Well-posedness of the Kolmogorov-Fokker-Planck equation on bounded domains

Lisa Valentini

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arXiv · 2606.19198

Well-posedness of the Kolmogorov-Fokker-Planck equation on bounded domains

Lisa Valentini

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Abstract

We establish well-posedness of the stationary Kolmogorov equation on a bounded domain, with either spherical or Gaussian velocities, subject to either inflow boundary conditions or specular reflection. For the sake of completeness, we also include the problem on the torus, which has already been solved by Albritton, Armstrong, Mourrat, and Novack (2024). Although the natural trace estimate with weight $|n_x\cdot v|$ does not hold in general in either setting, we provide a natural functional framework in which the trace is defined via the transport operator. We prove a Poincar\'e-type inequality with trace, which is an essential step towards the well-posedness of the inflow problem without damping. Moreover, we obtain a one-sided energy estimate for the trace with weight $|n_x\cdot v|$, in which the outflow (resp. inflow) flux is bounded by the energy inside the domain and the inflow (resp. outflow) flux. Finally, for the bounded-velocity model, we complement the existing well-posedness theory for the stationary Kolmogorov equation with inflow conditions on the hypoelliptic boundary and Dirichlet conditions on the velocity boundary.

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Lisa Valentini. 2026-06-17. Well-posedness of the Kolmogorov-Fokker-Planck equation on bounded domains. https://arxiv.org/abs/2606.19198

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