divConstraint_continuous
plain-language theorem explainer
For each fixed 2D Fourier mode the map from velocity coefficient vectors to the scalar divergence value is continuous. Researchers building Galerkin-to-continuum passages for incompressible fluids cite this when transferring the divergence-free condition across limits. The proof extracts the two coordinate projections via PiLp continuity and closes under constant multiplication and addition.
Claim. For each fixed mode $k=(k_1,k_2)inmathbb{Z}^2$, the map $vmapsto k_1v_1+k_2v_2$ from $mathbb{R}^2$ to $mathbb{R}$ is continuous.
background
Module ContinuumLimit2D (M5) supplies the pipeline shape that embeds finite Galerkin truncations into an infinite Fourier coefficient state and packages analytic steps as explicit hypotheses. Mode2 is the type of 2D Fourier modes, an integer pair. VelCoeff is the Euclidean 2-space of velocity Fourier coefficients at a mode. divConstraint k v is the real scalar $k_1v_1+k_2v_2$ that encodes the Fourier-side divergence constraint for that mode.
proof idea
The tactic proof first obtains continuity of each coordinate projection on the Euclidean space via PiLp.continuous_apply. It then combines the two projections with constant multiplications and addition, using the algebraic definition of divConstraint to finish.
why it matters
The lemma is invoked by divConstraint_eq_zero_of_forall, which transfers the divergence-free condition from approximants to the limit coefficient via continuity and uniqueness of limits. This supplies one concrete identification step inside the M5 continuum-limit pipeline and keeps the classical-bridge argument free of extra compactness assumptions beyond modewise convergence.
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