= Solution
Write $X(t,a)=X(t;0,a)$ and $A(t,x)=X(0;t,x)$. If $u_0\in C^1(\mathbb R^d)$, the unique global <classical solution> is
$$
\boxed{u(t,x)=u_0(A(t,x)).}
$$
It is $C^1$, and the <chain rule> makes its value constant along every <characteristic curve>. The same calculation proves classical uniqueness. No boundedness of $u_0$ is needed for this classical statement.
For $u_0\in L^\infty$, a bounded <weak solution> on the positive half-line is defined by
$$
\int_0^\infty\!\int_{\mathbb R^d}u\bigl[\varphi_t+\operatorname{div}_x(F\varphi)\bigr]dx\,dt+\int_{\mathbb R^d}u_0(x)\varphi(0,x)dx=0
$$
for every compactly supported $C^1$ <test function> on $[0,\infty)\times\mathbb R^d$. The analogous backward-time identity defines the solution on all real time. The $u\operatorname{div}F$ term is essential: this is transport, not the conservative equation $u_t+\operatorname{div}(Fu)=0$.
The same inverse-flow formula defines the unique weak solution, with $\|u(t)\|_\infty=\|u_0\|_\infty$, equality almost everywhere, and a representative continuous into $L^1$ on each compact spatial set. For a direct uniqueness argument, change variables $x=X(t,a)$ in the weak identity and put $w(t,a)=u(t,X(t,a))$. Using $J_t=(\operatorname{div}F)(t,X)J$ gives
$$
\int w\partial_t(J\zeta)\,da\,dt+\int u_0(a)\zeta(0,a)da=0,
$$
initially for $C^1$ compactly supported $\zeta$. Since $J>0$ and $J,J_t$ are continuous on compact sets, approximation in the spatial parameter permits $\zeta=\eta/J$: both $\zeta$ and $\zeta_t$ are uniformly approximable there, which are the only <derivatives> left in this identity. Hence $\int w\eta_t+\int u_0\eta(0)=0$ and $w=u_0$ almost everywhere. This also verifies existence by reversing the argument, and avoids assuming second <derivatives> of the $C^1$ field. Local $L^1$ continuity follows by approximating $u_0$ by continuous functions on compact flow tubes.
This is the <weak transport solution under characteristic coordinates>. The <weak-strong uniqueness principle> says that a weak solution in the specified class agrees with a classical solution having the same data for as long as that classical solution exists. Here it holds globally in the bounded class whenever the classical initial data are also bounded.
Back to article page
