Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Scope of the kinetic-to-fluid limit results
Remark
The Euler-passage theorem in this collection assumes weighted strong convergence to local Maxwellians and proves only the resulting conservation-law passage. The separate homogeneous BGK theorem gives an explicit relaxation solution and its limit. Neither result proves convergence for general inhomogeneous Boltzmann solutions, a Boltzmann–Grad limit from deterministic many-particle dynamics, or a Navier–Stokes–Fourier limit. Those questions require additional hypotheses and separate theorem chains.
Remark references
Facts & Assumptions
Given: The statements and hypotheses of the conditional Euler-passage theorem and homogeneous BGK relaxation theorem.
Euler equations under weighted local-equilibrium convergence: The Euler conclusion assumes weighted strong convergence and does not assert that such convergence occurs.
Exact homogeneous BGK fast relaxation: The explicit convergence result concerns the spatially homogeneous BGK relaxation model.
Verification
The first scope statement follows directly from the convergence hypotheses and conclusion of [F1]; its proof is a conditional passage of already assumed moments and fluxes.
The second theorem [F2] has no spatial dependence and uses the BGK operator, so its exact solution does not supply an existence or convergence argument for inhomogeneous Boltzmann dynamics. Neither supplier states a many-particle Boltzmann–Grad or Navier–Stokes–Fourier result; this remark records only the limits of these two carriers.