{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "HAZ0NKCAXS"
    },
    {
      "field": "uses",
      "target": "PPM7V7WWHQ"
    },
    {
      "field": "defines",
      "target": "P4RF8A1562"
    }
  ],
  "body": "\n## Summary\n\nBurgers' equation $u_t + u\\, u_x = \\nu\\, u_{xx}$ is the simplest equation that combines nonlinear transport with diffusion.\nFor $\\nu = 0$ it is the inviscid Burgers' equation, the model scalar [[conservation law|8RA7AMWSAF]], whose smooth solutions break down and form [[shock waves|JSDMGGTG7G]].\nFor $\\nu \u003e 0$ the [[Cole–Hopf transformation|VBMQ9SJ3DH]] turns it into the heat equation, so it can be solved exactly.\nIt is a standard test case for theory and for numerical methods in fluid dynamics.\nSee the [[definition|P4RF8A1562]].\n\n## Explanations\n\n### L1 human\n\nIn Burgers' equation each value of $u$ moves with speed equal to itself.\nSo high parts of a wave move faster than low parts, and the front of a wave gets steeper and steeper.\nWithout viscosity the profile eventually becomes vertical and a jump, a shock, appears.\nFor example, starting from $u(x, 0) = -x$, the solution is $u(x, t) = x/(t - 1)$, which becomes infinitely steep at time $t = 1$.\nWith a little viscosity the jump is smoothed out into a thin but steep layer.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "35Q35",
      "35L65",
      "35L67"
    ],
    "name": "Burgers' equation",
    "same_as": {
      "wikidata": "Q1014918",
      "wikipedia": "Burgers' equation"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Burgers%27_equation"
      }
    ],
    "specializes": [
      "8RA7AMWSAF"
    ],
    "tag": "NEGHXN8RE1"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/NEGHXN8RE1/",
  "links": [
    {
      "field": "specializes",
      "target": "8RA7AMWSAF"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/NEGHXN8RE1.md",
  "msc": [
    "35Q35",
    "35L65",
    "35L67"
  ],
  "name": "Burgers' equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NEG/NEGHXN8RE1.md",
  "status": null,
  "tag": "NEGHXN8RE1",
  "uri": "m2://NEGHXN8RE1",
  "url": "https://tamnd.github.io/math2/wiki/burgers-equation/",
  "writing": {
    "tag": "NEGHXN8RE1",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}