= Solution
A <characteristic curve> has $x-at$ constant. Backtracking reaches the initial line if $x\ge at$ and the inflow boundary otherwise. Hence
$$
\boxed{u(t,x)=\begin{cases}u_0(x-at),&x\ge at,\\ f(t-x/a),&0\le x<at.\end{cases}}
$$
Values on the dividing <characteristic curve> are immaterial for bounded <weak solutions>.
For a <classical solution> on the closed quadrant, choose $u_0,f\in C^1([0,\infty))$ and impose the <corner compatibility for constant-speed transport>
$$
\boxed{f(0)=u_0(0),\qquad f'(0)=-a u_0'(0).}
$$
These make the values and both first derivatives agree across $x=at$; each branch solves the <transport equation>. They are also necessary for a $C^1$ solution up to the initial and inflow boundaries.
Spatial $C^\infty$ regularity for every $t\ge0$, up to $x=0$, holds precisely for smooth data with all matching jets:
$$
\boxed{f^{(n)}(0)=(-a)^n u_0^{(n)}(0)\quad(n=0,1,2,\ldots).}
$$
Indeed the spatial derivatives of order $n$ at $x=at$ are $u_0^{(n)}(0)$ and $(-1/a)^n f^{(n)}(0)$. Necessity of smooth $u_0$ follows at $t=0$; smoothness of $f$ on any finite interval follows by reading the boundary branch in a spatial slice at a larger time.
Back to article page
