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  "body": "\n## Problem\n\nLet $F \\colon \\R^m \\to \\R^m$ be a $C^1$ function and consider the system of [[conservation laws|8RA7AMWSAF]] $u_t + F(u)_x = 0$ in one space dimension.\nLet $u_l, u_r \\in \\R^m$ be two constant states.\nFind an [[entropy solution|1A01A3EJRX]] $u \\colon \\R \\times (0, \\infty) \\to \\R^m$ with the initial values\n$$u(x, 0) = \\begin{cases} u_l \u0026 \\text{if } x \u003c 0, \\\\ u_r \u0026 \\text{if } x \u003e 0. \\end{cases}$$\n\n## Remarks\n\nThe equation and the data are unchanged by the scaling $(x, t) \\mapsto (\\lambda x, \\lambda t)$ for $\\lambda \u003e 0$, so one looks for self-similar solutions $u(x, t) = v(x / t)$.\n\nScalar case with convex flux.\nLet $m = 1$ and let $F$ be smooth and uniformly convex, and let $G$ be the inverse function of $F'$.\nIf $u_l \u003e u_r$, the entropy solution is the [[shock wave|JSDMGGTG7G]] equal to $u_l$ for $x \u003c \\sigma t$ and to $u_r$ for $x \u003e \\sigma t$, where $\\sigma = \\dfrac{F(u_l) - F(u_r)}{u_l - u_r}$.\nIf $u_l \u003c u_r$, the entropy solution is the [[rarefaction wave|MFN4Q5SH8H]] equal to $u_l$ for $x \u003c F'(u_l) t$, to $G(x/t)$ for $F'(u_l) t \\le x \\le F'(u_r) t$, and to $u_r$ for $x \u003e F'(u_r) t$.\n\nSystems.\nConsider a strictly hyperbolic system whose characteristic fields are each genuinely nonlinear or linearly degenerate.\nLax showed that its Riemann problem has a unique solution made of at most $m$ elementary waves when $\\abs{u_l - u_r}$ is small.\nThese waves are shocks, rarefaction waves and contact discontinuities, one for each characteristic family, separated by constant states.\nFor large data, or for more general systems, solutions may fail to exist or to be unique in this class.\nFor the Euler equations of gas dynamics, the problem posed by Riemann, the solution is known explicitly up to solving one scalar equation, and it may contain a vacuum.\n\nSolutions of Riemann problems are the building blocks of [[Godunov's scheme|43HZJ7DRT0]] and of the [[Glimm scheme|Y70AC2X30T]].\n\n## Explanations\n\n### L1 human\n\nPut a wall in a tube, with gas at one pressure on the left and another pressure on the right, and remove the wall at time zero.\nThe Riemann problem asks what happens next.\nWaves leave the place of the wall: some are sharp jumps, others are smooth fans that spread out.\nBecause the start has no length scale, the picture at time $2$ is the picture at time $1$ stretched by a factor $2$.\n",
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