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In this question, there is a nice intuitive explanation, why one uses weak solutions and what the intuition is behind the weak solutions: intuition behind weak solution

Is there a similar explanation for mild solutions?

Thank you in advance!

Best,

Luke

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Peter Wacken
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See Is a mild solution the same thing as a weak solution? , in which an example is given where a weak solution and mild solution for a particular problem coincide.

Mild solutions are weak solutions in the sense that they are not necessarily classical solutions. They are just a different way of getting a potential classical solution, or a substitute for a classical solution if none exist.

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  • There are probably also mild solutions, which are not weak solutions (I don't know an example, but otherwise there would only be one concept, I suppose). What is the difference of the two, if mild and weak solutions don't coincide? – Peter Wacken Feb 15 '17 at 20:19
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    It depends on what problem you're working on. Intuitively, it's a technicality. Probably the most common problem is uniqueness. E.g. $L^\infty$ weak solutions to the Navier--Stokes equations are nonunique but $L^\infty$ mild solutions are unique. In terms of definitions, mild solutions are solutions to integral equations, whereas weak solutions are distribution solutions to differential equations. – mathematician Feb 16 '17 at 12:51