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Justification of Assumption A (rigorous hydrodynamic limit regularity and convergence)

Establish that solutions of the scaled kinetic equation possess sufficient smoothness and compactness to validate all limiting procedures used in the hydrodynamic limit, thereby proving that Assumption A (smoothness and convergence needed for the limit) holds.

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Background

Assumption A posits the existence of sufficiently smooth solutions and strong convergences to justify the passage to the macroscopic limit. It underlies multiple steps in the derivation, including the identification of the limiting equilibria of the angular collision operator and the use of Generalized Collision Invariants.

A rigorous proof would likely require establishing uniform (in ε) a priori bounds, compactness, and convergence to Gibbs-type angular equilibria, enabling the rigorous derivation of the macroscopic system.

References

there are three outstanding open problems left out in this work, namely, the well-posedness of the kinetic equation, showing that Assumption \ref{as:A} holds, and obtaining an explicit lower bound for the operator $K$ eq:Ceta_coeff_porousmedium.

— Macroscopic effects of an anisotropic Gaussian-type repulsive potential: nematic alignment and spatial effects (2410.06740 - Merino-Aceituno et al., 9 Oct 2024) in Section “Conclusions and open questions”