Knowlpedia

Theorem

Localized difference-energy identity

A compact spatial weight exposes the transport and pressure fluxes in the energy of two flows.

Statement

For smooth solutions with common force and viscosity, let w=v−uw=v-u, π=P−p\pi=P-p, and let χ\chi be a smooth, time-independent compactly supported scalar weight. The gives

12ddt∫χ∣w∣2+ν∫χ∣∇w∣2=−∫χ (w⋅∇)u⋅w+ν2∫∣w∣2Δχ+12∫∣w∣2v⋅∇χ+∫πw⋅∇χ.\begin{aligned} \frac12\frac d{dt}\int\chi|w|^2+\nu\int\chi|\nabla w|^2 ={}&-\int\chi\,(w\cdot\nabla)u\cdot w +\frac\nu2\int|w|^2\Delta\chi\\ &+\frac12\int|w|^2v\cdot\nabla\chi +\int\pi w\cdot\nabla\chi. \end{aligned}

All integrals are over space. Compact support of χ\chi makes them local; no decay at infinity is needed.

Derivation

Pair the difference equation with χw\chi w. Integrating the Laplacian by parts twice gives the weighted gradient term and ν2∫∣w∣2Δχ\tfrac\nu2\int|w|^2\Delta\chi. Incompressibility moves the transport and pressure derivatives onto χ\chi. The reference-gradient term remains because the derivative acts on uu, not on the energy density.