Related papers: Remarks on graph-like forward self-similar solutio…
This paper studies Mullins' model of thermal grooving which consists of a surface diffusion flow equation with contact angle and no-flux boundary conditions. We consider this problem in a multi-dimensional half space and prove that if the…
We give a classification of all self-similar solutions to the curve shortening flow in the plane.
We consider four different models of nonlinear diffusion equations involving fractional Laplacians and study the existence and properties of classes of self-similar solutions. Such solutions are an important tool in developing the general…
In this article we classify solitons (equilibria, self-similar solutions and travelling waves) for the surface diffusion flow of entire graphs of function over the real line.
In this paper we prove short-time existence of a smooth solution in the plane to the surface diffusion equation with an elastic term and without an additional curvature regularization. We also prove the asymptotic stability of strictly…
We study the forward self-similar solutions to the $2$D hypodissipative Navier-Stokes equation with fractional diffusion $(-\Delta)^\alpha$ for $\frac{1}{2}<\alpha<1$. We first show that for arbitrarily large $(1-2\alpha)$-homogeneous…
We are interested in an anisotropic singular diffusion equation in the plane and in its regularization. We establish existence, uniqueness and basic regularity of solutions to both equations. We construct explicit solutions showing the…
This paper deals with the long-time behavior of solutions of nonlinear reaction-diffusion equations describing formation of morphogen gradients, the concentration fields of molecules acting as spatial regulators of cell differentiation in…
We consider a scalar parabolic equation in one spatial dimension. The equation is constituted by a convective term, a reaction term with one or two equilibria, and a positive diffusivity which can however vanish. We prove the existence and…
We study the Mullins' problem that was proposed by Mullins in 1957 and is one of the models of the thermal grooving by surface diffusion. Mathematically, this is the problem of evolving curves in the half space that is governed by the…
We show that self-similar solutions for the mean curvature flow, surface diffusion and Willmore flow of entire graphs are stable upon perturbations of initial data with small Lipschitz norm. Roughly speaking, the perturbed solutions are…
We consider a system of two reaction-diffusion-advection equations describing the one dimensional directed motion of particles with superimposed diffusion and mutual alignment. For this system we show the existence of traveling wave…
In this paper, we consider affine self-similar solutions for the affine curve shortening flow in the Euclidean plane. We obtain the equations of all affine self-similar solutions up to affine transformations and solve the equations or give…
We study the existence of particular traveling wave solutions of a nonlinear parabolic degenerate diffusion equation with a shear flow. Under some assumptions we prove that such solutions exist at least for propagation speeds c {\in}]c*,…
We survey the various constructions of forward self-similar solutions (and generalizations of self-similar solutions) to the Navier-Stokes equations. We also include and prove an extension of a recent result from [7].
We demonstrate the equivalence between the two popular models of nonlinear diffusion, the porous medium equation and the p-Laplacian equation. The equivalence is shown at the level of selfsimilar solutions.
In the description of transport phenomena an important aspect represents the diffusion. In certain cases the diffusion may appear together with convection. In this paper we study the diffusion equation with the self similar Ansatz. With an…
We advance an exact, explicit form for the solutions to the fractional diffusion-advection equation. Numerical analysis of this equation shows that its solutions resemble power-laws.
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 Schn\"urer and Schulze…
Extending the work of Jia and Sverak on self-similar solutions of the Navier-Stokes equations, we show the existence of large, forward, discretely self-similar solutions.