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Determining nonlinear balance laws in product-type domains by a single local passive boundary observation

Chaohua Duan, Hongyu Liu, Qingle Meng, Li Wang

TL;DR

This work addresses the inverse problem for nonlinear balance laws by recasting flux and source terms as nonlinear operators and showing that a single local passive boundary observation can nearly determine their action on the state, within product-type domains. The authors develop an operator-theoretic framework combined with complex geometrical optics and microlocal analysis to prove approximate unique identifiability of the flux and source operators from boundary data, with reconstruction errors scaling as powers of the microscale parameter $\varepsilon$. The main results establish quantitative bounds: $|\tilde{\mathscr{F}^1}(u^j)-\tilde{\mathscr{F}^2}(u^j)|\le C\varepsilon^{\tau}$ and, under a flux-equality condition, $|\mathfrak{f}^1(u^j)-\mathfrak{f}^2(u^j)|\le C'\varepsilon^{\tau_1}$; these bounds link interior nonlinear dynamics to boundary measurements in multiscale geometric settings. An application to reaction-diffusion-convection systems demonstrates the framework's practical relevance, providing explicit bounds for advection and reaction terms and revealing a holographic-like principle that interior dynamics are encoded in boundary observations at the macroscale.

Abstract

This paper introduces an operator-theoretic paradigm for solving inverse problems in nonlinear balance laws, shifting the focus from identifying specific functional forms to recovering the input-output actions of the associated flux and source operators. It is established that a single local passive boundary observation suffices to uniquely determine realizations of these operators for systems posed on product-type domains. This framework, which encompasses dynamical regimes, reveals a holographic-type principle where macroscopic boundary data encodes microscopic dynamical information, with broad implications for fluid dynamics and reaction-diffusion systems.

Determining nonlinear balance laws in product-type domains by a single local passive boundary observation

TL;DR

This work addresses the inverse problem for nonlinear balance laws by recasting flux and source terms as nonlinear operators and showing that a single local passive boundary observation can nearly determine their action on the state, within product-type domains. The authors develop an operator-theoretic framework combined with complex geometrical optics and microlocal analysis to prove approximate unique identifiability of the flux and source operators from boundary data, with reconstruction errors scaling as powers of the microscale parameter . The main results establish quantitative bounds: and, under a flux-equality condition, ; these bounds link interior nonlinear dynamics to boundary measurements in multiscale geometric settings. An application to reaction-diffusion-convection systems demonstrates the framework's practical relevance, providing explicit bounds for advection and reaction terms and revealing a holographic-like principle that interior dynamics are encoded in boundary observations at the macroscale.

Abstract

This paper introduces an operator-theoretic paradigm for solving inverse problems in nonlinear balance laws, shifting the focus from identifying specific functional forms to recovering the input-output actions of the associated flux and source operators. It is established that a single local passive boundary observation suffices to uniquely determine realizations of these operators for systems posed on product-type domains. This framework, which encompasses dynamical regimes, reveals a holographic-type principle where macroscopic boundary data encodes microscopic dynamical information, with broad implications for fluid dynamics and reaction-diffusion systems.

Paper Structure

This paper contains 12 sections, 22 theorems, 157 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical setup
  3. Geometric setup
  4. Technical developments and discussion
  5. Statement of main results and preliminaries
  6. Mathematical setup
  7. Proof of Theorem \ref{['main:thm']} in two dimensions
  8. Geometric Preparation
  9. Proof of Theorem \ref{['main:thm']} in three dimensions
  10. Nozzle domains
  11. Slab domains
  12. Application to Reaction-Diffusion-Convection Systems

Key Result

Theorem 1.2

Let $\Omega$ be a product-type domain defined in eq:NNN. Assume that $\mathbf{F}^j$ and $f^j$ belong to the admissible class $\mathcal{A}$, with the corresponding solution $u^j$ in the system eq:bl and the associated operators $\mathscr{F}^j$ and $\mathfrak{f}^j$ for $j=1,2$. Let $\Sigma \subset \wi then for any $(\mathbf{x},t) \in Q$, we obtain where $\tau, \tau_1 \in (0,1)$, and $C$ and $C'$ ar

Figures (4)

  • Figure 1: Schematic illustration of the nozzle and slab domains in $\mathbb{R}^{3}$
  • Figure 2: configuration for the 2D analysis showing (a) the original nozzle domain and (b) the transformed coordinate system with characteristic subdomain.
  • Figure 3: illustration of the nozzle domain in 3D
  • Figure 4: illustration of the geometry in 3D

Theorems & Definitions (46)

  • Definition 1.1: Nozzle domain
  • Definition 1.2: Slab domain
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1: Admissible class $\mathcal{A}$
  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • ...and 36 more