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On melting for the 3D radial Stefan problem

Chencheng Zhang

Abstract

We consider the three-dimensional radial Stefan problem which describes the evolution of a radial symmetric ice ball with free boundary \begin{equation*} \left\{\begin{aligned} &\partial_{t}u-\partial_{rr}u-\frac{2}{r}\partial_{r}u=0 \quad in\ r\geqλ(t),\\ &\partial_{r}u(t,λ(t))=-\dotλ(t),\\ &u(t,λ(t))=0,\\ &u(0,\cdot)=u_{0},\quad λ(0)=λ_{0}. \end{aligned}\right. \end{equation*} We prove the existence in the radial class of finite time melting with rates \begin{equation*} λ(t)=\left\{\begin{aligned} &4\sqrtπ\frac{\sqrt{T-t}}{|\log (T-t)|}(1+o_{t\rightarrow T}(1)),\\ &c(u_{0},k)(1+o_{t\rightarrow T}(1))(T-t)^{\frac{k+1}{2}},\quad k\in{\mathbb{N}}^{*}, \end{aligned}\right. \end{equation*} which respectively correspond to the fundamental stable melting rate and a sequence of codimension $k$ unstable rates. Our analysis mainly depend on the methods developed in [17] which deals with the similar problems in two dimensions and also the construction of both stable and unstable finite time blow-up solutions for the harmonic heat flow in [49],[50].

On melting for the 3D radial Stefan problem

Abstract

We consider the three-dimensional radial Stefan problem which describes the evolution of a radial symmetric ice ball with free boundary \begin{equation*} \left\{\begin{aligned} &\partial_{t}u-\partial_{rr}u-\frac{2}{r}\partial_{r}u=0 \quad in\ r\geqλ(t),\\ &\partial_{r}u(t,λ(t))=-\dotλ(t),\\ &u(t,λ(t))=0,\\ &u(0,\cdot)=u_{0},\quad λ(0)=λ_{0}. \end{aligned}\right. \end{equation*} We prove the existence in the radial class of finite time melting with rates \begin{equation*} λ(t)=\left\{\begin{aligned} &4\sqrtπ\frac{\sqrt{T-t}}{|\log (T-t)|}(1+o_{t\rightarrow T}(1)),\\ &c(u_{0},k)(1+o_{t\rightarrow T}(1))(T-t)^{\frac{k+1}{2}},\quad k\in{\mathbb{N}}^{*}, \end{aligned}\right. \end{equation*} which respectively correspond to the fundamental stable melting rate and a sequence of codimension unstable rates. Our analysis mainly depend on the methods developed in [17] which deals with the similar problems in two dimensions and also the construction of both stable and unstable finite time blow-up solutions for the harmonic heat flow in [49],[50].
Paper Structure (14 sections, 16 theorems, 435 equations)

This paper contains 14 sections, 16 theorems, 435 equations.

Table of Contents

  1. Introduction
  2. Spectral analysis in weighted spaces
  3. Properties of the harmonic oscillator
  4. Almost invertibility of the renormalised operator
  5. Diagonalisation of $H_{b}$
  6. Diagonalisation of $\mathcal{H}_{b}$
  7. Renormaliesd equations and initialisation
  8. Modulation equations
  9. Energy estimates
  10. Closing the bootstrap
  11. Proof of Proposition \ref{['pr3.3']}.
  12. Proof of Theorem \ref{['th1']}
  13. Coercivity estimates
  14. Cauchy theory for the radial Stefan problem

Key Result

Theorem 1.1

There exists a set of initial data $(u_{0},\lambda_{0})$ in $H^{2}\times \mathbb{R}^{*}_{+}$, such that the corresponding solution $(u,\lambda)\in C(\left[0,T\right);H^{2} )\times C^{1}([0,T);\mathbb{R}_{+})$ to the exterior Stefan problem (eq:2) melts in finite time $0<T(u_{0},\lambda_{0})<+\infty$

Theorems & Definitions (38)

  • Theorem 1.1: Melting dynamics
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1: Near inversion of $H_{b}-2k$
  • Lemma 2.2
  • proof
  • proof : Proof of Lemma \ref{['le:1']}
  • Proposition 2.3: Diagonalisation of $H_{b}$
  • ...and 28 more