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Existence and Uniqueness of Local Solutions for a Class of Partial Differential-Algebraic Equations

Seyyid Ali Benabdallah

TL;DR

The article investigates local existence and uniqueness of solutions for a class of nonlinear partial differential-algebraic equations by recasting them as abstract evolution problems in Banach spaces. The key idea is to exploit an invertible algebraic constraint operator to eliminate the algebraic part, yielding a reduced evolution $V_t = A(V) + K(t,V)$ where $K$ is Lipschitz in $V$ and continuous in $t$, allowing the use of $C_0$-semigroup theory to obtain a unique mild solution on a maximal interval, with the full state recovered via $w = L^{-1}G(t,V)$. The paper provides rigorous regularity checks for the reduced map and presents a concrete nonlinear PDAE application, demonstrating invertibility of $L$, dissipativity and maximality of $A$, and local Lipschitz continuity of $F$, which together guarantee local well-posedness. This framework lays a solid functional-analytic foundation for analyzing coupled PDE-DAE systems arising in physical and chemical processes.

Abstract

In this work, we present a result on the local existence and uniqueness of solutions to nonlinear Partial Differential-Algebraic Equations (PDAEs). By applying established theoretical results, we identify the conditions that guarantee the existence of a unique local solution. The analysis relies on techniques from functional analysis, semi-group theory, and the theory of differential-algebraic systems. Additionally, we provide applications to illustrate the effectiveness of this result.

Existence and Uniqueness of Local Solutions for a Class of Partial Differential-Algebraic Equations

TL;DR

The article investigates local existence and uniqueness of solutions for a class of nonlinear partial differential-algebraic equations by recasting them as abstract evolution problems in Banach spaces. The key idea is to exploit an invertible algebraic constraint operator to eliminate the algebraic part, yielding a reduced evolution where is Lipschitz in and continuous in , allowing the use of -semigroup theory to obtain a unique mild solution on a maximal interval, with the full state recovered via . The paper provides rigorous regularity checks for the reduced map and presents a concrete nonlinear PDAE application, demonstrating invertibility of , dissipativity and maximality of , and local Lipschitz continuity of , which together guarantee local well-posedness. This framework lays a solid functional-analytic foundation for analyzing coupled PDE-DAE systems arising in physical and chemical processes.

Abstract

In this work, we present a result on the local existence and uniqueness of solutions to nonlinear Partial Differential-Algebraic Equations (PDAEs). By applying established theoretical results, we identify the conditions that guarantee the existence of a unique local solution. The analysis relies on techniques from functional analysis, semi-group theory, and the theory of differential-algebraic systems. Additionally, we provide applications to illustrate the effectiveness of this result.

Paper Structure

This paper contains 4 sections, 9 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Existence and Uniqueness
  4. Application

Key Result

Theorem 2.1

1 Let $\mathcal{F}:[0,+\infty[\times \mathcal{X} \longrightarrow \mathcal{X}$ be continuous in $t$ for $t\geq 0$ and locally Lipchitz continuous in $U$, uniformly in $t$ on bounded intervals. If $\mathcal{A}$ is the infinitesimal generator of a $C_0$ semi-group $T(t)$ on $\mathcal{X}$ then for every has a unique mild solution $U$ on $[0,t_{max}[$. Moreover, if $t _{max} \leq +\infty$ then

Theorems & Definitions (18)

  • Theorem 2.1
  • Definition 2.2
  • Theorem 3.1
  • proof
  • proposition 4.1
  • proof
  • proposition 4.2
  • proof
  • proposition 4.3
  • proof
  • ...and 8 more