Inverse problems for quasi-linear elliptic systems modeling electrolysers

1. Problem Statement

The paper investigates the mathematical modeling and inverse problem associated with Proton Exchange Membrane (PEM) and Anion Exchange Membrane (AEM) electrolyser cells. These devices produce green hydrogen via electrochemical reactions.

• Governing Equations: The physical processes are modeled by a coupled system of second-order quasi-linear elliptic partial differential equations (PDEs). The system describes:

  1. Ion Concentrations ($c_i$): Governed by Fick's laws (diffusion) and source terms (reaction kinetics).
  2. Electric Potential ($\phi$): Governed by a Poisson-type equation coupled to the charge density of the ions.
  3. Temperature ($T$): Coupled to the system via the dependence of diffusion coefficients and source terms on temperature.

• The Inverse Problem: The goal is to uniquely reconstruct the unknown non-linear diffusion coefficients ($D_i$) and the phenomenological relation defining the electric potential ($\phi$) as functions of time, position, temperature, and ion concentrations.

• The Challenge: The authors demonstrate that boundary measurements alone are insufficient to uniquely determine these non-linear quantities within the interior of the domain. The system suffers from non-uniqueness when restricted to boundary data (Dirichlet-to-Neumann maps).

2. Methodology

The authors employ a combination of functional analysis, PDE theory, and a novel application of linearization techniques.

A. Mathematical Formulation

• Static Source-Free Case: The analysis focuses on the static equilibrium state ($\partial_t = 0$) with zero source terms ($g_i = 0$) to establish foundational results. The system reduces to:
  $$\text{div}(D_i(c, T, x)\nabla c_i) = 0, \quad \text{div}(\epsilon \nabla \phi(c, T, x)) = q \cdot c$$
• Measurements:
  • Boundary Data: Prescribed Dirichlet data ($c|_{\partial\Omega}, T|_{\partial\Omega}$) and measured Neumann data (fluxes $N \cdot \nabla c_i$, $N \cdot \nabla T$) and potential differences ($\phi(x) - \phi(y)$).
  • Interior Data: Temperature measurements $T(x)$ at points $x \in \Omega$.

B. Theoretical Tools

1. Existence and Uniqueness (Forward Problem):

   • The authors prove the existence of weak solutions using Schaefer's fixed point theorem. They transform the system into a symmetric form by introducing an inverse function for the potential $\phi$ (guaranteed by the monotonicity assumption $\partial_s \phi > 0$).
   • Uniqueness is established for the source-free case with constant boundary conditions.

2. Generalized Linearization (The Core Innovation):

   • The paper generalizes a linearization lemma by Sun (1996), originally developed for the scalar quasi-linear Calderón problem.
   • Standard Linearization: In scalar problems, linearization at constant boundary data "freezes" the coefficients, reducing the problem to a linear elliptic equation.
   • Systemic Challenge: In this coupled system, the non-linearity involves non-local dependencies (the potential depends on the global solution). Consequently, standard linearization does not freeze the coefficients; the linearized variables remain space-dependent.
   • The Solution: The authors show that interior measurements (specifically the temperature profile $T(x)$) are required to "freeze" the non-linearity. Once the interior state is known, the system decouples into standard linear elliptic equations, allowing the application of classical uniqueness results.

3. Key Contributions and Results

A. Impossibility of Boundary-Only Reconstruction

Theorem 3.3 establishes that boundary measurements (Dirichlet-to-Neumann map) alone cannot uniquely reconstruct the electric potential $\phi$ in the interior.

• Result: Boundary data can only determine the potential up to a constant and its gradient along the boundary.
• Counter-example: One can construct distinct potentials $\phi_1$ and $\phi_2$ that differ by a "bump function" in the interior but yield identical boundary measurements.

B. Unique Reconstruction with Hybrid Data

Theorem 3.5 is the main result, proving that a combination of boundary and interior measurements allows for the unique reconstruction of both the diffusion coefficients $D_i$ and the potential $\phi$ (up to a constant).

• Mechanism:
  1. Use boundary data to establish that the potentials agree on the boundary.
  2. Use interior temperature measurements ($T^1(x) = T^2(x)$) to fix the state variables.
  3. Apply the generalized linearization lemma (Lemma 4.6) to show that the difference in the Dirichlet-to-Neumann maps for the linearized systems must vanish.
  4. Invoke the uniqueness of the linear scalar Calderón problem to conclude $D_i^1 = D_i^2$ and $\phi^1 = \phi^2$.

C. The Linearization Lemma

Lemma 4.6 is a significant independent contribution. It extends Sun's linearization technique to systems of PDEs with non-local non-linearities. It proves that the normal traces of the solutions to the non-linear system converge to the normal traces of the solutions of a specific linearized system as the boundary perturbation tends to zero.

4. Significance and Implications

• Theoretical Advancement: The paper bridges the gap between scalar inverse problems (Calderón problem) and coupled systems. It resolves the issue of non-uniqueness in quasi-linear systems by demonstrating the necessity of internal data, a concept central to hybrid inverse problems (coupled-physics).
• Practical Application for Electrolysers:
  • The results provide a theoretical justification for the design of AEM electrolyser sensors.
  • It suggests that to optimize electrolyser design (e.g., extending membrane lifetime by understanding degradation), one cannot rely solely on external voltage/current monitoring. Internal temperature sensing is mathematically proven to be essential for uniquely identifying the internal material properties (diffusion coefficients) and reaction potentials.
• Modeling Insight: The work clarifies that while boundary data determines the behavior of the system "infinitesimally" near the boundary, the interior behavior is decoupled from the boundary unless internal constraints (measurements) are introduced.

5. Conclusion

The authors successfully demonstrate that the inverse problem for the quasi-linear elliptic system modeling electrolysers is ill-posed with boundary data alone but becomes well-posed (uniquely solvable) when augmented with interior temperature measurements. This is achieved through a rigorous generalization of linearization techniques, offering a robust framework for identifying solution-dependent parameters in coupled electrochemical systems.