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  "body": "\n## Statement\n\nLet $\\nu \u003e 0$.\nLet $u_0 \\colon \\R^3 \\to \\R^3$ be a [[smooth|0V33BNKZ9S]] vector field with [[divergence|0MWV6ZTK6S]] $\\nabla \\cdot u_0 = 0$.\nAssume that for every multi-index $\\alpha$ and every $K \u003e 0$ there is a constant $C_{\\alpha K}$ with $\\abs{\\partial^\\alpha u_0(x)} \\le C_{\\alpha K} (1 + \\abs{x})^{-K}$ for all $x \\in \\R^3$.\nThen there exist smooth functions $u \\colon \\R^3 \\times [0, \\infty) \\to \\R^3$ and $p \\colon \\R^3 \\times [0, \\infty) \\to \\R$ such that:\n\n1. $u$ and $p$ satisfy the [[Navier–Stokes equations|18KXZHVF4M]] $\\partial_t u + (u \\cdot \\nabla) u = \\nu \\Delta u - \\nabla p$ and $\\nabla \\cdot u = 0$ on $\\R^3 \\times [0, \\infty)$, with no external force.\n2. $u(x, 0) = u_0(x)$ for all $x \\in \\R^3$.\n3. There is a constant $E$ with $\\int_{\\R^3} \\abs{u(x, t)}^2 \\, dx \\le E$ for all $t \\ge 0$.\n\n## Remarks\n\nThe problem is open and is one of the seven Millennium Prize Problems of the Clay Mathematics Institute.\nThe official problem asks for a proof of this statement or of its analogue for periodic data on the torus, or for a counterexample showing a breakdown of smooth solutions.\nCharles Fefferman wrote the official problem description.\nSmooth solutions are known to exist for a short time for such data, and for all time when the data are small in a suitable norm.\nIn two space dimensions the corresponding statement is true.\nJean Leray proved that global weak solutions with finite energy exist, see [[existence of Leray weak solutions|DHHR5PZZ7E]].\nCaffarelli, Kohn and Nirenberg proved that the singular set of a suitable weak solution has one-dimensional parabolic Hausdorff measure zero.\n\n## Explanations\n\n### L1 human\n\nThe Navier–Stokes equations describe how water and air flow.\nThe question is whether a smooth, calm initial flow in all of space can develop an infinitely sharp feature in finite time.\nAn example of such a feature would be an infinite speed at a point.\nNobody knows, even though the equations are used every day in engineering.\nA proof either way earns a prize of one million dollars.\n",
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