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Well-posedness of the heat equation in domains with topological transitions

Maxim Olshanskii, Arnold Reusken

Abstract

We analyze a linear parabolic equation with homogeneous Dirichlet boundary conditions posed in domains whose evolution may involve topological transitions. The domains are described as sublevel sets of a smooth space-time level set function, allowing for transitions such as domain splitting and merging and the creation or vanishing of islands and holes. We introduce anisotropic space-time function spaces that extend the classical Bochner spaces used in cylindrical domains and establish key functional-analytic properties of these spaces, including the density of compactly supported smooth functions. This framework enables the application of the Babuška-Banach theorem, yielding existence, uniqueness, and a priori estimates for weak solutions. The analysis applies to domain evolutions generated by level set functions with isolated nondegenerate critical points, which correspond to the generic topology changes classified by Morse theory in two and three spatial dimensions.

Well-posedness of the heat equation in domains with topological transitions

Abstract

We analyze a linear parabolic equation with homogeneous Dirichlet boundary conditions posed in domains whose evolution may involve topological transitions. The domains are described as sublevel sets of a smooth space-time level set function, allowing for transitions such as domain splitting and merging and the creation or vanishing of islands and holes. We introduce anisotropic space-time function spaces that extend the classical Bochner spaces used in cylindrical domains and establish key functional-analytic properties of these spaces, including the density of compactly supported smooth functions. This framework enables the application of the Babuška-Banach theorem, yielding existence, uniqueness, and a priori estimates for weak solutions. The analysis applies to domain evolutions generated by level set functions with isolated nondegenerate critical points, which correspond to the generic topology changes classified by Morse theory in two and three spatial dimensions.
Paper Structure (10 sections, 14 theorems, 114 equations, 1 figure)

This paper contains 10 sections, 14 theorems, 114 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Domain evolution
  3. A classification result
  4. Uniform Poincaré and trace inequalities
  5. Model problem and space-time function spaces
  6. Function spaces
  7. Properties of the solution space $W$
  8. Well-posedness
  9. Proof of Theorem \ref{['densityH']}
  10. Proof of Lemma \ref{['lemA']}

Key Result

Lemma 2.1

\newlabelL:Morse Consider a nondegenerate critical point $(x_c,t_c)$ on the zero level of $\phi$. Without loss of generality, assume $(x_c,t_c)=(0,0)$. Then there exists a neighborhood $\widehat{X}= X \times (-\delta,\delta)$ of $(0,0)$ in $\mathbb{R}^{d+1}$ and a map $\psi:\, \widehat{X} \to \math with $0 \leq q \leq d$ and $v: (-\delta,\delta) \to \mathbb{R}$ a smooth map such that $v(0)=0$, $v'

Figures (1)

  • Figure 2.1: $d=2$. A splitting scenario with $\lambda_1<0<\lambda_2$, $\frac{\partial \phi}{\partial t}(x_c,t_c) > 0$. The left plot shows $\Gamma_{\!\mathcal{Q}}$ near $(x_c,t_c)$, while the right shows snapshots of $\Gamma(t)$ near $(x_c,t_c)$.

Theorems & Definitions (31)

  • Remark 2.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 21 more