publications

2024

1. ARMA

   Convergence Rates and Fluctuations for the Stokes–Brinkman Equations as Homogenization Limit in Perforated Domains

   Richard M. Höfer, and Jonas Jansen

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   We study the homogenization of the Dirichlet problem for the Stokes equations in \R^3 perforated by m spherical particles. We assume the positions and velocities of the particles to be identically and independently distributed random variables. In the critical regime, when the radii of the particles are of order m^-1, the homogenization limit u is given as the solution to the Brinkman equations. We provide optimal rates for the convergence u_m \to u in L^2, namely m^-β for all β< 1/2. Moreover, we consider the fluctuations. In the central limit scaling, we show that these converge to a Gaussian field, locally in L^2(\R^3), with an explicit covariance. Our analysis is based on explicit approximations for the solutions u_m in terms of u as well as the particle positions and their velocities. These are shown to be accurate in \dot H^1(\R^3) to order m^-β for all β< 1. Our results also apply to the analogous problem regarding the homogenization of the Poisson equations.

2. Nonlinearity

   Thermocapillary thin films: periodic steady states and film rupture

   Gabriele Bruell, Bastian Hilder, and Jonas Jansen

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   We study stationary, periodic solutions to the thermocapillary thin-film model which can be derived from the Bénard–Marangoni problem via a lubrication approximation. When the Marangoni number M increases beyond a critical value , the constant solution becomes spectrally unstable via a (conserved) long-wave instability and periodic stationary solutions bifurcate. For a fixed period, we find that these solutions lie on a global bifurcation curve of stationary, periodic solutions with a fixed wave number and mass. Furthermore, we show that the stationary periodic solutions on the global bifurcation branch converge to a weak stationary periodic solution which exhibits film rupture. The proofs rely on a Hamiltonian formulation of the stationary problem and the use of analytic global bifurcation theory. Finally, we show the instability of the bifurcating solutions close to the bifurcation point and give a formal derivation of the amplitude equation governing the dynamics close to the onset of instability.

3. arXiv

   Fast-moving pattern interfaces close to a Turing instability in an asymptotic model for the three-dimensional Bénard-Marangoni problem

   Bastian Hilder, and Jonas Jansen

   Publisher: arXiv

   Abs Link Supplementary Material

   We study the bifurcation of planar patterns and fast-moving pattern interfaces in an asymptotic long-wave model for the three-dimensional Bénard–Marangoni problem, which is close to a Turing instability. We derive the model from the full free-boundary Bénard–Marangoni problem for a thin liquid film on a heated substrate of low thermal conductivity via a lubrication approximation. This yields a quasilinear, fully coupled, mixed-order degenerate-parabolic system for the film height and temperature. As the Marangoni number M increases beyond a critical value M^*, the pure conduction state destabilises via a Turing(–Hopf) instability. Close to this critical value, we formally derive a system of amplitude equations which govern the slow modulation dynamics of square or hexagonal patterns. Using center manifold theory, we then study the bifurcation of square and hexagonal planar patterns. Finally, we construct planar fast-moving modulating travelling front solutions that model the transition between two planar patterns. The proof uses a spatial dynamics formulation and a center manifold reduction to a finite-dimensional invariant manifold, where modulating fronts appear as heteroclinic orbits. These modulating fronts facilitate a possible mechanism for pattern formation, as previously observed in experiments.

2023

1. arXiv

   Non-Newtonian thin-film equations: global existence of solutions, gradient-flow structure and guaranteed lift-off

   Peter Gladbach, Jonas Jansen, and Christina Lienstromberg

   Publisher: arXiv

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   We study the gradient-flow structure of a non-Newtonian thin film equation with power-law rheology. The equation is quasilinear, of fourth order and doubly-degenerate parabolic. By adding a singular potential to the natural Dirichlet energy, we introduce a modified version of the thin-film equation. Then, we set up a minimising-movement scheme that converges to global positive weak solutions to the modified problem. These solutions satisfy an energy-dissipation equality and follow a gradient flow. In the limit of a vanishing singularity of the potential, we obtain global non-negative weak solutions to the power-law thin-film equation }begin{equation*} }partial_t u + }partial_x}bigl(m(u) \textbar}partial_x^3 u - G^{}prime}prime}(u) }partial_x u\textbar^{α-1} }bigl(}partial_x^3 u - G^{}prime}prime}(u) }partial_x u}bigr)}bigr) = 0 }end{equation*} with potential \G in the shear-thinning (\α> 1\), Newtonian (\α= 1\) and shear-thickening case (\0 <α< 1\). The latter satisfy an energy-dissipation inequality. Finally, we derive dissipation bounds in the case \G}equiv 0 which imply that solutions emerging from initial values with low energy lift up uniformly in finite time.

2. SIMA

   Long-Time Behavior and Stability for Quasilinear Doubly Degenerate Parabolic Equations of Higher Order

   Jonas Jansen, Christina Lienstromberg, and Katerina Nik

   Publisher: Society for Industrial and Applied Mathematics

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   We deal with a system of partial differential equations describing a steady motion of an incompressible fluid with shear-dependent viscosity and present a new global existence result for 𝑝\textgreater2𝑑𝑑+2p\textgreater2dd+2 p\textgreater}frac{2d}{d+2} . Here p is the coercivity parameter of the nonlinear elliptic operator related to the stress tensor and d is the dimension of the space. Lipschitz test functions, a subtle splitting of the level sets of the maximal functions for the velocity gradients, and a decomposition of the pressure are incorporated to obtain almost everywhere convergence of the velocity gradients.

2022