A sharp rigidity/flexibility threshold for the isotropic Landau equation

We establish a sharp rigidity/flexibility threshold for stationary solutions of the Krieger–Strain equation, an isotropic model of the Landau–Coulomb equation. For every $1<p<\frac{6}{5}$, we use Nash iteration to construct nontrivial, nonnegative solutions in $L^1({\mathbb{R}}^3)\cap L^p({\mathbb{R}}^3)$ with arbitrarily strong exponential localization. Conversely, every stationary weak solution in $L^{\frac{6}{5}}({\mathbb{R}}^3)$ is trivial, identifying $L^{\frac{6}{5}}({\mathbb{R}}^3)$ as a new sharp integrability threshold. To our knowledge, this is the first use of Nash iteration for a nonlinear equation from collisional kinetic theory. The construction is based on a high–high–low cancellation within the Krieger–Strain operator and suggests that such mechanisms may occur more broadly in kinetic theory. The construction must accommodate kinetic features unusual for the method including a strongly nonlocal collision operator; an equation fundamentally posed on the whole space—not the periodic box; and a positive scalar unknown. At this low level of regularity, the usual formulations of the collision operator are not a priori well-defined, so a central part of the problem is specifying in what sense the constructed objects solve the equation. We isolate the notion of mollifier confluence, a simple and canonical way to interpret a nonlinearity below naive thresholds related to multiplying distributions. We complement this definition with a systematic treatment of weak solution notions and several explicit formal computations and clarifying examples that may be of independent interest.