Beyond Lagrangians: Noether's theorem in gradient flow PDEs

Noether's theorem is traditionally applied to Lagrangian systems to identify conserved quantities. In this work, we apply Noether's theorem instead to a broad class of non-Lagrangian gradient flow partial differential equations (PDEs) arising in physics, showing how continuous symmetries constrain the evolution of such systems and, in certain special cases, still give rise to conserved quantities. We demonstrate symmetry-induced evolutionary constraints numerically on the thin-film equation with capillary and van der Waals forces, and also theoretically derive a conserved quantity for a singular fast-diffusion equation. These results not only provide a tool to analyze gradient flow PDEs; they also demonstrate the utility of Noether's theorem beyond Lagrangians.