Pressure perturbations and logarithmic spectral response in Laplacian growth

We study the instantaneous velocity response to a small analytic boundary-pressure perturbation in Hele--Shaw flow without surface tension. For interfaces described by polynomial conformal maps, even a single pressure harmonic can make the sum of squared coefficients in the far-field velocity expansion diverge, although each coefficient remains finite. Under uniform analytic continuation conditions on the inverse map and pressure profile, we prove logarithmic growth when the nearest square-root branch points, related by rotational symmetry, reach the circle whose radius is the conformal radius. One linear combination of pressure harmonics per symmetry sector determines this growth. Profiles for which it vanishes have bounded coefficient norm. We verify the conditions for polynomial maps with positive coefficients along suitable paths and construct exact Polubarinova--Galin trajectories reaching the threshold with smooth interfaces. The velocity measured on a fixed circle enclosing the interface throughout the approach remains bounded.