Velocity-Law Elasticity and Stable Hole Closing in Dynamic Bernoulli Flows

This work studies finite-time collapse of nearly spherical holes for the exterior one-phase dynamic Bernoulli problemΔu = 0 in {u > 0},V = G(|∇u|) on ∂{u > 0},in dimensions n ≥ 3 under a fixed positive far-field potential.For an exactly spherical hole of radius a(t), the boundary gradient is inversely proportional to the radius, and the radius evolves according to the exact nonlinear lawa_t = −G(q).The main objective is to determine when this spherical collapse remains dynamically stable under small nonradial perturbations.The central quantity is the logarithmic elasticity of the kinetic law,μ(q) = qG′(q) / G(q).After extinction-scale and center modulation, the degree-ℓ spherical-harmonic shape mode evolves at the linear rate1 − μ(q)(ℓ − 1).The degree-zero and degree-one modes correspond to scale and center modulation, while the first genuine shape mode is ℓ = 2. This leads to a sharp structural distinction: μ > 1 gives spherical shape stability, μ = 1 places the quadrupole mode at criticality, and μ < 1 produces linearly unstable shape modes. 03_Velocity_Law_Elasticity_and_…Under a uniform elasticity gap μ(q) ≥ 1 + δ, suitable scale-invariant derivative bounds on G, and a finite-collapse condition, the paper proves nonlinear exponential convergence of sufficiently small star-shaped perturbations to the spherical normalized profile. The rescaled evolution exists for all intrinsic times, the physical hole closes in finite time, the modulation center converges to a uniquely selected point, and the entire normalized trajectory converges to the unit sphere. 03_Velocity_Law_Elasticity_and_…The analysis separates the geometry of the nearly spherical hole from the growth of the physical boundary gradient. After normalization to the unit sphere, the exterior harmonic problem defines a dimensionless boundary-gradient map. Its linearization on a spherical harmonic of degree ℓ is multiplication by ℓ − 1, which produces the elasticity spectrum once it is combined with the nonlinear velocity law. 03_Velocity_Law_Elasticity_and_…The nonlinear argument uses scale and center modulation to remove the coordinate modes, leaving the genuine shape subspace ℓ ≥ 2. The resulting projected evolution has the form of a non-autonomous order-one parabolic problem whose coefficients depend on the evolving elasticity μ(q(τ)). A uniform gap above the critical value 1 gives a stable spectral gap for every nontrivial shape mode. 03_Velocity_Law_Elasticity_and_… 03_Velocity_Law_Elasticity_and_…A key analytic component is a direct non-autonomous maximal-L2-regularity estimate in spherical harmonics. The mode-by-mode propagator yields uniform control of the solution, one derivative in space, and one derivative in intrinsic time. Combined with tame nonlinear estimates for the nearly spherical Dirichlet-to-Neumann operator and the modulation parameters, this gives a nonlinear weighted bootstrap and exponential convergence in the rescaled variables. 03_Velocity_Law_Elasticity_and_…The modulation parameters are integrable in intrinsic time. As a consequence, the rescaled radius has a well-defined asymptotic normalization and the center converges to a unique collapse point. Under the stated finite-collapse condition, the physical time remains finite while the intrinsic time tends to infinity, and the normalized free boundary converges along the full trajectory to the unit sphere. 03_Velocity_Law_Elasticity_and_…For the prototype dynamic Bernoulli law G(q) = q² − 1 with q > 1, the elasticity satisfies μ(q) > 2, so the model lies strictly inside the stable regime. The classical Newtonian law G(q) = q instead gives μ = 1, placing the quadrupole mode at neutrality and matching the known appearance of elliptic or ellipsoidal extinction behavior in classical Hele-Shaw contraction. 03_Velocity_Law_Elasticity_and_…The main structural conclusion is that the stability of a collapsing harmonic Bernoulli hole is controlled not by the absolute magnitude of the boundary speed, but by the logarithmic elasticity of the kinetic law. A uniform gap above the critical value μ = 1 converts the exact radial collapse law into a nonlinear all-dimensional spherical stability theorem.