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Parabolic PDEs on a fixed domain with evolving subdomains: function spaces and well-posedness

Van Chien Le, Karel Van Bockstal

TL;DR

The paper addresses parabolic PDEs on a fixed spatial domain containing evolving subdomains with a possibly discontinuous time-derivative coefficient across the interface. It develops a generalized space-time variational framework by introducing $W^{1,p,p'}_{\alpha}(J,V,V')$ to meaningfully define $\alpha\partial_t u$ via Reynolds transport, proves density of smooth functions, establishes embedding and integration-by-parts results, and proves well-posedness through the Banach–Nečas–Babuška theorem. The analysis covers equivalence with material derivatives and accommodates a tensor-product space structure that benefits numerical methods, including classical time stepping and space-time schemes. The results provide a rigorous foundation for simulations of phase transitions, mass transfer, heat, and EM problems on moving geometries, with explicit a priori estimates and a clear path to discretization.

Abstract

This paper develops the necessary ingredients for the variational approach of initial boundary-value problems of parabolic partial differential equations on a fixed spatial domain containing evolving subdomains. In particular, we introduce function spaces for the variational solution that extend standard Sobolev-Bochner spaces to account for a coefficient associated with the time derivative that may be discontinuous across the evolving interface. We further show the density of smooth functions in these spaces by extending the mollification technique and the Reynolds transport theorem, and establish the corresponding "embedding" theory and an integration by parts formula. Finally, we prove the well-posedness of the space-time variational formulation in the natural setting using the Banach-Necas-Babuska theorem.

Parabolic PDEs on a fixed domain with evolving subdomains: function spaces and well-posedness

TL;DR

The paper addresses parabolic PDEs on a fixed spatial domain containing evolving subdomains with a possibly discontinuous time-derivative coefficient across the interface. It develops a generalized space-time variational framework by introducing to meaningfully define via Reynolds transport, proves density of smooth functions, establishes embedding and integration-by-parts results, and proves well-posedness through the Banach–Nečas–Babuška theorem. The analysis covers equivalence with material derivatives and accommodates a tensor-product space structure that benefits numerical methods, including classical time stepping and space-time schemes. The results provide a rigorous foundation for simulations of phase transitions, mass transfer, heat, and EM problems on moving geometries, with explicit a priori estimates and a clear path to discretization.

Abstract

This paper develops the necessary ingredients for the variational approach of initial boundary-value problems of parabolic partial differential equations on a fixed spatial domain containing evolving subdomains. In particular, we introduce function spaces for the variational solution that extend standard Sobolev-Bochner spaces to account for a coefficient associated with the time derivative that may be discontinuous across the evolving interface. We further show the density of smooth functions in these spaces by extending the mollification technique and the Reynolds transport theorem, and establish the corresponding "embedding" theory and an integration by parts formula. Finally, we prove the well-posedness of the space-time variational formulation in the natural setting using the Banach-Necas-Babuska theorem.
Paper Structure (4 sections, 5 theorems, 36 equations, 1 figure)

This paper contains 4 sections, 5 theorems, 36 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Function spaces
  3. Variational formulation
  4. Conclusions

Key Result

Lemma 2.2

If $p \in [2, \infty)$, then $\mathcal{D}(J, \mathop{\mathrm{V}}\nolimits)$ is densely contained in $\mathop{\mathrm{W}}\nolimits^{1, p, p^\prime}_{\alpha}(J, \mathop{\mathrm{V}}\nolimits, \mathop{\mathrm{V}}\nolimits^\prime)$.

Figures (1)

  • Figure 1: Illustrations of a fixed domain $\Omega$ comprising two subdomains $\Omega_1(t)$ and $\Omega_2(t)$, separated by an evolving interface $\Gamma(t)$. Left: The interface intersects the boundary of $\Omega$. Right: The closed interface divides the domain into two subdomains, with one entirely enclosed within the other.

Theorems & Definitions (12)

  • Remark 1
  • Definition 2.1
  • Lemma 2.2
  • Remark 2
  • Remark 3
  • Lemma 2.3
  • Remark 4
  • Remark 5
  • Remark 6
  • Lemma 3.1: Inf-sup condition
  • ...and 2 more