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Inverse boundary value problem for the Convection-Diffusion equation with local data

Pranav Kumar, Anamika Purohit

TL;DR

This work addresses the inverse boundary value problem for the time-dependent convection-diffusion equation with partial boundary data. It develops a reflection-based CGO framework and derives a local DN-map identity, showing that the convection coefficient and density can be uniquely determined up to a gauge transform $\Psi$, with $A^{(2)}-A^{(1)}=\nabla_x\Psi$ and $q_2-q_1=\partial_t\Psi$. The gauge is the only obstruction to uniqueness when data are available on the accessible portion of the boundary, under a flat inaccessible boundary. The results extend Isakov's reflection approach to parabolic-type operators and provide a rigorous treatment of partial data recovery for time-dependent coefficients in convection-diffusion systems.

Abstract

We study a local data inverse problem for the time-dependent Convection-Diffusion Equation (CDE) in a bounded domain where a part of the boundary is treated to be inaccessible. Up on assuming the inaccessible part to be flat, we seek for the unique determination of the time-dependent convection and the density terms from the knowledge of the boundary data measured outside the inaccessible part. In the process, we show that there is a natural gauge in the perturbations, and we prove that this is the only obstruction in the uniqueness result.

Inverse boundary value problem for the Convection-Diffusion equation with local data

TL;DR

This work addresses the inverse boundary value problem for the time-dependent convection-diffusion equation with partial boundary data. It develops a reflection-based CGO framework and derives a local DN-map identity, showing that the convection coefficient and density can be uniquely determined up to a gauge transform , with and . The gauge is the only obstruction to uniqueness when data are available on the accessible portion of the boundary, under a flat inaccessible boundary. The results extend Isakov's reflection approach to parabolic-type operators and provide a rigorous treatment of partial data recovery for time-dependent coefficients in convection-diffusion systems.

Abstract

We study a local data inverse problem for the time-dependent Convection-Diffusion Equation (CDE) in a bounded domain where a part of the boundary is treated to be inaccessible. Up on assuming the inaccessible part to be flat, we seek for the unique determination of the time-dependent convection and the density terms from the knowledge of the boundary data measured outside the inaccessible part. In the process, we show that there is a natural gauge in the perturbations, and we prove that this is the only obstruction in the uniqueness result.
Paper Structure (7 sections, 3 theorems, 58 equations)

This paper contains 7 sections, 3 theorems, 58 equations.

Table of Contents

  1. Introduction and the main result
  2. Proof of the main theorem
  3. Preliminary results
  4. Construction of special solutions
  5. An integral identity
  6. Recovery of Convection coefficient
  7. Recovery of density coefficient

Key Result

Theorem 1.2

Let $Q=(0, T)\times \Omega$ where $T>0$ and $\Omega \subset \left\{x\in \mathbb{R}^{n}: x_{n}>0\right\}$, $n\geq 3,$ be a bounded simply connected domain with smooth boundary $\partial\Omega$, and let $\Gamma_{0} = \partial\Omega \cap \left\{x\in {\mathbb{R}}^{n}: x_{n}=0\right\}\neq \emptyset$, and Then there exists a function $\Psi\in C^{\infty}(\overline{Q})$ with $\Psi|_{\Sigma} = \partial_{\n

Theorems & Definitions (4)

  • Remark 1.1: Gauge invariance
  • Theorem 1.2
  • Proposition 2.1: Anamika-24
  • Proposition 2.2: Anamika-24