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Well-Posedness of the R13 Equations Using Tensor-Valued Korn Inequalities

Peter Lewintan, Lambert Theisen, Manuel Torrilhon

TL;DR

This work proves the well-posedness of the linearized $R_{13}$ moment system on bounded Lipschitz domains by formulating a grouped mixed (saddle-point) problem and establishing new tensor-valued Korn-type inequalities. The key technical advances are a tensor-valued $\mathbb{C}$-ellipticity result for the symmetric and trace-free gradient and a right-inverse construction for the matrix-valued divergence, enabling coercivity on the kernel and a robust inf-sup condition. Combining these with Brezzi theory yields existence and uniqueness of weak solutions, with regularity dictated by the boundary coupling parameter $\epsilon^{\mathrm{w}}$ and the Knudsen number $Kn$. The analysis also clarifies the relationship to classical Stokes and Poisson problems in the low-$Kn$ limit and paves the way for stable numerical discretizations of higher-order moment closures.

Abstract

In this paper, we finally catch up with proving the well-posedness of the linearized R13 moment model, which describes, e.g., rarefied gas flows. As an extension of the classical fluid equations, moment models are robust and have been frequently used, yet they are challenging to analyze due to their additional equations. By effectively grouping variables, we identify a 2-by-2 block structure, allowing the analysis of the well-posedness within the abstract LBB framework of saddle point problems. Due to the unique tensorial structure of the equations, in addition to an interesting combination of tools from Stokes' and linear elasticity theory, we also need new coercivity estimates for tensor fields. These Korn-type inequalities are established by analyzing the symbol map of the symmetric and trace-free part of tensor derivative fields. Together with the corresponding right inverse of the tensorial divergence, we obtain the existence and uniqueness of weak solutions.

Well-Posedness of the R13 Equations Using Tensor-Valued Korn Inequalities

TL;DR

This work proves the well-posedness of the linearized moment system on bounded Lipschitz domains by formulating a grouped mixed (saddle-point) problem and establishing new tensor-valued Korn-type inequalities. The key technical advances are a tensor-valued -ellipticity result for the symmetric and trace-free gradient and a right-inverse construction for the matrix-valued divergence, enabling coercivity on the kernel and a robust inf-sup condition. Combining these with Brezzi theory yields existence and uniqueness of weak solutions, with regularity dictated by the boundary coupling parameter and the Knudsen number . The analysis also clarifies the relationship to classical Stokes and Poisson problems in the low- limit and paves the way for stable numerical discretizations of higher-order moment closures.

Abstract

In this paper, we finally catch up with proving the well-posedness of the linearized R13 moment model, which describes, e.g., rarefied gas flows. As an extension of the classical fluid equations, moment models are robust and have been frequently used, yet they are challenging to analyze due to their additional equations. By effectively grouping variables, we identify a 2-by-2 block structure, allowing the analysis of the well-posedness within the abstract LBB framework of saddle point problems. Due to the unique tensorial structure of the equations, in addition to an interesting combination of tools from Stokes' and linear elasticity theory, we also need new coercivity estimates for tensor fields. These Korn-type inequalities are established by analyzing the symbol map of the symmetric and trace-free part of tensor derivative fields. Together with the corresponding right inverse of the tensorial divergence, we obtain the existence and uniqueness of weak solutions.
Paper Structure (14 sections, 13 theorems, 77 equations, 1 figure)

This paper contains 14 sections, 13 theorems, 77 equations, 1 figure.

Table of Contents

  1. Introduction
  2. State of the art, context, and our contribution
  3. The connection to Stokes' and Poisson's problem
  4. Additional notation
  5. A mixed formulation for the linear R13 equations
  6. Coercivity and Korn-type inequalities
  7. Tensor-valued Korn inequality
  8. Right-inverse of the matrix-valued divergence operator
  9. Existence of unique weak solutions for the linear R13 equations
  10. Coercivity on the kernel
  11. Sup-condition
  12. Proof of the main result
  13. Conclusion and future work
  14. Continuity of the bilinear forms

Key Result

Lemma 3.1

Let $d\ge2$, $\Omega\subset\mathbb{R}^d$ be a bounded Lipschitz domain, $q\in(1,\infty)$, $V$ and $\tilde{V}$ two finite-dimensional real inner product spaces, and $\mathbb{A}\coloneqq\sum_{j=1}^d\mathbb{A}_j\partial_j$ a linear homogeneous differential operator of first order with constant coeffici

Figures (1)

  • Figure 1: Visualization of the weak equation structure, in which for the first system (top), the two equations of the heat system only couple through the bilinear form $c(\boldsymbol{r},\boldsymbol{\sigma})$ to the three fluid equations. A reordering according to trivial diagonal terms yields the saddle point structure in the second system (bottom).

Theorems & Definitions (32)

  • Remark 2.1
  • Lemma 3.1: Aronszajn aronszajnCoerciveIntegrodifferentialQuadratic1955, Korn inequalities of the first type
  • Example 3.2
  • Example 3.3
  • Remark 3.4: Legendre--Hadamard ellipticity
  • Lemma 3.5: Nečas necasNormesEquivalentesDans1966, Korn inequalities of the second type
  • Example 3.6
  • Remark 3.7
  • Lemma 3.8
  • Proof 1
  • ...and 22 more