We are interested in the thin-film equation with zero-contact angle and quadratic mobility, modeling the spreading of a thin liquid film, driven by capillarity and limited by viscosity in conjunction with a Navier-slip condition at the substrate. This degenerate fourthorder parabolic equation has the contact line as a free boundary. From the analysis of the self-similar source-type solution, one expects that the solution is smooth only as a function of two variables (x, x β ) (where x denotes the distance from the contact line) with β = √ 13−1 4 ≈ 0.6514 irrational. Therefore, the well-posedness theory is more subtle than in case of linear mobility (coming from Darcy dynamics) or in case of the second -order counterpart (the porous medium equation).Here, we prove global existence and uniqueness for one-dimensional initial data that are close to traveling waves. The main ingredients are maximal regularity estimates in weighted L 2 -spaces for the linearized evolution, after suitable subtraction of a(t) + b(t)x β -terms.

The Brownian motion of a hot nanoparticle is described by an effective Markov theory based on fluctuating hydrodynamics. Its predictions are scrutinized over a wide temperature range using large-scale molecular dynamics simulations of a hot nanoparticle in a Lennard-Jones fluid. The particle positions and momenta are found to be Boltzmann distributed according to distinct effective temperatures T HBM and T k . For T HBM we derive a formally exact theoretical prediction and establish a generalised Einstein relation that links it to directly measurable quantities. * * *We gratefully acknowledge helpful discussions with Jean-Louis Barrat (Grenoble), Ramin Golestanian (Oxford), and Markus Selmke (Leipzig), and thank Hugo Brandao for a careful reading of the manuscript. This work was supported by the Alexander von Humboldt foundation, the Deutsche Forschungsgemeinschaft (DFG) via FOR 877 and, within the German excellence initiative, p-5

We derive generalized Langevin equations for the translational and rotational motion of a heated Brownian particle from the fluctuating hydrodynamics of its nonisothermal solvent. The temperature gradient around the particle couples to the hydrodynamic modes excited by the particle itself so that the resulting noise spectrum is governed by a frequency-dependent temperature. We show how the effective temperatures at which the particle coordinates and (angular) velocities appear to be thermalized emerge from this central quantity.