Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 30B (Partial Differential Equations)

Write $z=(x,v)$ and $B(z)=(v,-\nabla V(x))$. The characteristic flow map ${\Phi }_t$ solves $\dot{z}=B(z)$. Its vector field is globally Lipschitz continuous, so Picard-Lindelöf theorem gives a unique global flow with inverse ${\Phi }_{-t}$. Also ${\mathrm{div}}_{z}B=0$.

The proposed Dirac measure is the unit mass at ${\Phi }_t(z_0)$. For every compactly supported smooth test function $\psi$,This is the weak formulation of $f_t+\mathrm{div}(Bf)=0$, equal to the Liouville equation because $\mathrm{div}B=0$. The initial unit mass is recovered at $t=0$. The characteristic equations are precisely $\dot{\hat{x}}=\hat{v}$, $\dot{\hat{v}}=-\nabla V(\hat{x})$; the time-derivative dots, lost in the converted text, are present in the PDF.

Along a characteristic curve, the Liouville equation gives $d[f({\Phi }_t(z_0),t)]/dt=0$. HenceComposition preserves nonnegativity, so $f\ge 0$. For locally integrable data this formula defines the transported weak solution, and the Lipschitz invertible flow preserves null sets locally. Thus the same assertion holds almost everywhere without requiring pointwise smooth data.

Formally the Jacobian determinant of the characteristic flow map satisfies $J^{\prime }=\mathrm{div}B({\Phi }_t)J=0$, $J(0)=1$, so $J=1$. Changing variables $z={\Phi }_t(z_0)$ givesTaking the $p$th root proves $\boxed{\Vert f(t){\Vert }_{L^p}=\Vert f_I{\Vert }_{L^p}}$ for $1\le p<\infty$. Equivalently multiply the equation by $p{f}^{p-1}$ and integrate the divergence term, with vanishing boundary terms. Volume preservation gives the result without needing such boundary decay for general integrable data.

For the quadratic potential the characteristic curves are harmonic oscillations:The inverse is $x_0=x\cos t-v\sin t$, $v_0=v\cos t+x\sin t$. Therefore the Liouville equation has solutionThis formula works for smooth data or by transport for locally integrable data, and for measures by pushforward. Although the quadratic $V$ itself is not globally Lipschitz as in the initial sufficient hypotheses, its gradient is globally Lipschitz, which is the property actually needed for this global flow.