Uniqueness of DiPerna-Lions renormalized solutions of the hard-sphere Boltzmann equation

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Uniqueness of DiPerna-Lions renormalized solutions of the hard-sphere Boltzmann equation evidence for this event

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Uniqueness of DiPerna-Lions renormalized solutions of the hard-sphere Boltzmann equation parent of this event

VibeMathed record: Uniqueness of DiPerna-Lions renormalized solutions of the hard-sphere Boltzmann equation evidence for this event

Theorem 1.1 (principal): there are $R<\infty$ and a nonnegative $F_0$ with $\mathrm{supp}_vF_0\subset B_R(0)$ and $\int F_0(1+|v|^2+|\log F_0|)<\infty$ on $\mathbb T^3\times\mathbb R^3$ admitting two global entropy solutions with exact local conservation of mass, momentum and kinetic energy, differing on a set of positive measure, both in $C([0,\infty);L^1)$, with $\int_0^T\int\langle v\rangle^k(Q^++Q^-)<\infty$ for all $T,k$. The companion (September 23) proves nonuniqueness in a class with local mass and total momentum conservation. The datum has unbounded amplitudes at shrinking spatial scales. It does not give nonuniqueness for a dense or generic set of data (the Gismondi-Golding-Novack conjecture), nor for other kernels, the whole space, or the Landau equation. parent of this event

Silvestre, Regularity estimates and open problems in kinetic equations (2022) evidence for this event

Companion: Nonuniqueness for the periodic hard-sphere Boltzmann equation (Sept 23, 2026) evidence for this event

Lean (covers the September 23 companion's theorem): OAI/MathematicalPhysics/Boltzmann/Main.lean evidence for this event