Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 6 / 1

The characteristic equations for a transport equation on phase space areA sufficient global hypothesis is that $F$ is continuous in time, locally Lipschitz continuous in $(x,v)$ uniformly on compact time intervals, and, for every finite $T$, satisfies $|F(t,x,v)|\le C_T(1+|x|+|v|)$ for $|t|\le T$. These hypotheses give unique characteristic curves for all real times if the field is defined for all real times. It suffices for forward existence to impose them on nonnegative time. One may strengthen the spatial hypothesis to $C^1$ when differentiating the characteristic flow map below. Local Lipschitz continuity alone guarantees only local existence; the linear-growth bound prevents finite-time escape.

Put $z=(X,V)$ and $b(t,z)=(V,F(t,X,V))$. On a small closed time interval and a closed ball around the initial point, let $M$ bound $b$ and let $K$ be its spatial Lipschitz constant. The mapsends the corresponding closed ball of continuous paths into itself when the time length times $M$ is at most the ball radius. If the time length times $K$ is less than one, it is a contraction in the supremum norm. The Banach fixed-point theorem gives a local solution and uniqueness; differentiating its integral equation gives the ordinary differential equation. Overlapping local solutions agree by this uniqueness.

For continuation, the growth assumption gives, on any bounded time interval,for forward time, with the analogous reversed-time bound. The Gronwall inequality bounds the trajectory on that interval. A finite terminal time is impossible: within the resulting compact ball the field is bounded, so the path has a limit at the endpoint, and the local construction restarts there. This proves the global characteristic flow under linear growth. No differentiability of the flow with respect to its initial point is needed for this existence and uniqueness proof.

For this force, $\dot{V}=V$, hence $V(t)=e^{t}v$ and $X(t)=x+(e^t-1)v$. Invert the characteristic flow map: the initial velocity for an endpoint $(x,v)$ is $e^{-t}v$, and the initial position is $x-(1-e^{-t})v$. Constancy along each characteristic curve givesThere is no amplitude factor here: this is scalar advection, not the conservative continuity equation for a compressible density.

Differentiating the explicit characteristic flow map gives the block triangular Jacobian matrixIts Jacobian determinant is the product of its diagonal block determinants. Therefore

Let $A(t)=D_{(x,v)}(X,V)$ along one characteristic curve. Differentiating the characteristic equation gives $\dot{A}=D_{z}b(t,X,V)A$, $A(0)=I_6$, with $b=(V,F)$. The Liouville formula for a fundamental matrix then gives the phase-space flow JacobianIndeed $\mathrm{tr}{D}_{z}b={\mathrm{div}}_{x,v}(v,F)={\nabla }_v\cdot F$, since $v$ is independent of $x$. Differentiation proves $\dot{J}=({\nabla }_v\cdot F)J$. If that divergence vanishes, $J(0)=1$ gives $\boxed{J\equiv 1}$ and the flow preserves Lebesgue measure on phase space.

The finite-$p$ assertion needs the zero-divergence hypothesis from the preceding part, which is not repeated in this part's printed hypotheses. The norm is on ${\mathbb{R}}_x^3\times {\mathbb{R}}_v^3$, and finite conservation requires $f_0$ to belong to the corresponding Lp space in addition to its smoothness.

Writing ${\Phi }_t$ for the characteristic flow map, the general solution is $f_t=f_0\circ {\Phi }_t^{-1}$. A change of variables givesThus the Lp conservation for incompressible transport isWithout that condition the requested conclusion is false: the force in part (c), together with any nonzero smooth integrable initial datum given by a Gaussian function, gives $\Vert f_t{\Vert }_p=e^{3t/p}\Vert f_0{\Vert }_p$. The essential supremum is still conserved by a complete invertible flow, because composition does not change the range of values.