Hopf Obstruction and Transported Forced Brakke Motion in Ordered Viscoelastic Cores

We study topological relaxation in ordered viscoelastic conformation flows at finite epsilon.

In an ordered region, a positive spectral gap selects an oriented principal axis and hence an S^2-valued director with a Hopf class.

We show that a change of this class must be accompanied, before the sharp-interface limit, by one of a finite list of costs: exterior gap concentration, ordered-core mass, boundary flux, FENE/collar loss, or a topology exit.

The result is proved for a concrete Landau-de Gennes ordered-core closure coupled to an Oldroyd/FENE-type transport law.

The structural hypotheses used in the argument are verified up to the first typed exit time: the Morse-Bott ordered well, tubular soft coordinates, massive-mode coercivity, a projected transported Ginzburg-Landau equation, exterior gap control, and tame FENE/collar coefficients.

The projected Ginzburg-Landau equation separates translation modes from the remaining residuals.

The translation modes give the normal line force, while the orthogonal soft, massive, geometric, and collar terms are absorbed by coercivity or charged to the corresponding exit.

A modulated-energy argument propagates a nonempty class of vortex-tube data on regular intervals.

On each such interval, the normalized core measures converge to an integral one-varifold satisfying a transported forced Brakke inequality with the computed force.

The theorem therefore derives the force projection, open-basin propagation, and Brakke compactness estimates before invoking any limiting Brakke flow, and it records the finite-epsilon cost when the regular ordered-core description breaks down.