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Analysis of the weak formulation of a coupled nonlinear parabolic system modeling a heat exchanger

Ali Ouattara Kouma, Gossrin Jean-Marc Bomisso, Bérenger Akon Kpata, Kidjégbo Augustin Touré

TL;DR

The paper addresses a nonlinear, strongly coupled parabolic system $(\Sigma_T)$ modeling temperature evolution in a coaxial heat exchanger with spatially varying coefficients. It develops a tailored $\mathrm{Faedo-Galerkin}$ framework augmented by a basis-change transformation to handle cross-coupling and interfacial terms, proving existence and uniqueness of weak solutions and establishing time-space regularity. Under stronger data, it proves higher spatial regularity, providing comprehensive a priori estimates that support stable numerical approximations. The results extend classical scalar parabolic theory to a multi-component, heterogeneous setting, offering a rigorous foundation for simulations of complex heat-exchanger architectures and related coupled diffusion-convection-reaction systems.

Abstract

This paper establishes the existence, uniqueness and time-space regularity of the weak solution to a nonlinear coupled parabolic system modeling temperature evolution in a coaxial heat exchanger with source terms and spatially varying coefficients. The system is formulated in a weak sense and the analysis relies on a Faedo-Galerkin method tailored to handle the nonlinear coupling and heterogeneous domains. Under suitable assumptions on the initial data and source terms, enhanced regularity in both time and space is obtained. In contrast with classical scalar models, the study addresses a multi-component system with realistic boundary conditions and complex interfacial dynamics.

Analysis of the weak formulation of a coupled nonlinear parabolic system modeling a heat exchanger

TL;DR

The paper addresses a nonlinear, strongly coupled parabolic system modeling temperature evolution in a coaxial heat exchanger with spatially varying coefficients. It develops a tailored framework augmented by a basis-change transformation to handle cross-coupling and interfacial terms, proving existence and uniqueness of weak solutions and establishing time-space regularity. Under stronger data, it proves higher spatial regularity, providing comprehensive a priori estimates that support stable numerical approximations. The results extend classical scalar parabolic theory to a multi-component, heterogeneous setting, offering a rigorous foundation for simulations of complex heat-exchanger architectures and related coupled diffusion-convection-reaction systems.

Abstract

This paper establishes the existence, uniqueness and time-space regularity of the weak solution to a nonlinear coupled parabolic system modeling temperature evolution in a coaxial heat exchanger with source terms and spatially varying coefficients. The system is formulated in a weak sense and the analysis relies on a Faedo-Galerkin method tailored to handle the nonlinear coupling and heterogeneous domains. Under suitable assumptions on the initial data and source terms, enhanced regularity in both time and space is obtained. In contrast with classical scalar models, the study addresses a multi-component system with realistic boundary conditions and complex interfacial dynamics.
Paper Structure (10 sections, 6 theorems, 71 equations)

This paper contains 10 sections, 6 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definition of the weak solution
  4. Existence and uniqueness of the weak solution
  5. Existence of the weak solution
  6. Construction of approximate solutions
  7. A priori estimates
  8. Passage to limits
  9. Uniqueness of the weak solution
  10. Additional regularity results

Key Result

Lemma 4.1

There exists a sequence of functions $\left\{ \boldsymbol{\varPsi}_i^{(\alpha)} \right\}_{i=1} ^{\infty}$ for each $\alpha \in \mathcal{I}$, such that: where with $\delta_{\alpha k}$ the Kronecker symbol on $\mathcal{I}$.

Theorems & Definitions (14)

  • Remark 3.1
  • Definition 3.2
  • Lemma 4.1
  • proof
  • Theorem 4.2: Construction of approximate solutions
  • proof
  • Theorem 4.3
  • proof
  • Theorem 4.4: Existence of weak solution
  • proof
  • ...and 4 more