1.2 The transport equation
Theorem 1.2.1. Let \(c \in \mathbb R\) and \(f \in C^1(\mathbb R)\). The initial value problem \(u_t + cu_x = 0\), \(u(x, 0) = f(x)\), has the unique \(C^1\) solution \[ u(x, t) = f(x - ct) . \] More generally, \(u_t + cu_x = g(x, t)\) with \(g\) continuous and \(C^1\) in \(x\) has the unique solution \[ u(x, t) = f(x - ct) + \int _0^tg\bigl (x - c(t - s), s\bigr )\,ds . \]
Proof. Fix \((x, t)\) and set \(z(s) = u(x + c(s - t), s)\), the value of \(u\) along the line through \((x, t)\) with slope \(dx/ds = c\). By the chain rule, \(z'(s) = cu_x + u_t = g(x + c(s - t), s)\). So \(z(t) = z(0) + \int _0^tz'(s)\,ds\), that is \(u(x, t) = f(x - ct) + \int _0^tg(x - c(t - s), s)\,ds\). Any \(C^1\) solution must therefore be this function, which proves uniqueness. Conversely, the formula defines a \(C^1\) function, and differentiating under the integral sign shows it satisfies the equation and the initial condition. □
The lines \(x - ct = \text {const}\) are the characteristics. Along each one the homogeneous equation says that \(u\) is constant: the initial profile is carried to the right at speed \(c\) without change of shape. The inhomogeneous formula is Duhamel’s principle again: the source \(g\) at time \(s\) is transported from \(s\) to \(t\) and summed. The equation describes a pollutant carried by a river of speed \(c\), the density of cars in a traffic stream all moving at the same speed, or any quantity advected by a uniform flow.
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