Dice Question Streamline Icon: https://streamlinehq.com

Well-posedness of the scaled kinetic equation

Prove well-posedness (existence, uniqueness, and regularity) for the scaled kinetic equation ∂t f^ε − μ ∇x·((∇x W_{f^ε}) f^ε) − Dx Δx f^ε = (1/ε^a)(λ ∇u·((∇u W_{f^ε}) f^ε) + Du Δu f^ε) + lower-order terms, in particular for the rescaled form given in equation (\eqref{eq:rescaled_kinetic}).

Information Square Streamline Icon: https://streamlinehq.com

Background

The kinetic equation governs the evolution of the one-particle distribution f(t,x,u) under anisotropic Gaussian-type repulsion with angular diffusion and spatial diffusion. Its rescaled form introduces a singular perturbation in ε that concentrates the angular dynamics near equilibria.

A rigorous well-posedness theory for this nonlocal, nonlinear kinetic equation would provide a solid foundation for the subsequent hydrodynamic limit and justify the formal manipulations used to derive the macroscopic equations.

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”