Generalization of the Wasserstein-to-dual-norm estimate for radius distributions

Establish a generalization of the standard Wasserstein duality estimate that controls differences of measures on spatial position and particle radius in the mixed dual norm associated with $H_x^1W_\\sigma^{1,\\infty}$, thereby relating the empirical-measure Wasserstein distance to the Dirichlet energy available for the concentration error.

Background

In the critical homogenization argument, the empirical particle measures are atomic in the radius variable, whereas the concentration error is controlled primarily through its Dirichlet energy. The usual Kantorovich–Rubinstein estimate involving a Lipschitz norm is therefore insufficient, and the authors invoke a stronger Wasserstein estimate for absolutely continuous measures in the spatial variable.

Because the regularized empirical measure remains singular in the radius variable, the authors cannot directly apply the cited estimate in the mixed space–radius setting. They instead prove a weaker auxiliary estimate based on transported characteristics. A direct generalization of the standard proof would provide a sharper and more natural connection between the Wasserstein control of particle distributions and the energy estimates for the concentration.

References

Clearly $f_\varepsilon{\rm app}$ is not even absolutely continuous with respect to the Lebesgue measure due to the Dirac measures in the distribution of the radii. However its marginal with respect to the space variable $\rho_\varepsilon{\rm app} := \int_{[0,\infty)} f_\varepsilon{\rm app} d \sigma$ is indeed uniformly bounded in $L\infty$. Since, we only test with Lipschitz functions with respect to the $\sigma$ variable, one could therefore hope to relate $\mathcal W_2(f_\varepsilon,f)$ to the norm $\|f_\varepsilon{\rm app}(t) - f(t)\|{(H1_xW\sigma{1,\infty})\ast}$. It seems not obvious how to generalize the proof of \Lemma 5.33 in this way, though.

— Reactive Flow around Spherically Evolving Particles in Critical and Supercritical Dilute Regimes  (2608.28489 - Eden et al., 28 Aug 2026) in Section 5, subsection “Proof of Proposition 5.1: Estimate of $u_\\varepsilon-u$”, paragraph beginning “Finally, we turn to $I_{6,4}$”