Local Flatness Regularity for Uniformly Monotone Dynamic Bernoulli Laws

This work studies local regularity for one-phase free-boundary evolutions governed by the dynamic Bernoulli law Δu = 0 in {u > 0},V = G(|∇u|) on ∂{u > 0}, where the velocity law G is uniformly monotone on a fixed positive slope interval. The main result is a local epsilon-regularity theorem that does not assume in advance that the free boundary is graphical. If a viscosity solution is sufficiently flat with respect to an exact variable-slope planar front, then in a smaller space-time cylinder the free boundary becomes a spatial C¹,β graph, and its spatial unit normal is Hölder continuous in the natural order-one space-time metric. The quantitative constants are uniform over the prescribed velocity-law class and do not depend on a temporal modulus of the reference slope. A central feature of the proof is a law-independent tangent dynamics. For an exact planar front with slope q(t) and position s(t), satisfying s′(t) = G(q(t)), the linearized coefficient is q(t)G′(q(t)). Introducing the intrinsic time variable τ′(t) = q(t)G′(q(t)) transforms the blow-up equation into the universal fractional-parabolic equation H_τ + |D′|H = 0. Thus, at tangent scale, the particular nonlinear velocity law disappears and survives only through the intrinsic time reparametrization. After localization, the correct first-order tangent is generally not affine in time. Instead, it has the form Z(τ) + b · x′, with Z in W¹,∞. This scalar-path structure is necessary because localization produces time-dependent scalar forcing.A second key ingredient is an exact finite-scale nonlinear reconstruction. The localized tangent is converted back into an exact planar solution of the original nonlinear Bernoulli law through q*(t) = G⁻¹(S′(t)). The resulting first-order correction is q* = q + εqZ_τ + O(ε²), which is independent of the particular choice of G. This exact reconstruction is what allows the improvement-of-flatness scheme to close. The proof combines localized harmonic-cap barriers, Harnack oscillation reduction, full positive-level domain variations, intrinsic-time compactness, viscosity identification of the fractional-parabolic limit, exact corrected-front reconstruction, and an infinite improvement-of-flatness iteration. Cross-center compatibility of the limiting tangent planes then yields Hölder continuity of the spatial normal and establishes the local graph structure of the free boundary. The framework includes the prototype dynamic Bernoulli law G(q) = q² − 1 as a special case, but the theorem is formulated for a general uniformly monotone class. The main structural principle is the separation between universal tangent dynamics and law-specific finite-scale reconstruction. This preprint forms the local-regularity component of a broader study of dynamic Bernoulli free-boundary flows, including strong graphical evolution, stability of planar fronts, and singularity dynamics.