Normalized velocity-reset collision operator ($Bg=\rho (g)M-g$)

Let $M\ge 0$ have velocity integral one, and $\rho (g)(x)=\int g(x,v)dv$. On phase-space $L^1$, the normalized relaxation collision operator is $B=P-I$, where $Pg=\rho (g)M$. It obeys $\Vert B\Vert \le 2$ because $\Vert Pg{\Vert }_1\le \Vert g{\Vert }_1$. It preserves mass at each spatial position. The corresponding linear Boltzmann equation combines this bounded collision operator with the free-transport semigroup.

For $M>0$ with integral one, the Cauchy-Schwarz inequality makes the mass functional bounded on the weighted Hilbert space. The vector $M$ has norm one, so the velocity-reset projection is the orthogonal projection onto its span. Consequently $L=\Pi -I$ is bounded and self-adjoint. This Hilbert geometry differs from the norm-one positive reset projection on phase-space $L^1$.

For the normalized velocity-reset collision operator on the weighted Hilbert space, $L=\Pi -I$ vanishes on the equilibrium direction and equals $-I$ on its orthogonal complement. The spectral gap is exactly one. Conservation of the projected component and the displayed energy identity give exact exponential decay of the remaining component; the full solution need not decay to zero.

For $E=L_{x,v}^1$, let $(\tau f)(t)={\int }_0^t{U}_{t-s}(P-I)f(s)ds$. The factorial bound for a Volterra iterate is $\Vert {\tau }^{n}f(t){\Vert }_1\le (2t{)}^n{\sup }_{s\le t}\Vert f(s){\Vert }_1/n!$. Thus the Neumann series converges in $C([0,T];E)$ for every finite $T$ and solves $f=U_{\cdot }{f}_0+\tau f$. The same estimate applied to a difference proves uniqueness among locally time-bounded integral solutions. The whole collision term remains inside this undamped integral, rather than absorbing loss into the free propagator.

For a nonnegative normalized velocity density $M$, $Pg(x,v)=M(v)\int g(x,w)dw$ is a positive norm-one projection on phase-space $L^1$. Normalization gives $P^2=P$. Its range consists of fields $a(x)M(v)$. This Banach-space projection need not be an orthogonal projection in unweighted $L^2$; no such assertion is needed for the $L^1$ collision estimates.