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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 asymptotically self-similar as time tends to infinity. Our results are built upon the global analytic solutions constructed by Koch and Lamm cite{KochLamm}, the compactness arguments adapted by Asai and Giga cite{Giga2014}, and the spatial equi-decay properties on certain weighted function spaces. The proof for all of the above flows are achieved in a unified framework by utilizing the estimates of the linearized operator.
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We consider the nonlinear heat equation $u_t = Delta u + |u|^alpha u$ with $alpha >0$, either on ${mathbb R}^N $, $Nge 1$, or on a bounded domain with Dirichlet boundary conditions. We prove that in the Sobolev subcritical case $(N-2) alpha <4$, for every $mu in {mathbb R}$, if the initial value $u_0$ satisfies $u_0 (x) = mu |x-x_0|^{-frac {2} {alpha }}$ in a neighborhood of some $x_0in Omega $ and is bounded outside that neighborhood, then there exist infinitely many solutions of the heat equation with the initial condition $u(0)= u_0$. The proof uses a fixed-point argument to construct perturbations of self-similar solutions with initial value $mu |x-x_0|^{-frac {2} {alpha }}$ on ${mathbb R}^N $. Moreover, if $mu ge mu _0$ for a certain $ mu _0( N, alpha )ge 0$, and $u_0 Ige 0$, then there is no nonnegative local solution of the heat equation with the initial condition $u(0)= u_0$, but there are infinitely many sign-changing solutions.
We show the existence of self-similar solutions for the Muskat equation. These solutions are parameterized by $0<s ll 1$; they are exact corners of slope $s$ at $t=0$ and become smooth in $x$ for $t>0$.
This paper addresses the construction and the stability of self-similar solutions to the isentropic compressible Euler equations. These solutions model a gas that implodes isotropically, ending in a singularity formation in finite time. The existence of smooth solutions that vanish at infinity and do not have vacuum regions was recently proved and, in this paper, we provide the first construction of such smooth profiles, the first characterization of their spectrum of radial perturbations as well as some endpoints of unstable directions. Numerical simulations of the Euler equations provide evidence that one of these endpoints is a shock formation that happens before the singularity at the origin, showing that the implosion process is unstable.
We will give a new proof of a recent result of P.~Daskalopoulos, G.Huisken and J.R.King ([DH] and reference [7] of [DH]) on the existence of self-similar solution of the inverse mean curvature flow which is the graph of a radially symmetric solution in $mathbb{R}^n$, $nge 2$, of the form $u(x,t)=e^{lambda t}f(e^{-lambda t} x)$ for any constants $lambda>frac{1}{n-1}$ and $mu<0$ such that $f(0)=mu$. More precisely we will give a new proof of the existence of a unique radially symmetric solution $f$ of the equation $mbox{div},left(frac{ abla f}{sqrt{1+| abla f|^2}} right)=frac{1}{lambda}cdotfrac{sqrt{1+| abla f|^2}}{xcdot abla f-f}$ in $mathbb{R}^n$, $f(0)=mu$, for any $lambda>frac{1}{n-1}$ and $mu<0$, which satisfies $f_r(r)>0$, $f_{rr}(r)>0$ and $rf_r(r)>f(r)$ for all $r>0$. We will also prove that $lim_{rtoinfty}frac{rf_r(r)}{f(r)}=frac{lambda (n-1)}{lambda (n-1)-1}$.
We construct forward self-similar solutions (expanders) for the compressible Navier-Stokes equations. Some of these self-similar solutions are smooth, while others exhibit a singularity do to cavitation at the origin.
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