The concrete local Comparator-to-development conversion #
Compact initial vorticity remains in one compact set for a positive time. Elliptic recovery gives all spatial L² derivatives on that interval, and the genuine Euler pairings against dense compact solenoidal tests provide the time regularity needed for an ordinary Euler evolution.
Recover the actual all-order L² Fréchet tensors from scalar coordinate derivatives. The derivative hypotheses used here are consequences of compact vorticity, ordinary smoothness, and finite velocity energy.
Bounding a tensor by the energies of its coordinate evaluations #
The reconstruction map is fixed in each derivative order. Its operator norm therefore gives a finite constant converting the sum of the scalar coordinate energies into a bound for the literal tensor norm.
A tensor's squared operator norm is bounded by the sum of the squared scalar coordinate evaluations, with a constant depending only on its order.
List coordinate derivatives agree with the ordinary Fréchet tensor evaluated on the corresponding sequence of coordinate directions.
Coordinate projection commutes with the actual iterated Fréchet derivative.
Every coordinate evaluation of the Fréchet tensor is in L².
Smooth finite-energy divergence-free velocity with compact vorticity has all its actual Fréchet derivatives in L².
The derivative class used by the development follows from ordinary smoothness, finite energy, solenoidality, and compact vorticity.
Equations
- EulerComparatorRecovery.smoothL2FieldOfCurlCompact u hu hL2 hdiv hc = { field := u, smooth := hu, integrable := ⋯ }
Instances For
The literal tensor energy is controlled by finitely many scalar coordinate energies. The constant is fixed by the derivative order alone.
The L² class of a smooth field's tensor has exactly its ordinary integral energy, so quantitative recovery applies to the development's norm.
Uniform bounds for scalar coordinate-word energies yield uniform L² norms of the actual Fréchet tensors over an arbitrary parameter set.
Joint spatial coordinate derivatives and uniform energy bounds for families supported in a fixed compact set.
Uniform spatial derivative energies for a Comparator solution whose vorticity stays in one compact set on a finite time interval. Ordinary joint smoothness supplies the compact source bounds, and elliptic recovery supplies the velocity derivative bounds.
The literal Laplacian of the velocity is the negative curl of its vorticity, by solenoidality and the ordinary curl-curl identity.
Locality of curl places the Laplacian support inside the vorticity support.
Every scalar component of the actual velocity Laplacian remains jointly smooth through time zero.
Common compact support of vorticity and Comparator joint smoothness uniformly control every scalar coordinate derivative of the Laplacian.
A Comparator solution with common compact vorticity support has uniform energies for all genuine scalar coordinate derivatives of its velocity.
Actual ordinary smooth-L² velocity slices recovered from a common compact vorticity support. The fields are definitionally the Comparator velocity.
Equations
- Euler.ComparatorBridge.recoveredVelocity h T K hK hsupport t = EulerComparatorRecovery.smoothL2FieldOfCurlCompact (fun (x : EulerSmoothLimit.Space) => v x ↑t) ⋯ ⋯ ⋯ ⋯
Instances For
All genuine spatial L² tensor norms are uniformly bounded on the common compact-vorticity interval. No time regularity of these norms is assumed.
A Comparator Euler solution with a common compact vorticity support is represented by an actual ordinary evolution throughout that interval.
Compact initial vorticity alone supplies the complete local conversion: the truncations, support propagation, spatial recovery, and time regularity are all obtained from the actual Comparator solution assumptions.