Example worked problem (elliptic PDE: existence via Lax–Milgram; nonlinear: existence via monotone operator)
Navier-Stokes equations (one of the Millennium Prize problems) involve nonlinear convective terms. Functional analysis provides weak solutions (Leray), regularity theory, and the concept of attractors for dissipative PDEs.
. The linear models she relied on—which were only "first approximations"—are no longer enough . She must transition to nonlinear functional analysis Nonlinear functional analysis – Knowledge and References
The true power of both linear and nonlinear functional analysis lies in its ability to solve concrete problems in physics, engineering, and numerical computation. I. Partial Differential Equations (PDEs)