transportoperators

Transport operators are mathematical tools used to describe the movement of a quantity along a prescribed velocity field. They appear in several related settings, including partial differential equations, dynamical systems, and measure transport.

In the PDE setting, a common form is the first-order operator L_b acting on a function u(x,t) by (L_b u)(x,t) = ∂_t u(x,t) + b(x,t)·∇u(x,t), where b(x,t) is a velocity field. Equations of the form ∂_t u + b·∇u = 0 are called transport (or advection) equations; their solutions propagate along characteristics defined by dx/dt = b(x,t). Alternatively, for a fixed time, the action of transporting a spatial field along the flow Φ_t of b is given by the composition operator (T_t u)(x) = u(Φ_t(x)). The family {T_t}_{t≥0} forms a semigroup when Φ_0 is the identity and Φ_{t+s} = Φ_t∘Φ_s. In measure terms, transporting mass along the flow is described by the push-forward μ_t = Φ_t# μ_0.

Applications include modeling pollutants in air or water, heat transfer, population movement, and traffic flow. In numerical analysis, advection problems require stable schemes (for example, upwind or Godunov methods) that respect the CFL condition to prevent nonphysical oscillations. Related concepts include the advection equation, the material derivative, and transport maps in optimal transport theory.