Introduction to Nonlinear PDE - Applications

Physics: Nonlinear Schrödinger (Bose-Einstein condensates, nonlinear optics), Gross-Pitaevskii (superfluids), Yang-Mills (gauge theory, quantum chromodynamics), Einstein equations (general relativity—geometric PDEs), Navier-Stokes (turbulence, still open millennium problem).

Biology: Reaction-diffusion (morphogenesis, Turing patterns), chemotaxis (Keller-Segel: $u_t = \Delta u - \nabla \cdot (u\nabla v)$ , $v_t = \Delta v + u - v$ ), population dynamics (Lotka-Volterra with diffusion), tumor growth models (free boundary problems).

ExampleGeometric Flows

Mean curvature flow: $\frac{\partial X}{\partial t} = -\kappa \mathbf{n}$ (surfaces shrink according to curvature)

Ricci flow: $\frac{\partial g_{ij}}{\partial t} = -2R_{ij}$ (Perelman used this to prove Poincaré conjecture)

Harmonic map flow: $u_t = \Delta u + A(u)(\nabla u, \nabla u)$ (maps between manifolds)

These geometric PDEs are fully nonlinear and exhibit rich singularity formation.

Materials Science: Cahn-Hilliard ( $u_t = -\Delta(\Delta u - u^3 + u)$ , phase separation), Allen-Cahn ( $u_t = \Delta u - W'(u)$ , phase field), crystal growth (Stefan problem, Mullins-Sekerka), dislocations (Hamilton-Jacobi equations).

Engineering: Porous medium (groundwater flow, gas dynamics), Monge-Ampère (optimal transport, antenna design), traffic flow (hyperbolic conservation laws), image processing (total variation denoising: $u_t = \nabla \cdot \frac{\nabla u}{|\nabla u|}$ ).

Remark

Machine learning connections: Neural ODEs $\frac{d\mathbf{h}}{dt} = f_\theta(\mathbf{h}, t)$ are continuous-depth networks. Gradient flows for optimization ( $\frac{dw}{dt} = -\nabla Loss$ ) connect to PDEs. Diffusion models for generative AI solve SDEs/PDEs backward in time.

Modern research: Mean field games (Nash equilibria as PDE systems), optimal transport (Wasserstein geometry), geometric analysis (Yamabe, prescribed curvature), nonlinear dispersive (water waves, general relativity).