Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 8 / 2

Fix $t,x$. The Cauchy-Schwarz inequality in the integration variable $v_{\ast }$ givesIntegrate first in $v$, then in $x$. The uniform kernel hypothesis bounds the first factor after velocity integration by $C^2$, independently of $t,x$. HenceThe linear Boltzmann collision operator is linear by its integral definition, so this proves it is a bounded linear operator on $L_{x,v}^2$. The estimate is the Hilbert-Schmidt kernel bound applied at each spatial point. The printed real-kernel square is $|k{|}^2$ for a complex kernel. Measurability of the coefficients is understood so that the displayed integrals are defined.

Free transport equation is the measure-preserving shear $(x,v)\mapsto (x-vs,v)$. Its composition operator preserves the $L_{x,v}^2$ norm. Since $a\ge 0$, the damping multiplier has modulus at most one:Consequently the damped free-transport evolution is contractive:This needs no upper bound on $a$. Nonnegative integrals may be interpreted in the extended sense with $e^{-\infty }=0$; local integrability along characteristics additionally gives the usual continuous initial trace.

Denote the damped free-transport evolution from time $s$ to time $t$ byBy the argument in (b), $\Vert U_a(t,s)g{\Vert }_2\le \Vert g{\Vert }_2$. The integral operator is the Boltzmann Volterra operatorThe Minkowski integral inequality and (a) give the useful stronger pointwise boundTherefore the requested estimate isFor strongly measurable bounded $L^2$-valued $f$, these are Bochner integrals; their finite norm bounds establish existence of the integrals.

For fixed $t$, put $M_t={\sup }_{0\le s\le t}\Vert f(s){\Vert }_2$. A Cauchy-Schwarz inequality in time strengthens the organization of the estimate in (c):Start with $\Vert \tau f(t){\Vert }_2^2\le C^2{t}^2{M}_t^2$. If the printed bound holds for $n-1$, thenTaking square roots proves the iterated Cauchy-Schwarz bound for a Volterra operator:There is also a factorial bound for a Volterra iterate, obtained by iterating the unsquared integral estimate in (c):Its factor $t^n/n!$ is the volume of the time-ordered simplex $0<s_n<\cdots <s_1<t$. It is stronger than the printed estimate because ${\prod }_{j=1}^n{j}^2\ge {\prod }_{j=1}^n(2j-1)$.

Work in the Banach space ${\mathcal{B}}_T$ of bounded strongly measurable maps $[0,T]\to L_{x,v}^2$, with norm $\Vert g{\Vert }_{{\mathcal{B}}_T}={\sup }_{0\le t\le T}\Vert g(t){\Vert }_2$. Keeping actual representatives at every time matches the pointwise-in-time mild formulation. The Boltzmann Volterra operator is bounded on this space, and the factorial bound for a Volterra iterate givesThus the Volterra series for the linear Boltzmann equation converges in operator norm for every finite $T$, even when $CT\ge 1$. SetFor its partial sums, $(I-\tau ){\sum }_{n=0}^N{\tau }^{n}F=F-{\tau }^{N+1}F$. The remainder tends to zero by the factorial estimate. Hence $(I-\tau )f=F$, precisely the required characteristic integral equation.

The norm bound in (b) gives an explicit choice of the existence constant:Also $f(0)=f_0$, since every term with $n\ge 1$ vanishes at zero and the damping interval has length zero. This proves existence in the paper's weak, characteristic-integral sense. With merely measurable nonnegative $a$, that sense does not itself require a continuous initial trace; that trace follows under the additional local characteristic-integrability condition described in (b).

Let $f_1,f_2$ have the same initial data and satisfy the characteristic integral equation. Their difference $g=f_1-f_2$ satisfies $g=\tau g$, hence $g={\tau }^{n}g$ for every $n$.

Fix any $T_0<T$ and let $M={\sup }_{0\le s\le T_0}\Vert g(s){\Vert }_2<\infty$. The factorial bound for a Volterra iterate givesThe scalar factor tends to zero, so $g(t)=0$ throughout this interval. Since $T_0<T$ is arbitrary, the weak solution of the linear Boltzmann equation is unique on $[0,T)$. The same argument gives uniqueness at $T$ whenever solutions are defined there by the integral formula.