{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\Omega$ be an open set in $\\R^n$ and let $m \\ge 1$.\nA [[partial differential equation|GC5BDZ6QBC]] of [[order|E7JDFDPM1X]] $m$ in $\\Omega$ is *quasilinear* if it has the following form.\n$$\\sum_{\\abs{\\alpha} = m} a_\\alpha\\left(D^{m-1} u, \\dots, D u, u, x\\right) \\partial^\\alpha u + a_0\\left(D^{m-1} u, \\dots, D u, u, x\\right) = 0.$$\nHere the functions $a_\\alpha$ and $a_0$ are given.\nHere $\\alpha$ runs over [[multi-indices|0PJXBR6QA3]], and $D^j u$ denotes the collection of all partial derivatives of order $j$.\nSo the equation is affine in the derivatives of order $m$, and its coefficients depend at most on the derivatives of order less than $m$.\n\n## Remarks\n\nExample: the minimal surface equation $\\operatorname{div}\\left(\\nabla u / \\sqrt{1 + \\abs{\\nabla u}^2}\\right) = 0$ in two variables expands to $(1 + u_y^2) u_{xx} - 2 u_x u_y u_{xy} + (1 + u_x^2) u_{yy} = 0$.\nThe coefficients of $u_{xx}$, $u_{xy}$, $u_{yy}$ depend only on first derivatives, so it is quasilinear of order $2$.\nNon-example: the Monge–Ampère equation $u_{xx} u_{yy} - u_{xy}^2 = f(x, y)$ is not quasilinear, because it contains products of second derivatives; it is [[fully nonlinear|2D3N1ZJWBW]].\nEvery [[semilinear PDE|RCNY89A4NY]] is quasilinear.\nFor first-order equations, quasilinear means $\\sum_{i} a_i(u, x) \\partial_i u + a_0(u, x) = 0$.\n\n## Explanations\n\n### L1 human\n\nQuasilinear means linear in the highest derivatives, once the solution and its lower derivatives are frozen.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "7X9N9H3RK6",
    "kind": "definition",
    "msc": [
      "35A99",
      "35J62",
      "35K59"
    ],
    "name": "Quasilinear PDE",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Nonlinear_partial_differential_equation"
      }
    ],
    "tag": "DV92N2YWZJ",
    "uses": [
      "GC5BDZ6QBC",
      "E7JDFDPM1X",
      "0PJXBR6QA3"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/DV92N2YWZJ/",
  "links": [
    {
      "field": "defines",
      "target": "7X9N9H3RK6"
    },
    {
      "field": "uses",
      "target": "GC5BDZ6QBC"
    },
    {
      "field": "uses",
      "target": "E7JDFDPM1X"
    },
    {
      "field": "uses",
      "target": "0PJXBR6QA3"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/DV92N2YWZJ.md",
  "msc": [
    "35A99",
    "35J62",
    "35K59"
  ],
  "name": "Quasilinear PDE",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/DV9/DV92N2YWZJ.md",
  "status": null,
  "tag": "DV92N2YWZJ",
  "uri": "m2://DV92N2YWZJ",
  "url": "https://tamnd.github.io/math2/wiki/quasilinear-pde-definition/",
  "writing": {
    "tag": "DV92N2YWZJ",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}