Wasserstein gradient flows of Maximum Mean Discrepancy with energy kernels

We study the Wasserstein gradient flow of the squared Maximum Mean Discrepancy (MMD) generated by the nonsmooth energy kernels K(z)=-|z|^q, 0<q<2. In dimensions d\ge2, the corresponding energies are not displacement semiconvex, so standard Wasserstein-gradient-flow theory does not apply. When d+q-2>0, we prove global well-posedness on \mathbb{R}^d for probability densities in subcritical L^p spaces, with targets in the same integrability class and with finite moments. We also include the one-dimensional Coulomb endpoint d=q=1. For the associated N-particle system, we prove global noncollision and fixed-N convergence to the collision-free critical set, a particle-to-continuum criticality principle, and a modulated-energy mean-field estimate that yields convergence of the particle dynamics to the continuum flow as N\to\infty on every finite time interval. We also construct collision-free saddle equilibria, showing that deterministic particle trajectories need not approach global empirical minimizers. For 1\le q<2, every continuum solution in our class has a narrowly relatively compact orbit, every ω-limit point is Lagrangian critical, and the orbit approaches the Lagrangian critical set. For 0<q<1, the same conclusions hold under uniform-in-time moment and subcritical L^p bounds. We prove that an absolutely continuous Lagrangian critical point equals the target when the source and target have finite moments of order q, except when 0<q<1 and d\in{1,3}. Under the preceding uniform bounds, rigidity gives convergence of the continuum flow to the target throughout the rigid part of the well-posedness range. Finally, we show that no initial-data-independent multiplicative MMD decay modulus exists on \mathbb{R}^d, and that global Polyak--Łojasiewicz inequalities fail in several whole-space and periodic Riesz/Coulomb regimes.