🎬Dissecting the Deterministic Roles of Diffusivity Decay and Dimensionality in PDEs

These analyses collectively demonstrate that the fundamental structure and parameters of partial differential equations (PDEs) precisely govern the ultimate physical fate of mass, energy, or potential fields. Specifically, the long-term behavior of a system is determined by its boundary conditions: boundaries lead to a stable steady state, enforcing either total loss (Dirichlet to zero equilibrium) or perfect conservation (Neumann to an average equilibrium), in contrast to infinite domains where spreading is perpetual. For transport, convection dictates the overall displacement of the material, while diffusion controls its immediate shape and spreading, but the decay rate ultimately limits the material's total existence time. Finally, the dimensionality of a concentrated source dictates the field's behavior, proving that distributing a charge eliminates the dramatic singularity characteristic of a 3D point source.

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Dissecting the Deterministic Roles of Diffusivity Decay and Dimensionality in PDEs | Animated Resultsviadean on Notion