Global Stability and Smoothing of Planar Dynamic Bernoulli Fronts

This work studies the nonlinear dynamics of small periodic perturbations of planar traveling fronts for the one-phase dynamic Bernoulli problemΔu = 0 in {u > 0},V = G(|∇u|) on ∂{u > 0},where the velocity law G is uniformly monotone on a fixed positive slope interval.Fix a positive reference slope q̄ and the corresponding planar front moving with normal speed G(q̄). In a comoving frame, the free boundary is represented as a periodic graph. The main result proves that sufficiently small perturbations in the maximal-regularity trace space generate a unique global strong solution that converges exponentially, modulo vertical translation, to a planar front. The asymptotic translation differs from the initial spatial mean only by a quadratic correction, and if G is smooth, the interface becomes spatially C∞ for every positive time.The central structural identity is the linearization of the exact graph evolution:DF(0) = −q̄G′(q̄)|D|.This shows that vertical translations form the only neutral mode, while every nonconstant periodic shape mode is strictly stable. The resulting decay rate is controlled by q̄G′(q̄), not by the sign of the physical propagation speed G(q̄). Consequently, advancing, stationary, and retreating planar fronts share the same local orbital-stability mechanism under uniform monotonicity. 02_Global_Stability_and_Smoothi…The proof begins with an exact graph formulation of the moving free-boundary problem and a flattening transformation to a fixed half-space. The associated boundary-gradient operator is expanded through the Dirichlet-to-Neumann map, yielding the principal order-one operator and a quadratic nonlinear remainder.The nonlinear graph equation is then placed in a maximal-Lp-regularity framework. The family of vertical translates forms a one-dimensional equilibrium manifold, the zero eigenvalue is semisimple, and the complementary spectrum lies strictly in the stable half-plane. These properties allow the normal-stability mechanism to produce global existence and exponential convergence for sufficiently small initial perturbations. 02_Global_Stability_and_Smoothi… 02_Global_Stability_and_Smoothi…A modulation argument identifies the asymptotic vertical shift more precisely. Writing the solution as the sum of its spatial mean and a mean-zero component, the nonlinear phase drift is shown to be quadratic. In particular, the limiting translation a∞ satisfies|a∞ − mean(f₀)| ≤ C ||f₀ − mean(f₀)||².Thus the nonlinear dynamics select a nearby planar front whose vertical displacement differs from the initial average position only at second order. 02_Global_Stability_and_Smoothi…For smooth velocity laws, the maximal-regularity flow can be bootstrapped through higher Sobolev scales. As a consequence, the interface is spatially smooth for every positive time, even when the initial perturbation is posed only in the natural trace space. 02_Global_Stability_and_Smoothi…The result is complementary to local flatness theory. Local epsilon-regularity provides a mechanism for creating graph structure from geometric flatness, while the present theorem describes the subsequent strong nonlinear dynamics once the solution has entered a sufficiently small graphical neighborhood of a planar front. 02_Global_Stability_and_Smoothi…The main structural principle is the separation between propagation and shape stability: the physical front speed is G(q̄), whereas the stability of nonconstant shape modes is governed by q̄G′(q̄) > 0. This distinction allows a unified treatment of both advancing and retreating dynamic Bernoulli fronts.