Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.17811 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929230269382656 |
|---|---|
| author | Zhang, Chencheng |
| author_facet | Zhang, Chencheng |
| contents | 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17811 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On melting for the 3D radial Stefan problem Zhang, Chencheng Analysis of PDEs 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]. |
| title | On melting for the 3D radial Stefan problem |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2401.17811 |