Local linear transport has a unique solution from noncharacteristic Cauchy data

Apply [L2] to the space-time vector field $\tilde{a}((x,t),r):=(a(x,t),1)$ with initial time $r=0$. After shrinking near ${\eta }_{\ast }$, its flow $\mathrm{\Gamma }(s;z)$ is jointly continuous, $D_z\mathrm{\Gamma }=Y$ exists, and $Y^{\mathrm{\prime }}=D\tilde{a}(\mathrm{\Gamma })Y$ with $Y(0,z)=I_{n+1}$. The coefficient $D\tilde{a}(\mathrm{\Gamma }(s;z))$ is jointly continuous; applying [L6] to this linear matrix ODE, with $z$ as parameter, makes $Y$ jointly continuous. Also ${\mathrm{\partial }}_s\mathrm{\Gamma }=\tilde{a}(\mathrm{\Gamma })$ is jointly continuous. Thus $\mathrm{\Gamma }$ is $C^1$ in $(s,z)$. Since $\sigma$ is $C^1$, $\mathrm{\Phi }(s,\eta ):=\mathrm{\Gamma }(s;\sigma (\eta ))$ is $C^1$, and $\mathrm{\Phi }(0,\eta )=\sigma (\eta )$.

At $(0,{\eta }_{\ast })$, the $s$-derivative of $\mathrm{\Phi }$ is the transport vector ${\mathrm{\partial }}_s\mathrm{\Phi }(0,{\eta }_{\ast })=\tilde{a}(\sigma ({\eta }_{\ast }),0)=(a(x_{\ast },t_{\ast }),1)=B(\sigma ({\eta }_{\ast }))\mathrm{.}$ For each $j$, the ${\eta }_j$-derivative is ${\mathrm{\partial }}_{{\eta }_j}\mathrm{\Phi }(0,{\eta }_{\ast })={\mathrm{\partial }}_{{\eta }_j}\sigma ({\eta }_{\ast })$ because $\mathrm{\Phi }(0,\eta )=\sigma (\eta )$. Thus the columns of $D\mathrm{\Phi }(0,{\eta }_{\ast })$ are exactly the $n$ tangent vectors to the data surface together with the transport vector, so [L1] says that $D\mathrm{\Phi }(0,{\eta }_{\ast })$ is invertible.

By [L4], after shrinking domains there are neighborhoods $I$ of $0$, $V$ of ${\eta }_{\ast }$, and $W$ of $(x_{\ast },t_{\ast })$ such that $\mathrm{\Phi }:I\times V\to W$ is a $C^1$ diffeomorphism. Write $\mathrm{\Gamma }(\rho ;\sigma (\eta ))=(X(\rho ;\eta ),T(\rho ;\eta ))$. For each $\eta \in V$, [L3] gives the unique solution $z(\cdot ,\eta )$ of ${\mathrm{\partial }}_{s}z(s,\eta )+c(X(s;\eta ),T(s;\eta ))z(s,\eta )=f(X(s;\eta ),T(s;\eta )),z(0,\eta )=g(\eta ),$ namely $z(s,\eta )=e^{-{\int }_0^{s}c(X(\rho ;\eta ),T(\rho ;\eta ))d\rho }g(\eta )+{\int }_0^s{e}^{-{\int }_{\lambda }^{s}c(X(\rho ;\eta ),T(\rho ;\eta ))d\rho }f(X(\lambda ;\eta ),T(\lambda ;\eta ))d\lambda \mathrm{.}$ The $C^1$ integrands on the compact local interval may be differentiated in $s$ and $\eta$, so $z$ is $C^1$. Define $u:=z\circ {\mathrm{\Phi }}^{-1}$ on $W$.

Because $\mathrm{\Phi }(0,\eta )=\sigma (\eta )$, step 3.1 gives $u(\gamma (\eta ),\tau (\eta ))=g(\eta )$ for $\eta \in V$. Moreover $u(\mathrm{\Phi }(s,\eta ))=z(s,\eta )$, so [L5] and ${\mathrm{\partial }}_s\mathrm{\Phi }=(a,1)$ give ${\mathrm{\partial }}_{s}z(s,\eta )=u_t(\mathrm{\Phi }(s,\eta ))+a(\mathrm{\Phi }(s,\eta ))\cdot {\mathrm{\nabla }}_{x}u(\mathrm{\Phi }(s,\eta ))\mathrm{.}$ Substitution into the scalar ODE for $z$ proves the transport PDE throughout $W$. If two $C^1$ solutions shared the data, [L5] would restrict each one to the same scalar IVP on every characteristic, and uniqueness in [L3] would make them agree throughout $W$.