Propagation of half-derivative regularity for the Boltzmann equation

Let $s\in\left(0,\frac{1}{2}\right)$, let $f_0:\mathbb{R}^2\times\mathbb{R}^2\to\mathbb{R}$ satisfy the hypotheses of Theorem 1.1, and let $f(t)$ be the corresponding local mild solution on $[0,T_0)$. For any $T\in(0,T_0)$, the quantity

$$\left\Vert \left< v \right>^{\frac{1}{2}+}\left<\nabla_x\right>^{\frac{1}{2}+}f(t)\right\Vert_{L^\infty_{t\in[0,T]}L^2_{x,v}}$$

is finite.

Regularity propagation conjecture. The local solution $f(t)$ carries the $\frac{1}{2}+$ regularity of the initial data up to time $T_0$; in particular, for every $T\in(0,T_0)$,

$$\left\Vert \left< v \right>^{\frac{1}{2}+}\left<\nabla_x\right>^{\frac{1}{2}+}f(t)\right\Vert_{L^\infty_{t\in[0,T]}L^2_{x,v}}<\infty.$$

The authors can prove propagation of this regularity only for a time depending on the size of the initial $\frac{1}{2}+$ norm. The conjecture asserts persistence for a time depending instead on the lower-regularity $s$ norm appearing in the local well-posedness theorem.