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  "body": "\n## Definition\n\nLet $\\nu \\ge 0$ be a constant, the *viscosity*.\n*Burgers' equation* with viscosity $\\nu$ is the equation\n$$\\partial_t u + u\\, \\partial_x u = \\nu\\, \\partial_x^2 u$$\nfor an unknown function $u \\colon \\R \\times (0, \\infty) \\to \\R$, where $\\partial_t$ and $\\partial_x$ are [[partial derivatives|09EGC6F8PT]].\nFor $\\nu \u003e 0$ it is the *viscous* Burgers' equation.\nFor $\\nu = 0$ it is the *inviscid* Burgers' equation, usually written in the conservation form\n$$\\partial_t u + \\partial_x \\left( \\tfrac12 u^2 \\right) = 0.$$\n\n## Remarks\n\nThe inviscid equation is the scalar [[conservation law|8RA7AMWSAF]] with flux $F(u) = u^2/2$, which is uniformly convex.\nFor smooth $u$ the two forms agree, but for discontinuous weak solutions only the conservation form is meaningful, and it fixes the shock speed $(u_l + u_r)/2$.\nBy the method of characteristics, a smooth solution with initial value $g$ satisfies $u = g(x - ut)$; if $g$ is somewhere decreasing, the solution ceases to be smooth at time $1/\\max(-g')$.\n\nExample: $u(x, t) = x/(t + 1)$ solves the inviscid equation, and also the viscous one for every $\\nu$.\nCheck: $\\partial_t u = -x/(t + 1)^2$, $u\\, \\partial_x u = x/(t + 1)^2$, and $\\partial_x^2 u = 0$.\n\nNon-example: $u(x, t) = e^{-t} \\sin x$ solves the linear heat equation $u_t = u_{xx}$ but not Burgers' equation with $\\nu = 1$.\nIndeed for it $\\partial_t u - \\partial_x^2 u = 0$, while $u\\, \\partial_x u = e^{-2t} \\sin x \\cos x$ is not identically zero.\n",
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