We prove the existence of self-similar expanding solutions of the curvature flow on planar networks where the initial configuration is any number of half-lines meeting at the origin. This generalizes recent work by Schnürer and Schulze which treats the case of three half-lines. There are multiple solutions, and these are parametrized by combinatorial objects, namely Steiner trees with respect t...

The e ect of salt concentration Cs on the critical shear rate γ̇c required for the onset of shear thickening and apparent relaxation time λ of the shear-thickened phase, has been investigated systematically for dilute CTAB/NaSal solutions. Experimental data suggest a self-similar behavior of γ̇c and λ as functions of Cs. Speci cally, γ̇c ∼ Cs whereas λ∼ Cs such that an e ective Weissenberg numberW...

Self-similar reduction of an important class of two-dimensional conservation laws leads to boundary value problems for equations which change type. We have established a method for solving free boundary problems for quasilinear degenerate elliptic equations which arise when shocks interact with the subsonic (nonhyperbolic) part of the solution. This paper summarizes the principal features of th...

Simple, self-similar, elliptic solutions of non-relativistic fireball hydrodynamics are presented, generalizing earlier results for spherically symmetric fireballs with Hubble flows and homogeneous temperature profiles. The transition from one dimensional to three dimensional expansions is investigated in an efficient manner. Introduction. Recently, a lot of experimental and theoretical efforts...

We numerically solve a problem for the unsteady transonic small disturbance equations that describes the diffraction of a weak shock into an expansion wave. In the context of a shock moving into a semi-infinite wall, this problem describes the interaction between the reflected part of the shock and the part that is transmitted beyond the wall. We formulate the equations in self-similar variable...

We investigate the self-similar solutions to the isothermal compressible Navier–Stokes equations. The aim of this paper is to show that there exist neither forward nor backward self-similar solutions with finite total energy. This generalizes the results for the incompressible case in Nečas, J., Ru̇žička, M. & Šverák, V. (1996, On Leray’s self-similar solutions of the Navier-Stokes equations. Ac...

‎The concept of self-similarity on subsets of algebraic varieties‎ ‎is defined by considering algebraic endomorphisms of the variety‎ ‎as `similarity' maps‎. ‎Self-similar fractals are subsets of algebraic varieties‎ ‎which can be written as a finite and disjoint union of‎ ‎`similar' copies‎. ‎Fractals provide a framework in which‎, ‎one can‎ ‎unite some results and conjectures in Diophantine g...