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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^n$ be an open set and consider a [[partial differential equation|GC5BDZ6QBC]] $P(u) = f$ in $\\Omega$.\nLet $X$ be a space of functions on $\\Omega$ in which solutions are first sought, such as a [[Sobolev space|M4CEZ5K9HP]] for [[weak solutions|SJZX81A783]].\nLet $Y \\subseteq X$ be a smaller space of smoother functions.\nA *regularity result* for the equation is a statement of this form.\nIf $u \\in X$ solves $P(u) = f$, and $f$ and the coefficients of $P$ have a given smoothness, then $u \\in Y$.\nThe conclusion may hold on all of $\\Omega$ or on every compact subset of $\\Omega$.\n*Regularity theory* is the study of such results and of the estimates of the norm of $u$ in $Y$ that go with them.\n\n## Remarks\n\nExample: let $u \\in H^1_{\\mathrm{loc}}(\\Omega)$ be a weak solution of $-\\Delta u = f$ with $f \\in L^2_{\\mathrm{loc}}(\\Omega)$.\nThen $u \\in H^2_{\\mathrm{loc}}(\\Omega)$, and more generally $f \\in H^k_{\\mathrm{loc}}$ gives $u \\in H^{k+2}_{\\mathrm{loc}}$; this is [[elliptic regularity|2PVQY9PCT4]] with $X = H^1_{\\mathrm{loc}}$ and $Y = H^2_{\\mathrm{loc}}$.\n\nNon-example: for the transport equation $u_t + u_x = 0$ in $\\R^2$, the function $u(x, t) = g(x - t)$ is a weak solution for every $g \\in L^2_{\\mathrm{loc}}(\\R)$.\nSo smooth data, here $f = 0$, do not force smooth solutions, and no regularity result of this kind holds; singularities travel along characteristics.\n\nInterior regularity concerns compact subsets of $\\Omega$; regularity up to the boundary also needs a smooth boundary and smooth boundary data.\n\n## Explanations\n\n### L1 human\n\nRegularity theory asks how smooth a solution must be, once it is known to exist in a weak sense.\nElliptic equations improve smoothness, while hyperbolic equations only carry the roughness of the data along.\n",
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