Bastian Hilder - Research

Research: Multiscale dynamics, pattern formation and nonlinear waves

My main goal is to understand multiscale dynamics in pattern-forming systems and stochastic processes using techniques from nonlinear analysis, dynamical systems and partial differential equations with a focus on
- existence and stability of nonlinear waves and coherent structures, and
- reduction methods for high-dimensional multiscale systems.
I want to understand mathematical structures leading to observable phenomena and to develop techniques to study models derived from first principles
My research is motivated by wave phenomena in fluid dynamics, in particular in water waves, temperature-driven thin films, active fluids and turbulent flows

Keywords: partial differential equations; dynamical systems; pattern formation; travelling waves; nonlinear stability; reduction methods; center manifold reduction; spatial dynamics; coarse-graining

Current projects

Pattern formation in advective systems. Collaborator: Christian Kuehn
Global bifurcation of three-dimensional water waves with vorticity. Collaborators: Erik Wahlén, Giang To
Dimension breaking bifurcations in the three-dimensional water wave problem. Collaborators: Erik Wahlén, Jörg Weber
Instabilities and bifurcations in thermocapillary thin-film equations. Collaborators: Gabriele Brüll, Stefano Böhmer, Jonas Jansen, Evgeniy Lokharu
Nonlinear waves in periodic media. Collaborators: Xin Meng, Guido Schneider
Existence and stability of monostable front cascades. Collaborator: Louis Garénaux
Non-equilibrium functional inequalities for Markov processes with applications to coarse-graining. Collaborators: Patrick van Meurs, Upanshu Sharma

Preprints and Publications

Preprints

[PP3] Bastian Hilder, Christian Kuehn. Pattern formation and nonlinear waves close to a 1:1 resonant Turing and Turing–Hopf instability. submitted, 2025 [arXiv][GitHub repository]
[PP2] Stefano Böhmer, Bastian Hilder, Jonas Jansen. Pattern formation and film rupture in a two-dimensional thermocapillary thin-film model of the Bénard–Marangoni problem. submitted, 2025 [arXiv][GitHub repository]
[PP1] Louis Garénaux, Bastian Hilder. Linear convective stability of a front superposition with unstable connecting state. submitted, 2025. [arXiv][GitHub repository]

Publications

[J11] Bastian Hilder, Jonas Jansen. Fast-moving pattern interfaces close to a Turing instability in an asymptotic model for the three-dimensional Bénard–Marangoni problem. J. Nonlinear Sci., 35:105, 2025. [DOI] (open access) [arXiv] [GitHub repository]
[J10] Wolf-Patrick Düll, Bastian Hilder, Guido Schneider. Analysis of the embedded cell method in 2D for the numerical homogenization of metal-ceramic composite materials. J. Appl. Anal., 2024. [DOI] [arXiv]
[J9] Gabiele Brüll, Bastian Hilder, Jonas Jansen. Thermocapillary Thin Films: Periodic Steady States and Film Rupture. Nonlinearity,37(4):045016, 2024. [DOI] (open access) [arXiv] [GitHub repository]
[J8] Bastian Hilder, Upanshu Sharma. Quantitative coarse-graining of Markov chains. SIAM J. Math. Anal., 56(1):913-954, 2024. [DOI] [arXiv]
[J7] Bastian Hilder, Björn de Rijk, Guido Schneider. Moving modulating pulse and front solutions of permanent form in a FPU model with nearest and next-to-nearest neighbor interaction. SIAM J. Appl. Dyn. Syst., 22(2):1076-1113, 2023. [DOI] [arXiv]
[J6] Bastian Hilder, Björn de Rijk, Guido Schneider. Nonlinear stability of periodic roll solutions in the real Ginzburg-Landau equation against $C^m_{\mathrm{ub}}$-perturbations. Comm. Math. Phys., 400(1):277-314, 2023. [DOI] (open access) [arXiv]
[J5] Bastian Hilder. Modulating traveling fronts in a dispersive Swift-Hohenberg equation coupled to an additional conservation law. J. Math. Anal. Appl., 513(2), 2022. [DOI] [arXiv]
[J4] Bastian Hilder. Nonlinear stability of fast invading fronts in a Ginzburg-Landau equation with an additional conservation law. Nonlinearity, 34(8):5538--5575, 2021. [DOI]
[J3] Bastian Hilder. Modulating traveling fronts for the Swift-Hohenberg equation in the case of an additional conservation law. J. Differential Equations, 269(5):4353--4380, 2020. [DOI] [arXiv]
[J2] Bastian Hilder, Mark A. Peletier, Upanshu Sharma, Oliver Tse. An inequality connecting entropy distance, Fisher Information and large deviations. Stochastic Process. Appl., 130(5):2596--2638, 2020. [DOI] (open access)
[J1] Wolf-Patrick Düll, Bastian Hilder, Guido Schneider. Analysis of the embedded cell method in 1D for the numerical homogenization of metal-ceramic composite materials. J. Appl. Anal., 24(1):71--80, 2018. [DOI] [arXiv]

Thesis

Bastian Hilder. Invasion phenomena in pattern-forming systems admitting a conservation law structure. PhD thesis. University of Stuttgart, 2021. [pdf]
Bastian Hilder. An FIR inequality for Markov jump processes on discrete state spaces. Master's thesis. Eindhoven University of Technology / University of Stuttgart, 2017. [pdf]
Bastian Hilder. Analysis of the embedded cell method and of the underlying model equations in one and two dimensions. Bachelor's thesis, University of Stuttgart, 2015.