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Analysis of a one dimensional energy dissipating free boundary model with nonlinear boundary conditions. Existence of weak solutions

Benoît Merlet, Juliette Venel, Antoine Zurek

TL;DR

This work establishes the existence of maximal weak solutions for a one-dimensional diffusion model with a moving free boundary and nonlinear boundary conditions modeling oxide formation on nuclear waste containment. A Jordan–Kinderlehrer–Otto style minimizing movements scheme is constructed with a Wasserstein-like metric tailored to possibly unequal masses, augmented by a penalty to control boundary fluxes. Existence and regularity of one-step minimizers are proven, and as the time step vanishes, the discrete solutions converge to a weak maximal solution of the coupled PDE–ODE system with a Signorini-type boundary at the oxide/solution interface and a nonlinear balance at the oxide/metal interface. The approach combines unbalanced optimal transport, energy dissipation via Gibbs energy, and careful compactness arguments to handle the moving boundary and non-smooth boundary conditions, providing a mathematically rigorous basis for long-term predictions relevant to nuclear-waste safety assessments.

Abstract

This work is part of a general study on the long-term safety of the geological repository of nuclear wastes. A diffusion equation with a moving free boundary in one dimension is introduced and studied. The model describes some mechanisms involved in corrosion processes at the surface of carbon steel canisters in contact with a claystone formation. The main objective of the paper is to prove the existence of weak solutions to the problem which are maximal in time. For this, a time semidiscrete minimizing movements scheme based on a Wasserstein-like distance is introduced. The existence of solutions to the scheme is proved. Then, using a priori estimates, it is shown that as the time step goes to zero these solutions converge up to extraction towards a maximal weak solution to the free boundary model.

Analysis of a one dimensional energy dissipating free boundary model with nonlinear boundary conditions. Existence of weak solutions

TL;DR

This work establishes the existence of maximal weak solutions for a one-dimensional diffusion model with a moving free boundary and nonlinear boundary conditions modeling oxide formation on nuclear waste containment. A Jordan–Kinderlehrer–Otto style minimizing movements scheme is constructed with a Wasserstein-like metric tailored to possibly unequal masses, augmented by a penalty to control boundary fluxes. Existence and regularity of one-step minimizers are proven, and as the time step vanishes, the discrete solutions converge to a weak maximal solution of the coupled PDE–ODE system with a Signorini-type boundary at the oxide/solution interface and a nonlinear balance at the oxide/metal interface. The approach combines unbalanced optimal transport, energy dissipation via Gibbs energy, and careful compactness arguments to handle the moving boundary and non-smooth boundary conditions, providing a mathematically rigorous basis for long-term predictions relevant to nuclear-waste safety assessments.

Abstract

This work is part of a general study on the long-term safety of the geological repository of nuclear wastes. A diffusion equation with a moving free boundary in one dimension is introduced and studied. The model describes some mechanisms involved in corrosion processes at the surface of carbon steel canisters in contact with a claystone formation. The main objective of the paper is to prove the existence of weak solutions to the problem which are maximal in time. For this, a time semidiscrete minimizing movements scheme based on a Wasserstein-like distance is introduced. The existence of solutions to the scheme is proved. Then, using a priori estimates, it is shown that as the time step goes to zero these solutions converge up to extraction towards a maximal weak solution to the free boundary model.
Paper Structure (24 sections, 25 theorems, 370 equations, 4 figures)

This paper contains 24 sections, 25 theorems, 370 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Presentation of the reduced model
  3. Evolution of the Gibbs free energy
  4. Notion of weak solution and main result
  5. Formal discussion about the possible disappearance of the oxide layer
  6. A few words about the proof of Theorem \ref{['th.main']}
  7. Notation and conventions
  8. The optimisation problem defining one time step of the semidiscrete scheme
  9. The 2-Wasserstein distance in dimension 1
  10. An unbalanced optimal transport problem
  11. Optimal transport plans
  12. Definition of one step of the minimizing movement scheme
  13. Existence of a minimizer and first properties
  14. The optimization problem \ref{['OP']} as a time discretisation of \ref{['P']}
  15. Behavior of the minimizers at the fixed interface
  16. ...and 9 more sections

Key Result

Theorem 1.2

Let the following assumptions hold: Then, there exists $\overline T\in(0,+\infty]$ and a weak solution $(\rho,X)$ to the system P--P.bord on $(0,\overline T)$ in the sense of Definition def.sol.faible. Besides,

Figures (4)

  • Figure 1: The three possible cases of the boundary condition \ref{['P.bord2']}
  • Figure 2: The transport map $T_+$ in the case $0\le\mathbf{M}(\rho)-\mathbf{M}(\rho^0)=:m$.
  • Figure 3: The transport maps $T_-$ and $S_-$ in the case $0<\mathbf{M}(\rho^0)-\mathbf{M}(\rho)=:m$.
  • Figure 4: Possible behaviors of $\rho$ at $x=0$ according to \ref{['bcx=0.bis']}. Compare with Figure \ref{['figcbx=0']}.

Theorems & Definitions (53)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • proof : Proof of Theorem \ref{['theo.ExistenceJKO.1']}
  • Theorem 2.5: Existence of a minimizer in $\mathcal{A}$
  • proof
  • Theorem 2.6
  • ...and 43 more