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  "body": "\n## Definition\n\nLet $n \\ge 1$, let $f \\colon \\R \\to \\R^n$ be [[continuously differentiable|0VW90BBSJ4]], and let $g \\colon \\R^n \\to \\R$ be bounded and measurable.\nConsider the scalar conservation law\n$$u_t + \\operatorname{div} f(u) = 0 \\text{ in } \\R^n \\times (0, \\infty), \\qquad u(x, 0) = g(x).$$\nA bounded measurable function $u \\colon \\R^n \\times (0, \\infty) \\to \\R$ is an *entropy solution* in the sense of Kruzhkov if the following holds.\nFor every real number $k$ and every nonnegative test function $\\varphi \\in C_c^\\infty(\\R^n \\times \\R)$,\n$$\\int_0^\\infty \\!\\! \\int_{\\R^n} \\Big( \\abs{u - k} \\, \\varphi_t + \\sgn(u - k) \\big( f(u) - f(k) \\big) \\cdot \\nabla_x \\varphi \\Big) \\, dx \\, dt + \\int_{\\R^n} \\abs{g - k} \\, \\varphi(x, 0) \\, dx \\ge 0.$$\n\n## Remarks\n\nTake $k$ larger than $\\sup \\abs{u}$ and $\\sup \\abs{g}$, and then smaller than $-\\sup \\abs{u}$ and $-\\sup \\abs{g}$.\nThis gives two opposite inequalities whose combination says that $u$ is a [[weak solution|SJZX81A783]] of the initial value problem.\nSo every entropy solution is a weak solution.\nKruzhkov's theorem states that for each bounded measurable $g$ there is exactly one entropy solution, and that it depends continuously on $g$ in $L^1_{\\mathrm{loc}}$.\nLet $u$ be piecewise smooth in one space dimension, with $f$ strictly convex.\nThen the condition is equivalent to the Rankine–Hugoniot jump condition together with the Lax condition $u_- \u003e u_+$ across each shock.\nHere $u_-$ and $u_+$ are the values to the left and to the right of the shock.\n\nExample: for Burgers' equation, $n = 1$ and $f(u) = u^2/2$, with $g = 1$ for $x \u003c 0$ and $g = 0$ for $x \u003e 0$, the shock $u = 1$ for $x \u003c t/2$ and $u = 0$ for $x \u003e t/2$ is the entropy solution.\nIts speed $1/2$ is $(f(1) - f(0))/(1 - 0)$, as Rankine–Hugoniot requires, and $u_- = 1 \u003e 0 = u_+$.\n\nNon-example: with $g = 0$ for $x \u003c 0$ and $g = 1$ for $x \u003e 0$, the shock $u = 0$ for $x \u003c t/2$ and $u = 1$ for $x \u003e t/2$ is a weak solution but not an entropy solution, because $u_- = 0 \u003c 1 = u_+$.\nThe entropy solution is the rarefaction fan $u = 0$ for $x \u003c 0$, $u = x/t$ for $0 \\le x \\le t$, and $u = 1$ for $x \u003e t$.\n",
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