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Remedy for non-ellipticity and ill-posedness in the mean-field limit of the 1D Lagrangian zonal-flow model

Determine a mathematically sound remedy for the lack of ellipticity and associated ill-posedness in the mean-field Fokker–Planck limit of the 1D Lagrangian SDE model for zonal flows, namely the equation ∂_t ω + α ∂_x(ω ∂_x ω) + ∂_x(ω β) = κ Δ ω whose effective diffusion term ∂_x((κ − α ω) ∂_x ω) loses ellipticity when ω(x) > κ/α; develop a formulation that remains well-posed while still allowing aggregation phenomena.

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Background

Passing to the mean-field limit of the proposed 1D Lagrangian SDE model yields a nonlinear Fokker–Planck equation with an effective diffusion operator ∂_x((κ − α ω) ∂_x ω). This operator is elliptic only when ω(x) ≤ κ/α, which suppresses clustering if enforced and leads to loss of ellipticity (and ill-posedness) otherwise.

Potential routes mentioned include retaining a finite mollification scale (avoiding the ε → 0 limit) or introducing a suitable stochastic forcing at the PDE level to recover regularity while enabling aggregation. A rigorous, principled remedy that ensures well-posedness without eliminating aggregation remains to be found.

References

However, this equation presents problems of well posedness, because the second order differential operator, written on the right-hand-side, takes the form

\partial_x ((\kappa- \alpha\omega)\partial_x\omega)

which is elliptic only when $\omega(x)\leq\kappa/\alpha$ for every $x$. If we start with an initial condition $\omega_0$ satisfying this inequality, one can prove that also the solution does, but this prevents the formation of clusters with increasing strength, the solution will simply decay. If we take an initial condition which violates this inequality, we have a problem of lack of ellipticity.

It is not clear yet how this problem can be remedied.

— Aggregation of vortex structures in 2D: the blob-wave system and its role in zonal flows (2505.22700 - Flandoli et al., 28 May 2025) in Section 5: A 1D Lagrangian phenomenological model of the zonal flows structure — Subsection “Difficulties with the mean field limit”