
- Partial dissipation in the Boussinesq equations.
Publications
- (with Niklas Knobel) On Echoes in Magnetohydrodynamics with Magnetic Dissipation, Journal of Differential Equations, Volume 367, 15 September 2023, Pages 625-688, 2023.
- On stability estimates for the inviscid Boussinesq equations, Journal of Nonlinear Science, 33, Article number: 106 (2023), 2023.
- (with Xian Liao) On variable viscosity and enhanced dissipation, Nonlinearity, Volume 36, Number 11, 2023.
- On echo chains in the linearized Boussinesq equations around traveling waves, SIAM Journal on Mathematical Analysis, Vol. 55, Iss. 5 (2023), 2023.
- On the Boussinesq Equations with Non-monotone Temperature Profiles, Journal of Nonlinear Science volume 31, Article number: 64 (2021), 2021.
- (with Francesco Della Porta, Angkana Rüland, Jamie Taylor) On a probabilistic model for martensitic avalanches incorporating mechanical compatibility, Nonlinearity 34.7 (2021): 4844, 2021.
- On Enhanced Dissipation for the Boussinesq Equations, Journal of Differential Equations, Volume 282, Pages 407-445, 2021.
- On Echo Chains in Landau Damping: Traveling Wave-like Solutions and Gevrey 3 as a Linear Stability Threshold, Annals of PDE 7, 1 (2021), 2021.
- (with Yu Deng) Echo chains as a linear mechanism: Norm inflation, modified exponents and asymptotics, Archive for Rational Mechanics and Analysis 242(1), 643-700, 2021.
- (with Yu Deng) On the Smallness Condition in Linear Inviscid Damping:Monotonicity and Resonance Chains, Nonlinearity 33 6176, 2020.
- Linear Inviscid Damping in Sobolev and Gevrey Spaces, Nonlinear Analysis 213 (2021): 112492, 2021.
- (with Angkana Rüland, Barbara Zwicknagl) Higher Sobolev Regularity of Convex Integration Solutions in Elasticity: The Planar Geometrically Linearized Hexagonal-to-Rhombic Phase Transformation, Journal of Elasticity (2019): 1-76, 2019.
- (with Pierluigi Cesana, Francesco Della Porta, Angkana Rüland, Barbara Zwicknagl) Exact constructions in the (non-linear) planar theory of elasticity: From elastic crystals to nematic elastomers, Arch. Rational Mech. Anal. 237 (2020) 383–445, 2020.
- (with Michele Coti Zelati) On degenerate circular and shear flows: the point vortex and power law circular flows, Communications in Partial Differential Equations, 44:2,110-155, 2019.
- On the forced Euler and Navier-Stokes equations: linear damping and modified scattering, Journal of Mathematical Fluid Mechanics(2019) 21:49, 2019.
- On geometric and analytic mixing scales: comparability and convergence rates for transport problems, Pure and Applied Analysis 1.4 (2019): 543-570, 2019.
- (with Angkana Rüland, Jamie M Taylor) Convex integration arising in the modelling of shape-memory alloys: rigidity, flexibility and some numerical implementations, Journal of Nonlinear Science 29, p. 2137–2184 (2019), 2019.
- (with Angkana Rüland, Barbara Zwicknagl) Higher Sobolev Regularity of Convex Integration Solutions in Elasticity: The Dirichlet Problem with Affine Data in int(K^{lc), SIAM J. Math. Anal., 50(4), 3791–3841, 2018.
- On circular flows: linear stability and damping, J. Differential Equations 263, 7856–7899, 2017.
- Linear inviscid damping for monotone shear flows in a finite periodic channel, boundary effects, blow-up and critical Sobolev regularity, Archive for Rational Mechanics and Analysis, 221(3):1449-1509, 2016.
- Linear inviscid damping for monotone shear flows, Transactions of the AMS, 369, 8799–8855, 2016.
You can also find a list of my publications on the arxiv, in my google scholar profile and in my OrcId profile.
Preprints
- (with Roman Indergand, Dennis Kochmann, Angkana Rüland, Antonio Tribuzio) On a T3-Structure in Geometrically Linearized Elasticity: Qualitative and Quantitative Analysis and Numerical Simulations, arXiv preprint 2408.13110, 2024.
- On nonlinear Landau damping and Gevrey regularity, arXiv preprint 2307.14271, 2023.
- (with Niklas Knobel) On the Sobolev Stability Threshold for the 2D MHD Equations with Horizontal Magnetic Dissipation, arXiv preprint 2309.00496, 2023
Habilitation thesis
- On Mixing and Resonances in Fluid Systems, KIT, 2023
I obtained my PhD under the supervision of Herbert Koch at the University of Bonn and have been an assistant professor (NTT) at the University of Southern California and a postdoctoral fellow at BCAM.