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Weak solutions to the heat conducting compressible self-gravitating flows in time-dependent domains

Kuntal Bhandari, Bingkang Huang, Šárka Nečasová

TL;DR

The paper proves the global-in-time existence of weak solutions to the heat-conducting compressible self-gravitating fluids in a time-dependent domain governed by the Navier–Stokes–Fourier–Poisson system. The authors develop a penalization framework to handle moving boundaries, mollify coefficients, and introduce artificial pressure and ballistic energy to control boundary heat flux, density, and temperature. They establish uniform a priori bounds and perform a sequence of limit passages (first in the boundary-penalization parameter, then in mollification and penalty parameters) to obtain a weak solution in the original moving domain that satisfies a ballistic energy inequality. The approach extends the NSF–Poisson theory to moving domains with nonhomogeneous boundary heat flux, providing a rigorous foundation for modeling viscous, heat-conducting self-gravitating fluids such as gaseous stars under time-dependent geometry.

Abstract

In this paper, we consider the heat-conducting compressible self-gravitating fluids in time-dependent domains, which typically describe the motion of viscous gaseous stars. The flow is governed by the 3-D Navier-Stokes-Fourier-Poisson equations where the velocity is supposed to fulfil the full-slip boundary condition and the temperature on the boundary is given by a non-homogeneous Dirichlet condition. We establish the global-in-time weak solution to the system. Our approach is based on the penalization of the boundary behavior, viscosity, and the pressure in the weak formulation. Moreover, to accommodate the non-homogeneous boundary heat flux, the concept of {\em ballistic energy} is utilized in this work.

Weak solutions to the heat conducting compressible self-gravitating flows in time-dependent domains

TL;DR

The paper proves the global-in-time existence of weak solutions to the heat-conducting compressible self-gravitating fluids in a time-dependent domain governed by the Navier–Stokes–Fourier–Poisson system. The authors develop a penalization framework to handle moving boundaries, mollify coefficients, and introduce artificial pressure and ballistic energy to control boundary heat flux, density, and temperature. They establish uniform a priori bounds and perform a sequence of limit passages (first in the boundary-penalization parameter, then in mollification and penalty parameters) to obtain a weak solution in the original moving domain that satisfies a ballistic energy inequality. The approach extends the NSF–Poisson theory to moving domains with nonhomogeneous boundary heat flux, providing a rigorous foundation for modeling viscous, heat-conducting self-gravitating fluids such as gaseous stars under time-dependent geometry.

Abstract

In this paper, we consider the heat-conducting compressible self-gravitating fluids in time-dependent domains, which typically describe the motion of viscous gaseous stars. The flow is governed by the 3-D Navier-Stokes-Fourier-Poisson equations where the velocity is supposed to fulfil the full-slip boundary condition and the temperature on the boundary is given by a non-homogeneous Dirichlet condition. We establish the global-in-time weak solution to the system. Our approach is based on the penalization of the boundary behavior, viscosity, and the pressure in the weak formulation. Moreover, to accommodate the non-homogeneous boundary heat flux, the concept of {\em ballistic energy} is utilized in this work.
Paper Structure (26 sections, 4 theorems, 206 equations)

This paper contains 26 sections, 4 theorems, 206 equations.

Table of Contents

  1. Introduction and general setting
  2. Problem statement
  3. Description of the time-dependent domain.
  4. Boundary conditions.
  5. Initial conditions.
  6. Bibliographic comments and main goal of our work
  7. Hypothesis
  8. Weak formulations
  9. Main result
  10. Penalized problem
  11. Mollification of the coefficients and initial data
  12. Penalization in the fixed domain and weak formulations
  13. Uniform bounds
  14. Passing to the limit
  15. Penalization limit: passing with $\varepsilon\to 0$
  16. ...and 11 more sections

Key Result

Theorem 3.1

Assume that $\Omega_0 \subset \mathbb{R}^3$ is a bounded domain of the class $\mathcal{C}^{2+\nu_0}$ for some $\nu_0>0$ and, suppose that $\mathbf{V}\in \mathcal{C}^1([0,T]; \mathcal{C}^3_c(\mathbb{R}^3;\mathbb{R}^3))$ satisfying trajec-eq and the hypothesis in subsection Section-hypothesis are sati

Theorems & Definitions (8)

  • Definition 2.1
  • Theorem 3.1
  • Definition 4.1
  • Theorem 4.2
  • proof
  • Remark 1
  • Lemma A.1: Generalized Poincaré inequality
  • Lemma A.2: Korn-Poincaré inequality