arXiv subjects

Hans Knuepfer

Publications and source records attributed to Hans Knuepfer.

4 recordsLinked to original sources

arXiv2025

Solutions of the thin film equation obtained in the limit of vanishing slip

We analyze the evolution of thin liquid droplets in the lubrication approximation with different slip conditions at the liquid-solid interface. Motivated by the classical no-slip paradox which states that the Navier-Stokes equations with a no-slip boundary condition require unphysical infinite dissipation during droplet spreading, we focus on the limit of vanishing slip. We show that in the no-slip limit three fundamentally different classes of limiting solutions are approached, each of them corresponding to a different scaling of the microscopic contact angle as the regularization parameter vanishes. These findings suggest that the thin-film equation with no slip supports a rich family of physically admissible solutions, provided one interprets the no-slip thin film equation as the asymptotic limit of models which regularized slip conditions. Even though the large apparent contact angles in some of these solutions seem incompatible with the lubrication approximation, a refined analysis shows that the underlying physical variables remain consistent with the assumptions for the lubrication approximation.

math.AP↗

arXiv2015

Low density phases in a uniformly charged liquid

This paper is concerned with the macroscopic behavior of global energy minimizers in the three-dimensional sharp interface unscreened Ohta-Kawasaki model of diblock copolymer melts. This model is also referred to as the nuclear liquid drop model in the studies of the structure of highly compressed nuclear matter found in the crust of neutron stars, and, more broadly, is a paradigm for energy-driven pattern forming systems in which spatial order arises as a result of the competition of short-range attractive and long-range repulsive forces. Here we investigate the large volume behavior of minimizers in the low volume fraction regime, in which one expects the formation of a periodic lattice of small droplets of the minority phase in a sea of the majority phase. Under periodic boundary conditions, we prove that the considered energy $Γ$-converges to an energy functional of the limit "homogenized" measure associated with the minority phase consisting of a local linear term and a non-local quadratic term mediated by the Coulomb kernel. As a consequence, asymptotically the mass of the minority phase in a minimizer spreads uniformly across the domain. Similarly, the energy spreads uniformly across the domain as well, with the limit energy density minimizing the energy of a single droplets per unit volume. Finally, we prove that in the macroscopic limit the connected components of the minimizers have volumes and diameters that are bounded above and below by universal constants, and that most of them converge to the minimizers of the energy divided by volume for the whole space problem.

math-ph↗

arXiv2012

On an isoperimetric problem with a competing non-local term. II. The general case

This paper is the continuation of [H. Knüpfer and C. B. Muratov, Commun. Pure Appl. Math. (2012, to be published)]. We investigate the classical isoperimetric problem modified by an addition of a non-local repulsive term generated by a kernel given by an inverse power of the distance. In this work, we treat the case of general space dimension. We obtain basic existence results for minimizers with sufficiently small masses. For certain ranges of the exponent in the kernel we also obtain non-existence results for sufficiently large masses, as well as a characterization of minimizers as balls for sufficiently small masses and low spatial dimensionality. The physically important special case of three space dimensions and Coulombic repulsion is included in all the results mentioned above. In particular, our work yields a negative answer to the question if stable atomic nuclei at arbitrarily high atomic numbers can exist in the framework of the classical liquid drop model of nuclear matter. In all cases the minimal energy scales linearly with mass for large masses, even if the infimum of energy may not be attained.

math.AP↗

arXiv2011

On an isoperimetric problem with a competing non-local term. I. The planar case

This paper is concerned with a study of the classical isoperimetric problem modified by an addition of a non-local repulsive term. We characterize existence, non-existence and radial symmetry of the minimizers as a function of mass in the situation where the non-local term is generated by a kernel given by an inverse power of the distance. We prove that minimizers of this problem exist for sufficiently small masses and are given by disks with prescribed mass below a certain threshold, when the interfacial term in the energy is dominant. At the same time, we prove that minimizers fail to exist for sufficiently large masses due to the tendency of the low energy configuration to split into smaller pieces when the non-local term in the energy is dominant. In the latter regime, we also establish linear scaling of energy with mass suggesting that for large masses low energy configurations consist of many roughly equal size pieces far apart. In the case of slowly decaying kernels we give a complete characterization of the minimizers.

math.AP↗