Transport Equation
1 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 \(c\), the transport equation is given by
\[ \begin{cases} \partial_t u + c\cdot \nabla u = 0 \\ u(x,o) = u_0(x). \end{cases} \]
Solution: The equation says that \(u\) is constant along the lines \(x-ct = \text{constant}\) (the characteristics of the PDE), to the initial profile simple translates
\[ u(x,t) = u_0(x-ct). \]
The following are some generalizations of this simple case:
Variable velocity: \(\partial_t u + b(x,t) \cdot \nabla u = 0\).
Characteristics solve tdhe ODE \(\partial_t x = b(x,t)\), and \(u\) is constant along these characteristics.Conservative form: \(\partial_t\rho + \nabla \cdot (\rho b) = 0\). This it the continuity equation for a density \(\pho\) being advected by the field \(b\). It differs from the form above when \(\nabla \cdot b \neq 0\), since the density can then compress or expand.
Source / decay terms: \(\partial_t u + c \partial x u = f(x,t)\) or \(\partial_t u + c \partial_x + \lambda_u = 0\). Along characteristics, these reduce to ODEs.
Non-linear case: \(\partial t + c(u) \partial_x u = 0\). Characteristics can cross, producing shocks, which is where weak solutions and entropy conditions come in.
2 Calculus Formula for Transport Equation
For a vector \(\vec V\) and \(u=u(x)\) be a multivariable function with \(x \coloneqq (x_1, \ldots , x_n)\). Then the derivative of \(u\) along the vector \(\vec V\) is defined as
\[ \partial_{\vec V} \coloneqq \nabla u \cdot \vec V = \sum_{i=1}^n \partial_{x_i} u V_i. \]
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