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

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.

💡 Research Summary

This paper establishes the well‑posedness of the linearized regularized 13‑moment (R13) equations, which are a higher‑order extension of the Navier–Stokes–Fourier system used to model rarefied gas flows. After a concise physical motivation—highlighting applications such as Knudsen pumps and micro‑channel flows—the authors present the governing equations: mass, momentum, and energy balance (1.1) together with evolution equations for the heat‑flux vector s, the deviatoric stress tensor σ, and the third‑order moments m, R, Δ (1.2). The closure relations (1.3) express the highest‑order moments in terms of lower‑order fields, yielding a closed system of thirteen unknowns.

Boundary conditions are taken from Onsager’s reciprocity principle (1.4). They couple normal fluxes (uₙ, σₙₜ, Rₙₜ, sₙ, mₙₙₙ) with thermodynamic driving forces (jumps of pressure, velocity, temperature) through a symmetric positive map involving the accommodation factor χ̃ and a parameter ε_w that controls the strength of velocity prescription. These conditions guarantee non‑negative entropy production and are compatible with the second law of thermodynamics.

The core of the analysis is a mixed variational formulation. The authors group the variables into two blocks: U = (σ, s, p) ∈ V and P = (u, θ) ∈ Q, where V = H¹(Ω; stf‑tensors) × H¹(Ω; ℝ³) × ˜H¹(Ω; ℝ) and Q = L²(Ω; ℝ³) × L²(Ω; ℝ). The spaces ˜H¹ incorporate the mean‑zero constraint when ε_w = 0. The weak problem reads: find (U,P) such that

 A(U,V) + B(V,P) = F(V) ∀ V ∈ V,
 B(U,Q) = G(Q)    ∀ Q ∈ Q,

with bilinear forms A and B defined in (2.6)–(2.8). The form A contains the symmetric gradient of s, the deviatoric part of Dσ, and various boundary contributions; B couples σ and s with the primal variables u and θ through divergence operators.

To prove existence and uniqueness, the authors place the problem in the abstract saddle‑point framework of Brezzi (the LBB condition). The main difficulty lies in establishing coercivity of A on the kernel of B, which requires new Korn‑type inequalities for tensor‑valued fields. Classical Korn’s inequality controls the H¹‑norm of a vector field by its symmetric gradient; here the authors need an estimate for the operator “stf D” acting on symmetric trace‑free tensors and for the third‑order tensor derivative. By analyzing the Fourier symbol of stf D, they prove that its kernel consists only of rigid‑body motions (which are eliminated by the boundary conditions), leading to the inequalities stated in Lemmas 3.9 and 3.12.

Furthermore, they construct a right inverse of the tensorial divergence (Div) that maps L²‑vector fields to H¹‑tensor fields, ensuring that the inf‑sup condition for B holds. With these tools, they verify the Brezzi conditions: (i) A is continuous and coercive on ker B, (ii) B satisfies the inf‑sup condition, and (iii) the linear functionals F and G are bounded. Consequently, Theorem 4.9 guarantees a unique weak solution (U,P) that depends continuously on the data (mass source m, body force b, wall data u_w, p_w, θ_w).

The paper concludes by discussing the implications for numerical analysis. The well‑posedness result provides the theoretical foundation for finite‑element, discontinuous‑Galerkin, and finite‑volume discretizations of the linear R13 system, ensuring stability and convergence of the schemes. Moreover, the tensor‑valued Korn inequalities introduced here are of independent interest and may be applicable to other problems involving incompatible tensor fields, such as micromorphic elasticity or gradient plasticity. Future work is suggested on extending the analysis to the fully nonlinear R13 equations, time‑dependent problems, and more general boundary conditions.