First-Order Transport and Characteristics

Definition Let $c$ be a real constant. The transport equation with speed $c$ is

posed for all real $x$ and $t \ge 0$ with the initial condition $u(x, 0) = f(x)$. If $f$ is continuously differentiable on $\mathbb{R}$, this problem has exactly one continuously differentiable solution,

It is a solution because $u_t = -c\,f'(x - ct)$ and $u_x = f'(x - ct)$. It is the only one because, for any continuously differentiable solution, the function $t \mapsto u(\xi + ct, t)$ has the derivative $u_t + c\,u_x$ evaluated on the line $x = \xi + ct$, which is zero, so $u$ keeps the value $f(\xi)$ along that whole line. The sign of $c$ gives the direction. For $c > 0$ the profile moves right, and for $c < 0$ it moves left.

Definition Consider a linear first-order equation

where $a$ and $q$ are continuous and $a$ is continuously differentiable in $x$. A characteristic curve is the graph in the $(x, t)$ plane of a solution $x(t)$ of the ordinary differential equation

Along such a curve the chain rule gives

The method of characteristics solves the path equation with $x(0) = \xi$, integrates the value equation starting from $u = f(\xi)$ at $t = 0$, and finally expresses $\xi$ in terms of the point $(x, t)$. The hypothesis on $a$ makes the characteristic through each point unique, so two characteristics of a linear equation never cross. This determines $u$ at every point whose characteristic, traced backward, reaches the initial line without leaving the region where the problem is posed.

Definition Pose the transport equation $u_t + c\,u_x = 0$ with a constant $c \ne 0$ on the quarter plane $x \ge 0$, $t \ge 0$, with initial data $u(x, 0) = f(x)$ for $x \ge 0$. If $c > 0$, the characteristics $x = \xi + ct$ move right and enter the region through the edge $x = 0$. That edge is an inflow boundary, and one condition $u(0, t) = g(t)$ is prescribed there. The solution is $u = f(x - ct)$ in the region $x \ge ct$, whose characteristics start on the initial line, and

in the region $0 \le x < ct$, whose characteristics start on the boundary at the time $t - x/c$. If $c < 0$, every characteristic through a point of the quarter plane traces back to the initial line, the values $u(0, t) = f(-ct)$ are already determined, and $x = 0$ is an outflow boundary where no independent condition can be imposed. On an interval $0 \le x \le L$ with $c > 0$, data go at the left end only. The two pieces join continuously across the line $x = ct$ exactly when $g(0) = f(0)$. If $f$ and $g$ are continuously differentiable, the solution is continuously differentiable exactly when also

which is the transport equation evaluated at the corner, where $u_t = g'(0)$ and $u_x = f'(0)$.

Definition In the equation $u_t + a(x, t)\,u_x = 0$, with a speed that depends on position or time and is continuous and continuously differentiable in $x$, the characteristics solve $dx/dt = a(x, t)$ and are curves rather than parallel lines. The value $u$ is still constant along each of them, so $u(x, t) = f(\xi)$ at every point whose characteristic traces back to a starting point $\xi$ on the initial line. Where characteristics spread apart the profile is stretched, and where they draw together it is compressed, but every value it carries is kept. This advective form differs from the conservation law $\rho_t + (a\rho)_x = 0$ for a density $\rho$, which expands to $\rho_t + a\rho_x = -a_x\rho$ and changes the carried value wherever $a_x \ne 0$.

Definition A source term is a right side $q$ in the equation

with $q$ continuous. Along a characteristic $x(t)$ the carried value $U(t) = u(x(t), t)$ is no longer constant but obeys the ordinary differential equation

When $q$ does not depend on $u$, integration gives the carried value at time $t$ as the starting value plus the accumulated source,

with $q$ evaluated at $(x(\tau), \tau)$ along the characteristic. The speed decides where information travels, and the source decides how the carried value changes on the way.

Definition The inviscid Burgers’ equation is

a transport equation whose speed is the unknown itself. For smooth solutions it can be written in the conservation form $u_t + \left(\tfrac{1}{2}u^2\right)_x = 0$. Take $u(x, 0) = f(x)$ with $f$ continuously differentiable. Each characteristic carries the constant value $f(\xi)$ and therefore moves on the straight line

While these lines do not cross, the solution is classical, with $u_x = f'(\xi)/(1 + f'(\xi)\,t)$ at the point reached by the line from $\xi$. If $f' \ge 0$ everywhere, they never cross, and a classical solution exists for all $t \ge 0$. If $\inf f' = -m$ with $0 < m < \infty$, the largest interval $[0, T)$ on which a classical solution exists has

and $u_x$ becomes unbounded as $t$ increases to $T$. A weak solution of the conservation form with initial data $f$ is a locally bounded function $u$ with

for every continuously differentiable $\varphi$ that vanishes outside a bounded set, where $Q$ is the half plane $t > 0$. Classical solutions are weak solutions, and weak solutions may contain jumps. Across a jump along a smooth curve $x = s(t)$, with the value $u_-$ on its left and $u_+$ on its right, the weak form forces the Rankine-Hugoniot condition

Weak solutions with the same data need not be unique. An admissibility condition, such as the entropy condition $u_- > u_+$ that the limit of vanishing viscosity satisfies, selects the physical one.