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  "body": "\n## Definition\n\nLet $F \\colon \\R \\to \\R$ be a $C^1$ function and consider the scalar [[conservation law|8RA7AMWSAF]] $u_t + F(u)_x = 0$ on $\\R \\times (0, \\infty)$.\nLet $u$ be a [[weak solution|SJZX81A783]] and let $C = \\set{(s(t), t) : t \\in I}$ be a curve, with $I$ an open interval and $s \\colon I \\to \\R$ of class $C^1$.\nSuppose that $u$ is $C^1$ on each side of $C$ near $C$.\nLet $u_l(t)$ and $u_r(t)$ be its one-sided limits from the left and from the right of $(s(t), t)$.\nThe curve $C$ is a *shock* of $u$ if $u_l(t) \\ne u_r(t)$ for all $t \\in I$, so that $u$ jumps across $C$.\nThe shock is *admissible*, or satisfies the *Lax entropy condition*, if\n$$F'(u_l(t)) \u003e s'(t) \u003e F'(u_r(t)) \\qquad \\text{for all } t \\in I.$$\n\n## Remarks\n\nBecause $u$ is a weak solution, the shock speed is forced by the [[Rankine–Hugoniot condition|S4XMYJ3V42]] $s'(t) = \\dfrac{F(u_l) - F(u_r)}{u_l - u_r}$.\nThe entropy condition says that characteristics run into the shock from both sides, so the shock does not create information.\nFor uniformly convex $F$ it is equivalent to $u_l \u003e u_r$.\nFor systems a shock of the $k$-th family is admissible when $\\lambda_k(u_l) \u003e s' \u003e \\lambda_k(u_r)$ together with conditions on the other eigenvalues.\n\nExample: for [[Burgers' equation|NEGHXN8RE1]], $F(u) = u^2/2$, the function $u = 1$ for $x \u003c t/2$ and $u = 0$ for $x \u003e t/2$ has an admissible shock along $s(t) = t/2$.\nCheck: the speed $1/2 = (\\tfrac12 - 0)/(1 - 0)$ satisfies the Rankine–Hugoniot condition, which makes $u$ a weak solution, and $F'(1) = 1 \u003e \\tfrac12 \u003e 0 = F'(0)$.\n\nNon-example: the function $u = 0$ for $x \u003c t/2$ and $u = 1$ for $x \u003e t/2$ is also a weak solution of Burgers' equation with a jump along $s(t) = t/2$.\nIt fails the entropy condition, since $F'(0) = 0 \u003c \\tfrac12$.\nThe admissible solution with these initial values is a [[rarefaction wave|MFN4Q5SH8H]] instead.\n",
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