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arXiv · 2506.20775

The $\mathcal{M}$-Operator and Uniqueness of Nonlinear Kinetic Equations

Ricardo Alonso·Maria Pia Gualdani·Weiran Sun
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Abstract

We introduce an $\mathcal{M}$-operator approach to establish the uniqueness of continuous or bounded solutions for a broad class of Landau-type nonlinear kinetic equations. The specific $\mathcal{M}$-operator, originally developed in [3], acts as a negative fractional derivative in both spatial and velocity variables and interacts in a controllable manner with the kinetic transport operator. The novelty of this method is that it bypasses the need for bounds on the derivatives of the solution - an assumption typically required in uniqueness arguments for non-cutoff equations. As a result, the method enables working with solutions with low regularity.

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Ricardo Alonso, Maria Pia Gualdani, Weiran Sun. 2025-06-25. The $\mathcal{M}$-Operator and Uniqueness of Nonlinear Kinetic Equations. https://arxiv.org/abs/2506.20775

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