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

SIAM Journal on Applied Mathematics, vol. 86, pp. 2299–2320

Abstract

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.

Authors 3

1. Peter Lewintan

   Karlsruhe Institute of Technology

   Affiliation as printed

   Institute for Analysis, Karlsruhe Institute of Technology, Englerstr. 2, 76131 Karlsruhe, Germany

2. Lambert Theisen Aachen Corresponding author. Applied and Computational Mathematics

   RWTH Aachen University

   Affiliation as printed

   Corresponding author. Applied and Computational Mathematics, RWTH Aachen University, Schinkelstr. 2, 52062 Aachen, Germany

3. Manuel Torrilhon Aachen Applied and Computational Mathematics

   RWTH Aachen University

   Affiliation as printed

   Applied and Computational Mathematics, RWTH Aachen University, Schinkelstr. 2, 52062 Aachen, Germany