Knowlpedia

Definition

Linearized Navier–Stokes operator

The first variation of the momentum residual with respect to velocity and pressure.

At a smooth reference velocity uu, the linearized Navier–Stokes momentum operator acting on a velocity-pressure increment (w,π)(w,\pi) is

Lu(w,π)=∂tw+(u⋅∇)w+(w⋅∇)u+∇π−νΔw.\mathcal L_u(w,\pi)=\partial_tw+(u\cdot\nabla)w +(w\cdot\nabla)u+\nabla\pi-\nu\Delta w.

It is the coefficient of ε\varepsilon in the of (u+εw,p+επ)(u+\varepsilon w,p+\varepsilon\pi) with the force held fixed. For an incompressible perturbation one also imposes ∇⋅w=0\nabla\cdot w=0.

Exact remainder and spaces

The remainder is ε2(w⋅∇)w\varepsilon^2(w\cdot\nabla)w. Thus linearizing about a non-solution still leaves its original residual as an inhomogeneous term. To regard Lu\mathcal L_u as a bounded derivative between function spaces, choose source norms controlling the displayed time and spatial derivatives and a target norm in which the products are bounded; the algebraic formula alone does not specify such spaces.