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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^3$ be an open set and let $T \u003e 0$.\nThe *incompressible Euler equations* are the system\n$$\\partial_t u + (u \\cdot \\nabla) u = - \\nabla p, \\qquad \\nabla \\cdot u = 0,$$\nfor an unknown [[vector field|0299VCW1TH]] $u \\colon \\Omega \\times (0, T) \\to \\R^3$, the velocity, and an unknown function $p \\colon \\Omega \\times (0, T) \\to \\R$, the pressure.\nHere $\\nabla p$ is the [[gradient|0ABK9V9Y64]] of $p$ in $x$, $\\nabla \\cdot u$ is the [[divergence|0MWV6ZTK6S]] of $u$, and $(u \\cdot \\nabla) u$ has components $\\sum_{j=1}^3 u_j \\partial_j u_i$.\nThe density is normalised to $1$.\nOn a bounded domain one adds the boundary condition $u \\cdot n = 0$, where $n$ is the outer unit normal.\n\n## Remarks\n\nExample: on $\\Omega = \\R^3$, the rigid rotation $u = (-y, x, 0)$ with pressure $p = (x^2 + y^2)/2$ is a solution.\nIndeed $\\nabla \\cdot u = 0$, $(u \\cdot \\nabla) u = (-x, -y, 0)$ and $-\\nabla p = (-x, -y, 0)$.\n\nNon-example: $u = (x, 0, 0)$ is not a solution for any pressure, because $\\nabla \\cdot u = 1 \\ne 0$.\n\nThe *compressible Euler equations* have the density $\\rho$ and the pressure as further unknowns:\n$$\\partial_t \\rho + \\nabla \\cdot (\\rho u) = 0, \\qquad \\partial_t (\\rho u) + \\nabla \\cdot (\\rho u \\otimes u) + \\nabla p = 0,$$\ntogether with an energy equation or an equation of state that relates $p$ to $\\rho$.\nThey form a system of hyperbolic conservation laws, whose solutions can develop shock waves.\n",
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