Construction of convex kinetic entropies for nonlinear hyperbolic systems

Construct suitable convex kinetic entropies for nonlinear hyperbolic systems, including the compressible Euler equations, to enable entropy-based stability analysis of lattice Boltzmann methods.

Background

The paper proves weighted L2L^2-stability only for linear hyperbolic systems with periodic boundary conditions. For nonlinear systems, comparable weighted L2L^2-stability and convergence results remain difficult to establish. The authors discuss entropy-stable lattice Boltzmann approaches as an alternative, but these approaches depend on the availability of suitable convex kinetic entropies.

The paper explicitly identifies the construction of such entropies as unresolved for several systems, specifically including the compressible Euler equations. Resolving this issue would broaden the applicability of entropy-based stability methods to nonlinear hyperbolic systems.

References

These approaches, however, require suitable convex kinetic entropies, whose construction remains unresolved for several systems, , the compressible Euler equations.

— A Lattice Boltzmann Method with Adaptive Relaxation Parameter and Lax--Friedrichs-Type Equilibrium for Hyperbolic Systems  (2609.18177 - Zhang et al., 16 Sep 2026) in Remark following Theorem 4.3, Section 4 (Weighted $L^2$-stability)