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  "body": "\n## Summary\n\nThe transport equation $u_t + b \\cdot \\nabla u = 0$ says that a quantity $u$ is carried along by a flow of velocity $b$ without changing.\nIt is the simplest first-order [[partial differential equation|GC5BDZ6QBC]], also called the advection equation.\nIts solutions are the initial profile moved along the flow.\nIt is the model problem for the [[method of characteristics|7XAPBCJK3B]] and for numerical schemes of hyperbolic equations.\nSee the [[definition|R3X5WFHV8P]].\n\n## Explanations\n\n### L1 human\n\nThink of dye in a river that flows at constant speed.\nIf the dye does not spread out, the picture of the dye at a later time is the first picture shifted downstream.\nThe transport equation is the formula for this.\nIn one dimension, with speed $c$, it reads $u_t + c\\, u_x = 0$, and its solutions are $u(x, t) = g(x - ct)$.\nFor example, a bump centred at $x = 0$ at time $0$ is centred at $x = 3c$ at time $3$.\n",
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