Linear Boltzmann equation (${\partial }_{t}f+v\cdot {\nabla }_{x}f=K_{t}f-af$)

A linear kinetic transport equation with velocity-integral gain $K_t$ and nonnegative loss coefficient $a$. Its characteristic integral formulation combines the damped free-transport evolution with the Boltzmann Volterra operator. A uniform square-integrable velocity kernel yields bounded gain on $L_{x,v}^2$ and a globally convergent Volterra series for the linear Boltzmann equation on every finite time interval.

Nonnegative gain kernels and positive damping preserve pointwise order. The Volterra series for the linear Boltzmann equation has nonnegative terms when its initial datum is nonnegative. Applying this to the difference of two solutions proves comparison, with uniqueness identifying that difference with the positive series. This order property is distinct from positivity of a quadratic form on a Hilbert space.

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.

The factorial bound for a Volterra iterate gives convergence in the space of bounded strongly measurable $L^2$-valued functions on $[0,T]$. The sum solves $(I-\tau )f=F$, where $F=U_a(t,0)f_0$, and satisfies ${\sup }_{t\le T}\Vert f(t){\Vert }_2\le e^{CT}\Vert f_0{\Vert }_2$. Applying the same iterate bound to a difference $g=\tau g$ proves uniqueness on every finite interval.

Combine the damped free-transport evolution and the linear Boltzmann collision operator in a time-ordered Bochner integral. Its norm obeys $\Vert \tau f(t){\Vert }_2\le C{\int }_0^t\Vert f(s){\Vert }_{2}ds$. This reduces the characteristic integral equation to $(I-\tau )f=U_a(t,0)f_0$.

For nonnegative measurable $a$, define $(U_a(t,s)g)(x,v)=e^{-{\int }_s^{t}a(r,x-v(t-r),v)dr}g(x-v(t-s),v)$. The shear preserves Lebesgue measure and the multiplier lies in $[0,1]$, so this is a contraction on Lp spaces. Extended nonnegative integrals use $e^{-\infty }=0$. Characteristic local integrability gives the usual continuous trace at the starting time.

The gain term integrates over the incoming velocity. If $C^2=\int {\sup }_{t,x}|k(t,x,v,v_{\ast }){|}^{2}dvd{v}_{\ast }<\infty$, the Hilbert-Schmidt kernel bound at each $x$ gives $\Vert K_{t}g{\Vert }_{L_{x,v}^2}\le C\Vert g{\Vert }_{L_{x,v}^2}$, uniformly in time. The existence proof uses boundedness; kernel positivity is not needed for that estimate.

A bounded continuous factor $k_1$ and an integrable factor $k_2$ give a bounded velocity-integral gain on bounded continuous functions, with bound $\Vert Kf{\Vert }_{\infty }\le \Vert k_1{\Vert }_{\infty }\Vert k_2{\Vert }_1\Vert f{\Vert }_{\infty }$. The dominated convergence theorem proves joint continuity of the velocity integral. Combined with damped transport, the factorial bound for a Volterra iterate proves existence on arbitrary finite time slabs.