The specific proof may be in pure math but Navier-Stokes problems did originate in applied math. A lot of what we consider pure math has it’s origins in something applied. As one of my math professors implied, a lot of mathematicians liked to pretend they came up with theorems from scratch in their final published form, but in practice it was often from seeing examples, often practical ones in applied math, then making conjectures that eventually led to more generalized theorems.