Solvability of a moving contact-line problem with interface formation for an incompressible viscous fluid

We investigate the free-boundary problem of a steadily advancing meniscus in a circular capillary tube. The problem is described using the “interface formation model,” which was originally introduced with the aim of avoiding the singularities that arise when classical hydrodynamics is applied to problems with a moving contact line. We prove the existence of an axially symmetric solution in weighted Hölder spaces for low meniscus speeds.

Properties of integrals which have the type of derivatives of volume potentials for one ultraparabolic arbitrary order equation

In weighted Hölder spaces it is studied the smoothness of integrals, which have the structure and properties of derivatives of volume potentials which generated by fundamental solutions of the Cauchy problem for one ultraparabolic arbitrary order equation of the Kolmogorov type. The coefficients in this equation depend only on the time variable. Special distances and norms are used for constructing of the weighted Hölder spaces. The results of the paper can be used for establishing of the correct solvability of the Cauchy problem and estimates of solutions of the given non-homogeneous equation in corresponding weighted Hölder spaces.

Mean curvature self-shrinkers of high genus: Non-compact examples

We give the first rigorous construction of complete, embedded self-shrinking hypersurfaces under mean curvature flow, since Angenent’s torus in 1989. The surfaces exist for any sufficiently large prescribed genus g, and are non-compact with one end. Each has {4g+4} symmetries and comes from desingularizing the intersection of the plane and sphere through a great circle, a configuration with very high symmetry. Each is at infinity asymptotic to the cone in {\mathbb{R}^{3}} over a {2\pi/(g+1)} -periodic graph on an equator of the unit sphere {\mathbb{S}^{2}\subseteq\mathbb{R}^{3}} , with the shape of a periodically “wobbling sheet”. This is a dramatic instability phenomenon, with changes of asymptotics that break much more symmetry than seen in minimal surface constructions. The core of the proof is a detailed understanding of the linearized problem in a setting with severely unbounded geometry, leading to special PDEs of Ornstein–Uhlenbeck type with fast growth on coefficients of the gradient terms. This involves identifying new, adequate weighted Hölder spaces of asymptotically conical functions in which the operators invert, via a Liouville-type result with precise asymptotics.

On a viscous two-fluid channel flow including evaporation

In this contribution a particular plane steady-state channel flow including evaporation effects is investigated from analytical point of view. The channel is assumed to be horizontal. The motion of two heavy viscous immiscible fluids is governed by a free boundary value problem for a coupled system of Navier-Stokes and Stephan equations. The flow domain is unbounded in two directions and the free interface separating partially both liquids is semi-infinite, i.e. infinite in one direction. The free interface begins in some point Q where the half-line Σ1 separating the two parts of the channel in front of Q ends. Existence and uniqueness of a suitable solution in weighted HÖLDER spaces can be proved for small data (i.e. small fluxes) of the problem.