Conceptual Understanding of Tao’s Post

Interpret each term in the forced inviscid Boussinesq equations and identify transport, incompressibility, buoyancy, pressure, and external forcing.

Derive the two-dimensional Boussinesq vorticity equation from its momentum equation.

Distinguish smooth preterminal evolution, forcing that remains smooth through the terminal time, and finite-time blowup of derivative norms in the Boussinesq theorem.

For a single plane-wave temperature–vorticity ansatz, derive the Fourier-mode Biot–Savart velocity and compute how the temperature gradient and velocity scale with frequency.

Verify that shared-phase temperature and vorticity waves do not advect themselves because of phase–velocity orthogonality.

Derive the wavevector transport equation on an affine incompressible background.

Derive the coupled temperature–vorticity amplitude equations by cancelling the leading Boussinesq residuals.

Diagonalize the frozen-coefficient amplitude system and calculate its exponentially growing mode.

Analyze how controlled rotation resets a layer’s vorticity amplitude while retaining its temperature-gradient gain.

Construct a toy frequency-and-time schedule in which infinitely many layers fit before a finite terminal time while gradients diverge and field amplitudes remain bounded.

Explain how spatial cutoffs create compact nested affine cores and identify the localization residuals they introduce.

Use schematic derivative bounds to explain how increasing the correction order with frequency can make the total forcing converge in every mixed space–time derivative.

Derive the circulation and reduced-vorticity coupling for axisymmetric Euler flow with swirl in the Euler paper’s meridional variables.

Map the Boussinesq amplitude mechanism onto the Euler amplitude law, identifying the effects of material coordinates and spatially varying coefficients.

Describe how convex integration uses high-frequency corrections to cancel low-frequency Reynolds stress, and contrast that feedback with the predominantly low-to-high amplification in these blowup constructions.

Annotate Tao’s post claim by claim and create a targeted cross-paper reading map linking each claim to the relevant theorem statement, model or heuristic section, or technical assembly estimate in the Boussinesq and Euler papers.