Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 1 / 30A (Partial Differential Equations)

In phase-space coordinates $z=(x,v)$, the characteristic equations for a transport equation are the Hamilton equationsSmoothness of $H$ gives a unique local Hamiltonian flow ${\Phi }_t(\xi ,\eta )=(X(t),V(t))$. Along it the chain rule gives $d[f(X,V,t)]/dt=0$. ThusThe inverse is the backward autonomous flow, so this formula defines the unique local classical solution wherever the characteristic reaches the initial hypersurface. A smooth Hamiltonian on all of ${\mathbb{R}}^{2n}$ need not have a globally complete flow; the formula is understood on the interval and region where both flows exist. A common global time interval requires the corresponding completeness or growth assumption.

Every value of the solution is transported unchanged from its initial characteristic endpoint. Therefore the formula from (i) immediately givesMore generally the Hamiltonian transport equation preserves the range of the initial values while its characteristic solution exists: it neither creates negative values nor changes values by a source term.

The characteristic vector field $b=({\nabla }_{v}H,-{\nabla }_{x}H)$ is a divergence-free vector field:The Jacobian determinant $J_t=\det D{\Phi }_t$ obeys ${\dot{J}}_t=(\nabla \cdot b)({\Phi }_t)J_t=0$, with $J_0=1$. Thus the flow preserves phase-space volume, the Liouville theorem in Hamiltonian mechanics. Changing variables $(x,v)={\Phi }_t(\xi ,\eta )$ and using (i) givesThis applies on intervals where the flow is a diffeomorphism of the phase space and the integrals are defined; for finite conservation statements require integrability. For instance compactly supported initial data and $F(0)=0$ give the usual setting. An arbitrary smooth $F$ need not make these integrals finite on infinite-volume space, and merely smooth $H$ does not exclude escape of characteristics to infinity. Equivalently $F(f{)}_t+{\nabla }_z\cdot [bF(f)]=0$, with no flux through infinity, proves the same Casimir invariant of a transport equation conservation.

If $H=a(v)$, the characteristics have constant velocity variable $V(t)=\eta$ and $X(t)=\xi +t\nabla a(\eta )$. This flow is globally defined for every finite $t$ at each $(\xi ,\eta )$. ThereforeDirect differentiation gives $f_t+\nabla a(v)\cdot {\nabla }_{x}f=0$ and the initial value at $t=0$.