Free transport equation (${\partial }_{t}f+v\cdot {\nabla }_{x}f=0$)

Particles keep their velocity and move along straight trajectories. The method of characteristics gives $f(t,x,v)=f_0(x-tv,v)$. The shear preserves Lebesgue measure and every Lp norm. Compact initial support remains compact at each finite time, but need not remain in one compact set for all times.

For the free transport equation, substitute $y=x-tv$ into $\int |f_0(x-tv,v)|dv$. The resulting factor $|t{|}^{-d}$ and the bound by ${\sup }_v|f_0(y,v)|$ prove this mixed Lebesgue norm estimate. The right side contains initial data. The same-time mixed norm is not a conserved replacement: prescribing a product $\psi (x)\chi (v)$ at a later time and dilating $\chi$ disproves such a claim.

A distributional weak solution of ${\partial }_{t}f+v\cdot {\nabla }_{x}f=g$, with locally integrable $f,g$, becomes ${\partial }_{t}F=G$ after the shear $F(t,x,v)=f(t,x+tv,v)$. For almost every characteristic label, there is an absolutely continuous function representing $F$, and its increments are the integrals of $G$ for every pair of times. Arbitrary modifications on time slices can destroy this every-time identity without changing the distributional solution.

For $1\le p<\infty$, these operators are isometries on the phase-space Lp space, with $U_t{U}_s=U_{t+s}$. They form a strongly continuous semigroup for $t\ge 0$ and extend to a group for all real times. Strong continuity follows by smooth compactly supported approximation and measure preservation. They preserve the essential supremum as well, but strong continuity is not claimed on the whole $L^{\infty }$ space.