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Global well-posedness of a nonlinear Boussinesq-fluid-structure interaction system with large initial data

J. Zhang, S. Wang, L. Shen

TL;DR

This work analyzes a moving-interface, nonlinear Boussinesq-fluid-structure system with heat exchange, coupling incompressible fluid dynamics with elastic structure dynamics and thermal diffusion. It establishes global well-posedness by first constructing global weak solutions via Galerkin approximations, uniform energy estimates, and compactness, and then proving uniqueness in two dimensions. In two dimensions, it further obtains global strong and, under smoother data and compatibility, global smooth solutions for large initial data. The results extend classical FSI theory to a heat-affected, nonlinear setting with a moving interface, providing a rigorous framework for both weak and strong global solutions.

Abstract

In this paper, we consider the global well-posedness of the initial-boundary value problem to a nonlinear Boussinesq-fluid-structure interaction system, which describes the motion of an incompressible Boussinesq-fluid surrounded by an elastic structure with the heat exchange and is one coupled incompressible Boussinesq equations with the wave equation and heat equation by physical interface boundary conditions. Firstly, the global existence of weak solutions to this problem in two/three-dimension is proven by introducing one class of its suitable weak solution and using the compactness method. Then, the uniqueness of the weak solution to this problem in two-dimension is established. Finally, the existence and uniqueness of the global strong and smooth solution to this problem in two-dimension is obtained for any smooth large initial data under the assumptions of suitable compatibility conditions by establishing a priori higher order derivative estimates.

Global well-posedness of a nonlinear Boussinesq-fluid-structure interaction system with large initial data

TL;DR

This work analyzes a moving-interface, nonlinear Boussinesq-fluid-structure system with heat exchange, coupling incompressible fluid dynamics with elastic structure dynamics and thermal diffusion. It establishes global well-posedness by first constructing global weak solutions via Galerkin approximations, uniform energy estimates, and compactness, and then proving uniqueness in two dimensions. In two dimensions, it further obtains global strong and, under smoother data and compatibility, global smooth solutions for large initial data. The results extend classical FSI theory to a heat-affected, nonlinear setting with a moving interface, providing a rigorous framework for both weak and strong global solutions.

Abstract

In this paper, we consider the global well-posedness of the initial-boundary value problem to a nonlinear Boussinesq-fluid-structure interaction system, which describes the motion of an incompressible Boussinesq-fluid surrounded by an elastic structure with the heat exchange and is one coupled incompressible Boussinesq equations with the wave equation and heat equation by physical interface boundary conditions. Firstly, the global existence of weak solutions to this problem in two/three-dimension is proven by introducing one class of its suitable weak solution and using the compactness method. Then, the uniqueness of the weak solution to this problem in two-dimension is established. Finally, the existence and uniqueness of the global strong and smooth solution to this problem in two-dimension is obtained for any smooth large initial data under the assumptions of suitable compatibility conditions by establishing a priori higher order derivative estimates.

Paper Structure

This paper contains 5 sections, 2 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1.1
  3. Proof of Theorem 1.2
  4. Acknowledgements
  5. Conflict of Interest

Key Result

Theorem 1.1

Let $\mathrm{d}=2 \;{\rm or }\; 3$. Assume that the domain $\Omega_{f}$, $\Omega_{s}$ are smooth and bounded with the smooth interface boundary $\Gamma=\partial\Omega_{f}\cap\partial\Omega_{s}$. And for the initial date satisfy $(v_{0},d_{0})\in L^{2}(\Omega)$, $w_{0}\in V_{s}$, $\text{div} v_{0}=0$ Moreover, when $\mathrm{d}=2$, the global weak solution of 1.3 is unique.

Theorems & Definitions (4)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Theorem 1.2