IndisputableMonolith.NavierStokes.DiscreteMaximumPrinciple
This module supplies the OneStepData record for a single discrete incompressible Navier-Stokes time step on a finite lattice together with monotonicity lemmas for gradients and J-cost. Discrete fluid modelers and lattice NS analysts cite it to obtain stability bounds without continuous limits. The module consists of a core definition followed by direct algebraic applications of the imported operator surface and vortex-stretching estimates.
claimOneStepData encodes a velocity field $u$ on a finite three-direction lattice satisfying discrete incompressibility, together with the advection and viscous-absorption fields derived from the CoreNSOperator; the module then proves gradient nonincreasing and J-cost monotonicity under these data.
background
The module sits inside the discrete incompressible Navier-Stokes development on a finite lattice. DiscreteNSOperator supplies the three-direction lattice topology, the CoreNSOperator carrying only physical flow data, and the concrete derivations of pair-budget and viscous-absorption fields from the velocity gradient and Laplacian. VortexStretching closes the remaining analytic gaps by importing the finite-volume rigidity result of Thapa & Washburn (J. Math. Phys. 2026), the canonical reciprocal cost uniqueness of Washburn & Zlatanovic (Mathematics 2026), and the coherent comparison estimates of Pardo-Guerra et al. (Foundations 2026). OneStepData is the central record that packages a single time step for these operators.
proof idea
This is a definition module whose lemmas are established by direct appeal to the operator identities in DiscreteNSOperator and the dissipation bounds in VortexStretching; each monotonicity statement (gradient_nonincreasing, Jcost_nonincreasing, subK_preserved) follows from algebraic reduction of the discrete advection and viscous terms without additional analytic machinery.
why it matters in Recognition Science
The module supplies the discrete maximum-principle layer that supports lattice regularity statements in the Navier-Stokes domain. It directly feeds master_lattice_regularity and unconditional_Jcost_monotonicity, thereby embedding the Recognition Science J-cost and phi-ladder structure into the discrete fluid setting. The construction closes the discrete-step portion of the NS treatment by inheriting the zero-sorry estimates already established in the upstream VortexStretching module.
scope and limits
- Does not treat the continuous Navier-Stokes limit or convergence to weak solutions.
- Does not incorporate boundary conditions or forcing terms beyond the finite lattice.
- Does not address compressible or non-Newtonian flows.
- Does not supply numerical schemes or implementation details.
depends on (2)
declarations in this module (10)
-
structure
OneStepData -
theorem
advection_dominated_by_viscosity -
theorem
one_step_factor_le_one -
theorem
gradient_nonincreasing -
theorem
subK_preserved -
theorem
subK_propagation -
theorem
unconditional_gradient_bound -
theorem
unconditional_Jcost_monotonicity -
theorem
Jcost_nonincreasing -
theorem
master_lattice_regularity