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What's in here: why each moving boundary layer needs its own PDE — matched asymptotics, inner vs outer problems, Stefan conditions, why one monolithic equation isn't uniformly well-posed. Second half is the more interesting claim: multiscale coupling destroys predictability, via error backscatter from small to large scales, closure problems, and non-commuting limits. Cell-biology twin: multiscale.

Short answer: because the “action” near a moving interface lives on a different balance of forces, time scales, and even PDE type than the bulk. If you don’t split them, you smear the physics and lie to yourself with pretty plots. And once you go multiscale, tiny errors at the fast/small layer ricochet up the hierarchy, so predictability evaporates like your free time.

Why separate PDEs for each moving boundary layer

Think “inner vs outer” problems. A thin layer of thickness ε hugging an interface has different dominant terms than the bulk. You rescale coordinates by ε, do matched asymptotics, and new leading-order equations pop out that are not the same as the outer PDE.

Common consequences:

• Different equation type. Classic: Navier–Stokes in the outer flow, but in the viscous boundary layer the leading equations are Prandtl-like (streamwise advection vs wall-normal diffusion). In transonic flow one region is elliptic, another hyperbolic. One giant PDE won’t be uniformly well-posed across regions.
• Singular perturbations and sharp balances. Curvature, surface tension, latent heat, or reaction rates dominate in the inner layer. Fronts, shocks, and phase boundaries obey their own evolution laws (e.g., eikonal/Hamilton–Jacobi for a flame front speed, Stefan condition for melting/freezing).
• Jump/kinetic conditions live on the interface. Mass, momentum, energy fluxes must match across the moving boundary. Writing a dedicated interface PDE with jump conditions makes those constraints explicit and enforceable.
• Numerical sanity and efficiency. If you keep everything in one mesh with one PDE, you get stiffness, grid-scale numerical diffusion, and 10× compute for half the accuracy. Splitting lets you resolve the thin bits with the right scalings and couple them to the coarse outer solver cleanly.
• Topology changes. Coalescence, breakup, pinching, contact lines. Interface-aware PDEs (level set, phase-field, front-tracking) handle these. Monolithic bulk models pretend not to notice until it’s too late.

Concrete templates you see in the wild:

• Stefan problems: heat equation in each phase; interface moves with V_n \propto [\partial_n T] (latent heat).
• Flames/fronts: advection–diffusion–reaction in the bulk; interface speed s_n = s_L + D\,\kappa + u\!\cdot\!n + \dots.
• Boundary layers: outer Navier–Stokes + inner Prandtl; match shear and pressure.
• Porous media wetting fronts: outer Richards’ eq + sharp-front kinematics when capillary number is small.

Why multiscale slashes predictability

Welcome to the error cascade.