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  "body": "\n## Summary\n\nA weak solution of a [[partial differential equation|GC5BDZ6QBC]] is a function that satisfies the equation after multiplication by test functions and integration by parts.\nIn this form the derivatives fall on the test functions.\nIt need not be differentiable, and so it can describe shocks, corners and other rough behaviour.\nExistence theory usually first produces a weak solution, often in a [[Sobolev space|M4CEZ5K9HP]].\nThen regularity theory shows that it is smoother.\n\n## Explanations\n\n### L1 human\n\nA classical solution must have all the derivatives that appear in the equation.\nA weak solution only has to satisfy the equation \"on average\" against every smooth bump function.\nFor the transport equation $u_t + u_x = 0$, a moving step $u(x, t) = 1$ for $x \u003e t$ and $0$ for $x \u003c t$ is a weak solution.\nIt is not differentiable along the line $x = t$, but it describes a profile moving to the right with speed $1$.\n",
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