Well-Posedness of the Linear Regularized 13-Moment Equations Using Tensor-Valued Korn Inequalities

TL;DR

This paper proves the well-posedness of the linearized R13 moment model for rarefied gas flows using tensor-valued Korn inequalities and an abstract saddle point framework.

Contribution

It introduces a novel analysis approach combining tensorial Korn inequalities with saddle point theory to establish well-posedness of the R13 moment equations.

Findings

Established existence and uniqueness of weak solutions.

Developed new coercivity estimates for tensor fields.

Provided a foundation for future numerical analysis.

Abstract

In this paper, we finally prove 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 us to analyze well-posedness within the abstract LBB framework for 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. This result also serves as the basis for future numerical analysis of corresponding discretization schemes.