= Spatially nonintegrable forcing in transport
{title2=$h=x^2+v^2$}
Smooth forcing need not be integrable in phase space. For the <hyperbolic characteristic flow for an inverted oscillator>, the source $x^2+v^2$ accumulates as $q_t=\tfrac12\sinh(2t)(x^2+v^2)-(\cosh(2t)-1)xv$. Its smallest quadratic-form eigenvalue is $(1-e^{-2t})/2>0$ for $t>0$. Starting with a nonnegative integrable <Gaussian function>, the solution is therefore unbounded and outside every finite <Lp space> at positive times. Locally time-integrable $L^p$ forcing prevents this failure by the <Minkowski integral inequality>.
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