IndisputableMonolith.NavierStokes.JcostMonotonicity
The JcostMonotonicity module shows that J-cost decreases or remains constant under the discrete incompressible Navier-Stokes operator by absorbing lattice stretching into the viscous budget. Researchers working on Recognition Science fluid models would cite it to establish dissipation. The argument sums sitewise RCL pair bounds once the upstream operator fields are in place.
claimFor a CoreNSOperator on the three-direction lattice, $\partial_t J_{\rm cost} \le 0$ once sitewise RCL bounds are summed to absorb stretching into the viscous-absorption field.
background
The module sits inside the NavierStokes domain and imports the DiscreteNSOperator module. That upstream module supplies a finite three-direction lattice topology, the CoreNSOperator carrying only physical flow data, and concrete derivations of pair-budget and viscous-absorption fields from the velocity gradient and Laplacian.
J-cost is built from the Recognition Science J function whose composition law (RCL) supplies the sitewise bounds. The module doc states that lattice stretching is absorbed by the operator's viscous budget after these bounds are summed.
proof idea
This is a definition module whose internal lemmas first isolate the absorption step (stretching_absorbed_by_viscosity) and then conclude the sign of the time derivative (dJcost_dt_nonpos_of_operator). Both steps rest directly on the pair-budget and viscous-absorption fields supplied by DiscreteNSOperator.
why it matters in Recognition Science
The module supplies the monotonicity step required for any Recognition Science treatment of discrete fluid dissipation. It feeds the NavierStokes analysis by certifying that J-cost is non-increasing once the operator is applied, closing the stretching-to-viscosity absorption argument stated in its own documentation.
scope and limits
- Does not treat the continuous-space limit of the lattice model.
- Does not address compressible or non-incompressible variants.
- Does not include boundary conditions outside the supplied lattice topology.
- Does not prove global existence or uniqueness for the discrete system.