Knowlpedia

Theorem

Integration cancellation for divergence-free transport and pressure

Divergence-free fields remove transport and pressure terms from unweighted whole-domain energy integrals.

Statement

For smooth fields with compact support on Rn\mathbb R^n, or smooth periodic fields integrated over a fundamental cell, ∇⋅a=0\nabla\cdot a=0 implies

∫(a⋅∇)b⋅b=0.\int(a\cdot\nabla)b\cdot b=0.

Also, if ∇⋅b=0\nabla\cdot b=0, then ∫∇p⋅b=0\int\nabla p\cdot b=0, provided the pressure is smooth on the support in the compact-support case and periodic in the periodic case. These are energy cancellations.

Proof

The transport integrand is a⋅∇(∣b∣2/2)a\cdot\nabla(|b|^2/2). Integration by parts moves the derivative onto aa, and its divergence is zero. Similarly, ∫∇p⋅b=−∫p ∇⋅b\int\nabla p\cdot b=-\int p\,\nabla\cdot b. Compact support removes boundary terms; on a periodic cell opposite faces cancel. Multiplying by a nonconstant spatial cutoff leaves flux terms involving derivatives of that cutoff.