MathOverflow Asked on December 18, 2021
There are two closely related concepts and I am not sure exactly how close. Consider a partial differential equation. (The coefficients need not be constant but assume they are functions, and not more general distributions):
A weak solution is a function that satisfies the weak form of the equation.
A distributional solution is a distribution that satisfies the equation itself, when derivatives are understood as distributional derivatives. (As Alexandre Eremenko remarks in a comment, this will not always make sense since you cannot always multiply distributions. But it does make sense in the case that the distribution is a function, which is the case I ask about.)
The second concept is actually broader since, for example the Dirac delta $delta$ is a distributional solution of the ordinary differential equation (in variable $x$) $xf’=-f$ but not a weak solution since it is not a function.
What I do not know is this: When a function is a distributional solution to an equation then is it always a weak solution? And vice versa?
I believe I can show the answer is yes in both cases for linear partial differential equations, by fairly direct calculation. But I fear I may be missing some nuance. And I am not at all sure how to proceed for non-linear PDEs.
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