The Role of Symmetry and Separation in Surface Evolution and Curve Shortening

Abstract

Abstract. With few exceptions, known explicit solutions of the curve shortening flow (CSE) of a plane curve, can be constructed by classical Lie point symmetry reductions or by functional separation of variables. One of the functionally separated solutions is the exact curve shortening flow of a closed, convex "oval"-shaped curve and another is the smoothing of an initial periodic curve that is close to a square wave. The types of anisotropic evaporation coefficient are found for which the evaporation-condensation evolution does or does not have solutions that are analogous to the basic solutions of the CSE, namely the grim reaper travelling wave, the homothetic shrinking closed curve and the homothetically expanding grain boundary groove. Using equivalence classes of anisotropic diffusion equations, it is shown that physical models of evaporation-condensation must have a diffusivity function that decreases as the inverse square of large slope. Some exact separated solutions are constructed for physically consistent anisotropic diffusion equations.

Enhanced group classification of nonlinear diffusion–reaction equations with gradient-dependent diffusivity

“…Then one acts by the operators ∂t, ∂x and ∂ũ on the equation (6) and the operators ∂t, ∂x and ∂ũx on the equation (7). The determining equations derived in this way are appropriate in order to solve the problem of describing the equivalence groupoid G ∼ R of the class R. This problem is reduced to the classification of admissible transformations, which is similar to but more complicated than the classification of Lie symmetries in Sections 3 and 5 below.…”

“…We use the other way, which gives more compact determining equations for equivalence transformations. We solve the equations (6) and (7) with respect to f and g, respectively,…”

“…Each equivalence transformation of any subclass of the class R is completely defined by its the t-, x-and u-components. Indeed, if these components are known, then the u x -component is computed by the chain rule, and the expressions for f -and g-components follow from the equations (6) and (7). This is why we do not discuss the derivation of the latter expressions below.…”

“…Lie reductions of the latter equations were studied in [30]. Exact solutions of both nonlinear diffusion equations and nonlinear filtration equations were found using Lie and quasilocal (potential) symmetries [1,2], generalized separation of variables [7,11], nonclassical reductions and nonclassical potential reductions [5,15,42] and generalized conditional symmetries [43]. See also [3,Section 10.2], [34,Chapter 5] and references therein.…”

The fundamental solutions of the curve shortening problem via the Schwarz function

“…[5,6]. Curve shortening (1), and its variants, also has practical application; for example in the late-time evolution of Hele-Shaw free boundary flow in the presence of surface tension [3]. In the time reversed sense when the curve lengthens, the solutions have relevance to viscous fingering and crystal growth e.g.…”