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Recovering the Polytropic Exponent in the Porous Medium Equation: Asymptotic Approach

Hagop Karakazian, Toni Sayah, Faouzi Triki

TL;DR

This work studies the inverse problem of recovering the polytropic exponent $\gamma>1$ in the porous medium equation $u_t = \Delta u^\gamma$ from a large-time observation $u_T$ under Dirichlet conditions. Using the known asymptotic behavior $u(x,t) \sim (1+t)^{-1/(\gamma-1)} f(x)$ and introducing $w = (-\Delta)^{-1} u_T$, the authors derive a sharp $L^\infty$ error bound $\| (\gamma-1)u_T^\gamma - \frac{w}{1+T} \|_{L^\infty} \le \frac{C}{(1+T)^{2+1/(\gamma-1)}}$ and show that any exponent producing a similar bound must satisfy $|\gamma-\alpha| \le \hat{C}/(1+T)$. They then formulate a practical reconstruction by minimizing $\|F(\alpha)\|_{L^1}$ with $F(\alpha)=(\alpha-1)(1+T)u_T^\alpha - w$, yielding a convergent estimate $\gamma_m$ with the same $O(1/(1+T))$ rate; extensions to $L^p$ norms are discussed. The method is validated in 2D numerical simulations, demonstrating convergence of $\gamma_m$ to the true $\gamma$ as $T$ grows, with faster convergence for smaller exponents. This provides a rigorous and implementable framework for parameter recovery in nonlinear diffusion models from large-time data, with potential applications to porous media and related systems.

Abstract

In this paper we consider the time dependent Porous Medium Equation, $u_t = Δu^γ$ with real polytropic exponent $γ>1$, subject to a homogeneous Dirichlet boundary condition. We are interested in recovering $γ$ from the knowledge of the solution $u$ at a given large time $T$. Based on an asymptotic inequality satisfied by the solution $u(T)$, we propose a numerical algorithm allowing us to recover $γ$. An upper bound for the error between the exact and recovered $γ$ is then showed. Finally, numerical investigations are carried out in two dimensions.

Recovering the Polytropic Exponent in the Porous Medium Equation: Asymptotic Approach

TL;DR

This work studies the inverse problem of recovering the polytropic exponent in the porous medium equation from a large-time observation under Dirichlet conditions. Using the known asymptotic behavior and introducing , the authors derive a sharp error bound and show that any exponent producing a similar bound must satisfy . They then formulate a practical reconstruction by minimizing with , yielding a convergent estimate with the same rate; extensions to norms are discussed. The method is validated in 2D numerical simulations, demonstrating convergence of to the true as grows, with faster convergence for smaller exponents. This provides a rigorous and implementable framework for parameter recovery in nonlinear diffusion models from large-time data, with potential applications to porous media and related systems.

Abstract

In this paper we consider the time dependent Porous Medium Equation, with real polytropic exponent , subject to a homogeneous Dirichlet boundary condition. We are interested in recovering from the knowledge of the solution at a given large time . Based on an asymptotic inequality satisfied by the solution , we propose a numerical algorithm allowing us to recover . An upper bound for the error between the exact and recovered is then showed. Finally, numerical investigations are carried out in two dimensions.
Paper Structure (6 sections, 10 theorems, 51 equations, 1 figure, 1 table)

This paper contains 6 sections, 10 theorems, 51 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Inverse Problem: The Asymptotic Method
  4. A Numerical Algorithm
  5. Numerical Simulations
  6. 2D Simulations

Key Result

Theorem 1

(Well-Posedness - Theorem 6.12 in vazquez07) For any non-negative $u_0 \in L^1(\Omega)$, the direct (PME) problem has a unique and stable weak solution $u \in C([0,\infty),L^1(\Omega))$ with $u^\gamma \in L^2_{loc}(\mathbb{R}^+,H_0^1(\Omega))$ satisfying: for all $\eta \in C_0^1([0,\infty) \times \overline \Omega)$ which vanishes everywhere for $0<t<\tau$ for some $\tau >0$. Furthermore when $u_0

Figures (1)

  • Figure 1: Right: The function $F$ with respect to $\alpha$ for $N=10$, $T=1000$ and $\gamma=3.5$ in $2D$. Left: a zoom.

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Remark 1
  • ...and 7 more