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Ill-posedness of the Boltzmann-BGK model in the exponential class

Donghyun Lee, Sungbin Park, Seok-Bae Yun

Abstract

BGK (Bhatnagar-Gross-Krook) model is a relaxation-type model of the Boltzmann equation, which is popularly used in place of the Boltzmann equation in physics and engineering. In this paper, we address the ill-posedness problem for the BGK model, in which the solution instantly escapes the initial solution space. For this, we propose two ill-posedness scenarios, namely, the homogeneous and the inhomogeneous ill-posedness mechanisms. In the former case, we find a class of spatially homogeneous solutions to the BGK model, where removing the small velocity part of the initial data triggers ill-posedness by increasing temperature. For the latter, we construct a spatially inhomogeneous solution to the BGK model such that the local temperature constructed from the solution has a polynomial growth in spatial variable. These ill-posedness properties for the BGK model pose a stark contrast with the Boltzmann equation for which the solution map is, at least for a finite time, stable in the corresponding solution spaces.

Ill-posedness of the Boltzmann-BGK model in the exponential class

Abstract

BGK (Bhatnagar-Gross-Krook) model is a relaxation-type model of the Boltzmann equation, which is popularly used in place of the Boltzmann equation in physics and engineering. In this paper, we address the ill-posedness problem for the BGK model, in which the solution instantly escapes the initial solution space. For this, we propose two ill-posedness scenarios, namely, the homogeneous and the inhomogeneous ill-posedness mechanisms. In the former case, we find a class of spatially homogeneous solutions to the BGK model, where removing the small velocity part of the initial data triggers ill-posedness by increasing temperature. For the latter, we construct a spatially inhomogeneous solution to the BGK model such that the local temperature constructed from the solution has a polynomial growth in spatial variable. These ill-posedness properties for the BGK model pose a stark contrast with the Boltzmann equation for which the solution map is, at least for a finite time, stable in the corresponding solution spaces.
Paper Structure (15 sections, 31 theorems, 486 equations, 1 figure)

This paper contains 15 sections, 31 theorems, 486 equations, 1 figure.

Table of Contents

  1. Introduction
  2. BGK model of the Boltzmann equation
  3. Two instability scenarios for the BGK model
  4. Spatially homogeneous Ill-posedness mechanism:
  5. Ill-posedness from spatial inhomogeneity:
  6. Comparison with the well-posedness result of the Boltzmann equation
  7. Ill-posedness theory of spatially homogeneous BGK
  8. The proof of Theorem \ref{['thm:2_1']}
  9. Technical Lemmas for inhomogeneous Ill-posedness
  10. The case of $|x|\leq \max\{(18 C_T\alpha)^{\frac{1}{1-\gamma}}, 1\}$
  11. The case of $|x|\geq \max\{(18 C_T\alpha)^{\frac{1}{1-\gamma}}, 1\}$ and $0\leq r\leq t^{-1}|x|$
  12. The case of $|x|\geq \max\{(18 C_T\alpha)^{\frac{1}{1-\gamma}}, 1\}$ and $r\geq t^{-1}|x|$
  13. Ill-posedness theory of spatially inhomogeneous BGK
  14. Well-Posedness theory of the Boltzmann equation
  15. Asymptotic blow-up of the Boltzmann equation

Key Result

Theorem 1.1

For any given $N\geq 1$, $\alpha>0$, $\beta>0$, $\varepsilon > 0$, and $1\leq p\leq \infty$, the following statements hold:

Figures (1)

  • Figure 1: Shape of $C_a$.

Theorems & Definitions (68)

  • Theorem 1.1: homogeneous ill-posedness for the BGK model
  • Theorem 1.2: inhomogeneous Ill-posedness of the BGK model
  • Remark 1.3
  • Theorem 1.4: Well-posedness of the Boltzmann
  • Remark 1.5
  • Remark 1.6
  • Theorem 2.1
  • Remark 2.2
  • proof : Proof of Theorem \ref{['thm:1']}
  • Proposition 2.1
  • ...and 58 more