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

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.

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.