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Local well-posedness of a coupled Jordan-Moore-Gibson-Thompson-Pennes model of nonlinear ultrasonic heating

Imen Benabbas, Belkacem Said-Houari

TL;DR

The paper addresses the local well-posedness of a coupled nonlinear acoustic-heat model for ultrasonic heating, combining the hyperbolic JMGT equation with temperature-dependent coefficients and the semilinear Pennes bioheat equation. The authors develop a rigorous energy framework, proving first and higher-order energy estimates for the linearized JMGT problem and a regularity theory for the linearized Pennes equation, then employ a Banach fixed-point argument to handle the nonlinear coupling. Under regular, small initial data and short time, they obtain existence, uniqueness, and continuous dependence of solutions in the natural energy spaces, with explicit energy bounds that quantify the coupling effects. The results provide a mathematically solid foundation for locally well-posed simulations of nonlinear ultrasound heating in biological tissues, relevant to applications such as HIFU.

Abstract

In this work, we investigate a mathematical model of nonlinear ultrasonic heating based on the Jordan-Moore-Gibson-Thompson equation (JMGT) with temperature-dependent medium parameters coupled to the semilinear Pennes equation for the bioheat transfer. The equations are coupled via the temperature in the coefficients of the JMGT equation and via a nonlinear source term within the Pennes equation, which models the absorption of acoustic energy by the surrounding tissue. Using the energy method together with a fixed point argument, we prove that our model is locally well-posed, provided that the initial data are regular, small in a lower topology and the final time is short enough.

Local well-posedness of a coupled Jordan-Moore-Gibson-Thompson-Pennes model of nonlinear ultrasonic heating

TL;DR

The paper addresses the local well-posedness of a coupled nonlinear acoustic-heat model for ultrasonic heating, combining the hyperbolic JMGT equation with temperature-dependent coefficients and the semilinear Pennes bioheat equation. The authors develop a rigorous energy framework, proving first and higher-order energy estimates for the linearized JMGT problem and a regularity theory for the linearized Pennes equation, then employ a Banach fixed-point argument to handle the nonlinear coupling. Under regular, small initial data and short time, they obtain existence, uniqueness, and continuous dependence of solutions in the natural energy spaces, with explicit energy bounds that quantify the coupling effects. The results provide a mathematically solid foundation for locally well-posed simulations of nonlinear ultrasound heating in biological tissues, relevant to applications such as HIFU.

Abstract

In this work, we investigate a mathematical model of nonlinear ultrasonic heating based on the Jordan-Moore-Gibson-Thompson equation (JMGT) with temperature-dependent medium parameters coupled to the semilinear Pennes equation for the bioheat transfer. The equations are coupled via the temperature in the coefficients of the JMGT equation and via a nonlinear source term within the Pennes equation, which models the absorption of acoustic energy by the surrounding tissue. Using the energy method together with a fixed point argument, we prove that our model is locally well-posed, provided that the initial data are regular, small in a lower topology and the final time is short enough.
Paper Structure (11 sections, 7 theorems, 203 equations)

This paper contains 11 sections, 7 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. Statement of the problem and main result
  3. Main result
  4. Preliminaries
  5. Notation
  6. Inequalities and embedding results
  7. Analysis of the linearization of the JMGT equation
  8. First Energy estimate
  9. Higher-order energy estimate
  10. Analysis of the bioheat equation
  11. Well-posedness of the nonlinear problem \ref{['coupled_problem_JMGT_Pennes']}--Proof of Theorem \ref{['local_Existence_Thoerem']}

Key Result

Theorem 2.1

Given $\tau>0$. Let Assumptions Assumption_nonlinear and Assumption_Initial data hold. There exist a final time $T>0$, small $\eta_0=\eta_0(T)>0$ and a constant $R>0$ such that if the initial data satisfy the system Main_system_JMGT admits a unique solution $(p, \Theta) \in \mathcal{X}_p^2 \times \mathcal{X}_\Theta^2$ that satisfies the estimate Moreover, the solution depends continuously on the

Theorems & Definitions (12)

  • Theorem 2.1
  • Lemma 3.1
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Proposition 5.1
  • proof
  • Lemma 6.1
  • proof
  • ...and 2 more