Transport Equation
1 What is the Transport Equation?
The transport equation is a first-order PDE that describes how a quantity is carried along by a flow without being created or destroyed. In its simplest form, with constant velocity \(c\in\mathbb{R}^n\), the transport equation with initial data \(u_0\) is given by \[ \begin{cases} \partial_t u + c\cdot \nabla u = 0, & t>0,\ x\in\mathbb{R}^n, \\ u(0,x) = u_0(x). \end{cases} \]
In one space dimension this reads \(u_t+cu_x=0\).
1.1 Derivative Along a Vector
Let \(\vec V\) be a vector and \(u=u(x)\) a differentiable function of \(x=(x_1,\ldots,x_n)\). The derivative of \(u\) along \(\vec V\) is \[ \partial_{\vec V}u : = \nabla u \cdot \vec V = \sum_{i=1}^n V_i\,\partial_{x_i} u. \]
The transport equation is therefore the statement that \(u\) has zero derivative along the space-time vector \((1,c)\), i.e. \(\partial_{(1,c)}u=0\) for \(u=u(t,x)\). See directional derivative.
2 Transport Equation Solution
If \(u_0\in C^1\), the unique \(C^1\) solution of the transport equation is \[ u(t,x)=u_0(x-ct). \]
Proof. Fix \(x_0\) and consider the line \(x(s)=x_0+cs\) in space. By the chain rule \[ \frac{d}{ds}u(s,x(s))=u_t+c\cdot\nabla u=0, \] so \(u\) is constant along the line, hence \(u(t,x_0+ct)=u(0,x_0)=u_0(x_0)\). Every point \((t,x)\) lies on exactly one such line, namely the one through \(x_0=x-ct\), which gives \(u(t,x)=u_0(x-ct)\) and shows uniqueness. Conversely this function satisfies the equation by the chain rule.
The lines \(x=x_0+ct\) are the characteristics of the PDE (see Method of Characteristics). The solution is the initial profile translated at speed \(c\) without changing shape. In the \((x,t)\) plane these are parallel lines of slope \(1/c\), and the value of \(u\) at a point is read off by following the characteristic back to \(t=0\).
Rescaled equation. For \(a\,u_t+b\,u_x=0\) with \(a\neq0\) divide by \(a\) to get speed \(c=b/a\), so \(u(t,x)=u_0\!\left(x-\tfrac{b}{a}t\right)\). For \(2u_t+3u_x=0\) the characteristics are \(x=\tfrac32t+x_0\).
3 Generalizations
3.1 Damping and Source Terms
For \(u_t+cu_x+\lambda u=f(t,x)\) the PDE becomes an ODE along each characteristic, with \(U(s)=u(s,x_0+cs)\): \[ U'(s)+\lambda U(s)=f(s,x_0+cs),\qquad U(0)=u_0(x_0). \] Solving with the integrating factor \(e^{\lambda s}\) and setting \(x_0=x-ct\), \[ u(t,x)=e^{-\lambda t}u_0(x-ct)+\int_0^t e^{-\lambda(t-s)}f\big(s,\,x-c(t-s)\big)\,ds. \]
In particular, pure decay (\(f=0\)) gives \(u=e^{-\lambda t}u_0(x-ct)\): the profile translates and its amplitude decays exponentially, and a source (\(\lambda=0\)) adds the accumulated contributions along the characteristic.
3.2 Variable Velocity
For \(u_t+b(t,x)\cdot\nabla u=0\) the characteristics solve the ODE \[ \dot x(s)=b(s,x(s)), \] and \(u\) is constant along each of them. They are no longer straight lines, and the solution is \(u(t,x)=u_0(X_0)\) where \(X_0\) is the starting point of the characteristic passing through \((t,x)\).
3.3 Conservative Form
\[ \partial_t\rho+\nabla\cdot(\rho b)=0 \] is the continuity equation for a density \(\rho\) advected by the field \(b\). Expanding the divergence, \[ \partial_t\rho+b\cdot\nabla\rho=-\rho\,\nabla\cdot b, \] so it differs from the form above when \(\nabla\cdot b\neq0\), since the density can then compress or expand.
3.4 Non-Linear Case
For \(u_t+c(u)u_x=0\) the characteristics are still straight lines, with slope determined by the data, \[ x=x_0+c(u_0(x_0))\,t, \] and \(u\) is constant on each, giving the implicit solution \(u=u_0\big(x-c(u)t\big)\). Characteristics can cross, producing shocks, at which point classical solutions fail and weak solutions and entropy conditions are needed.
4 Derivation From Conservation Laws
Let \(u(t,x)\) be the density of a quantity in a medium moving with velocity \(V\). Conservation of mass on an arbitrary region \(\Omega\) states that the rate of change of the amount in \(\Omega\) equals minus the flux through \(\partial\Omega\): \[ \frac{d}{dt}\int_\Omega u\,dx=-\int_{\partial\Omega}\vec J\cdot\vec n\,dS =-\int_\Omega\nabla\cdot\vec J\,dx, \] by the divergence theorem. Since \(\Omega\) is arbitrary, \[ u_t+\nabla\cdot\vec J=0. \]
- Pure transport: \(\vec J=u\vec V\) gives \(u_t+\nabla\cdot(u\vec V)=0\), which is \(u_t+\vec V\cdot\nabla u=0\) for constant \(\vec V\).
- Transport and diffusion: adding the diffusion flux \(-k\nabla u\) (Fourier’s law, see Heat Equation) gives \(\vec J=u\vec V-k\nabla u\) and so \[ u_t+\vec V\cdot\nabla u=k\,\Delta u \] for constant \(\vec V\).