arXiv:2610.33173v1 Announce Type: new Abstract: We develop a convex-analytic $L^2$-framework for gradient flows of pumping-energy functionals coupled with elliptic state equations and an evolving nonnegative conductivity $a\geq 0$. The main difficulty is that the conductivity may be merely square-integrable, or degenerate, and formal differentiation need not produce an $L^2$-subgradient. After imposing the strictly positive lower cutoff $r+a\geq r>0$, we define the pumping-energy functional by a variational formula, prove convexity and lower semicontinuity, and show that its $L^2$-subdifferential is nonempty if and only if the pressure gradient belongs to $L^4$. We also obtain a unique maximal-monotone $L^2$-gradient flow satisfying an energy identity. Moreover, we prove Mosco convergence of the pumping-energy functionals as the cutoff vanishes. An inverse-moment conductivity constraint provides a weak alternative to a uniform positive lower bound. For nonlinear laws $a=\mu(b)$, primal and flux representations distinguish two complementary regimes. If $\mu$ is concave, the pumping-energy functional remains convex, and we characterize its $L^2$-subdifferential and the induced gradient flow. If $1/\mu$ is concave, the pumping-energy functional is concave, and therefore its negative is convex. We construct its pointwise gradient flow on a uniformly elliptic box. Admissible stationary points for power laws yield $p$-Laplacian equations with $p>2$ and $1 < p < 2$, respectively.
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