{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "441JWPAV3W"
    },
    {
      "field": "defines",
      "target": "RGWYQCS5A1"
    }
  ],
  "body": "\n## Summary\n\nA mild solution of a linear evolution equation $u' = A u + f$ in a [[Banach space|06GZ956657]] is a continuous function.\nIt is given by the variation of constants formula $u(t) = T(t) u_0 + \\int_0^t T(t - s) f(s) \\, ds$, where $T(t)$ is the [[C0-semigroup|EVF180QJXY]] generated by $A$.\nIt needs no differentiability in time and no assumption that $u_0$ lies in the domain of $A$.\nEvery classical solution is a mild solution, so mild solutions extend the notion of solution to rough data.\nMany semilinear parabolic and hyperbolic equations are solved by finding a mild solution as a fixed point of this integral formula.\n\n## Explanations\n\n### L1 human\n\nFor a scalar equation $u' = a u + f(t)$, the solution is $u(t) = e^{at} u_0 + \\int_0^t e^{a(t - s)} f(s) \\, ds$.\nA partial differential equation such as the heat equation can be seen as such an equation, with $a$ replaced by an operator and $e^{ta}$ by a semigroup of operators.\nThe same formula then makes sense even when the initial state is too rough to be differentiated.\nThe function it defines is called the mild solution.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "47D06",
      "35K90",
      "34G10"
    ],
    "name": "Mild solution",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/C0-semigroup"
      }
    ],
    "tag": "3VBRS94V1G"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/3VBRS94V1G/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/3VBRS94V1G.md",
  "msc": [
    "47D06",
    "35K90",
    "34G10"
  ],
  "name": "Mild solution",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3VB/3VBRS94V1G.md",
  "status": null,
  "tag": "3VBRS94V1G",
  "uri": "m2://3VBRS94V1G",
  "url": "https://tamnd.github.io/math2/wiki/mild-solution/",
  "writing": {
    "tag": "3VBRS94V1G",
    "kind": "concept",
    "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
      }
    ]
  }
}