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  "body": "\n## Definition\n\nLet $n, m \\ge 1$ and let $F = (F_1, \\dots, F_n)$ with each $F_j \\colon \\R^m \\to \\R^m$ a $C^1$ function, called the *flux*.\nThe *system of conservation laws* with flux $F$ is the [[partial differential equation|GC5BDZ6QBC]]\n$$\\partial_t u + \\sum_{j=1}^n \\partial_{x_j} \\big( F_j(u) \\big) = 0$$\nfor an unknown $u \\colon \\R^n \\times (0, \\infty) \\to \\R^m$, where $\\partial_t$ and $\\partial_{x_j}$ are [[partial derivatives|09EGC6F8PT]].\nFor $m = 1$ it is called a *scalar conservation law*.\nThe components of $u$ are the *conserved quantities*.\n\n## Remarks\n\nThe name comes from the integral form.\nLet $u$ be a $C^1$ solution, and let $V \\subseteq \\R^n$ be a bounded region with smooth boundary and outer unit normal $\\nu$.\nThen\n$$\\frac{d}{dt} \\int_V u \\, dx = - \\int_{\\partial V} \\sum_{j=1}^n F_j(u)\\, \\nu_j \\, dS,$$\nby the divergence theorem, so the total amount of $u$ in $V$ changes only through the flux across $\\partial V$.\nThe integral form, or the equivalent weak form with test functions, still makes sense for discontinuous $u$, and it is the basis of weak solutions.\n\nExample: the inviscid [[Burgers' equation|NEGHXN8RE1]] $u_t + (u^2/2)_x = 0$ is a scalar conservation law with $n = m = 1$ and $F(u) = u^2/2$.\nThe Euler equations of gas dynamics are a system with $m = n + 2$ conserved quantities: mass, momentum and energy.\n\nNon-example: the equation $u_t + x\\, u_x = 0$ is not of conservation form with a flux depending on $u$ only.\nIndeed it equals $u_t + (x u)_x = u$, so the amount of $u$ is not conserved: $\\int_\\R u(x, t)\\, dx$ grows like $e^{t}$ for integrable solutions, as $u(x, t) = g(x e^{-t})$ shows.\n",
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