Distributions and weak derivatives

Many functions that matter have corners or jumps (a shock, a plucked string, \( |x| \)), and some "functions" that matter are not functions at all (a point source). Distributions handle all of them by testing. A test function \( \varphi \) is infinitely differentiable and vanishes outside a bounded set. Every locally integrable \( f \) defines the functional \( \varphi \mapsto \int f\varphi\,dx \), and a distribution is any continuous linear functional on test functions. The Dirac delta \( \delta(\varphi) = \varphi(0) \) is one that comes from no function.

The definition of the derivative is forced by integration by parts. For smooth \( f \), \( \int f'\varphi = -\int f\varphi' \) because \( \varphi \) vanishes at the ends. So for any distribution \( T \) we define \( T'(\varphi) = -T(\varphi') \). Every distribution now has derivatives of every order. The Heaviside step \( H \) has \( H'(\varphi) = -\int_0^\infty\varphi' = \varphi(0) \), so \( H' = \delta \).

A function \( g \in L^1_{\text{loc}} \) is the weak derivative of \( f \) when \( \int f\varphi'\,dx = -\int g\varphi\,dx \) for every test function. The weak derivative of \( |x| \) is \( \operatorname{sign}(x) \), as the example finds: split the integral at 0 and integrate by parts on each side, and the boundary terms at 0 cancel because \( |x| \) is continuous. The weak derivative of \( \operatorname{sign}(x) \) is not a function; as a distribution it is \( 2\delta \), and the jump is exactly what produced it.

Picture it: you are not allowed to look at \( f \) directly, only at its blurred averages \( \int f\varphi \) through every possible smooth lens \( \varphi \). Sharp corners and spikes are invisible to each single lens, but the collection of all of them determines the object, and moving the derivative onto the lens costs nothing.

Think it: a weak solution of \( -\Delta u = f \) is a \( u \) with \( \int\nabla u\cdot\nabla\varphi = \int f\varphi \) for all test \( \varphi \). Existence of weak solutions follows from Hilbert space arguments (the Riesz representation theorem), and a separate regularity theory then asks whether they are smooth. That two-step strategy is exactly the shape of the Navier-Stokes problem: Leray proved in 1934 that weak solutions exist for all time, for all finite-energy divergence-free data. Whether smooth data always give smooth solutions is the open question; whether Leray's solutions are unique is also unsettled without a force (with a suitably chosen force, Albritton, Brué and Colombo showed in 2022 that uniqueness fails).