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Math-2851 · mathematics · Pde

Uniqueness of admissible periodic hard-sphere Boltzmann evolution

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every nonnegative initial density on $T^3×R^3$ with compact velocity support and finite mass, kinetic energy and absolute entropy, must the periodic hard-sphere Boltzmann equation have at most one global admissible renormalized solution satisfying the source mass, momentum, energy and entropy conditions, strong L1 continuity and integrable gain and loss on a common initial interval?

Short English statement

For every nonnegative initial density on $T^3×R^3$ with compact velocity support and finite mass, kinetic energy and absolute entropy, must the periodic hard-sphere Boltzmann equation have at most one global admissible renormalized solution satisfying the source mass, momentum, energy and entropy conditions, strong L1 continuity and integrable gain and loss on a common initial interval?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2851-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2851

Periodic Boltzmann nonuniqueness: The announced pair shares one bounded-velocity datum satisfying the required mass, energy and entropy bounds. Both evolutions meet the renormalized admissibility and initial-interval integrability conditions, so their distinctness directly contradicts the uniqueness statement.

Boltzmann nonuniqueness with local conservation: The announced pair shares one bounded-velocity datum satisfying the required mass, energy and entropy bounds. Both evolutions meet the renormalized admissibility and initial-interval integrability conditions, so their distinctness directly contradicts the uniqueness statement.

The claimed counterexample lies within the question’s hypotheses and contradicts its universal conclusion.

Uniqueness of admissible periodic hard-sphere Boltzmann evolution

OpenAI's claim

Periodic Boltzmann nonuniqueness: The manuscript claims two global renormalized hard-sphere Boltzmann solutions on $\mathbb T^3\times\mathbb R^3$ with the same initial density; they satisfy the stated local-mass, total-momentum, energy, and entropy conditions, with strong $L^1$ continuity and integrable gains and losses on an initial interval.

Boltzmann nonuniqueness with local conservation: The refinement claims two distinct global entropy solutions for one periodic hard-sphere initial datum while preserving mass, momentum, and kinetic energy locally; polynomially weighted gain and loss integrability and strong $L^1$ continuity hold on bounded time intervals.

Mathematical statement of the claimed result

Mathematical claim 1.
On $\Omega=\mathbb T^3_x\times\mathbb R^3_v$, consider $(\partial_t+v\cdot\nabla_x)F=Q(F,F)$ with $a=(v-v_*)\cdot\omega$, $v^\prime=v-a\omega$, $v_*^\prime=v_*+a\omega$ and $Q=\int_{\mathbb R^3\times S^2}|a|(F^\prime F_*^\prime-FF_*)\,dv_*\,d\omega$. The claim gives a nonnegative datum $F_0$ of bounded velocity support and finite $\int_\Omega F_0(1+|v|^2+|\log F_0|)$ with two distinct global admissible solutions. These are weakly $L^1$ continuous against bounded tests, retain bounded mass, energy and absolute entropy on finite intervals, conserve local mass and total momentum, and obey global energy and relative-entropy dissipation inequalities at every time. For $M=(2\pi)^{-3/2}e^{-|v|^2/2}$ and every $C^1$ renormalization $\beta$ with $\beta(0)=0$, $|\beta^\prime(z)|\le C_\beta/(1+z)$, they solve $(\partial_t+v\cdot\nabla_x)[M\beta(F/M)]=\beta^\prime(F/M)Q(F,F)$ distributionally, with the separate renormalized gains and losses integrable. For some initial time interval shared by the two evolutions, continuity holds in the strong $L^1$ norm and their unrenormalized gains and losses are integrable. Local momentum and energy conservation are not assumptions of this first solution class.
Mathematical claim 2.
On $\Omega=\mathbb T^3_x\times\mathbb R^3_v$, consider $(\partial_t+v\cdot\nabla_x)F=Q(F,F)$ with $a=(v-v_*)\cdot\omega$, $v^\prime=v-a\omega$, $v_*^\prime=v_*+a\omega$ and $Q=\int_{\mathbb R^3\times S^2}|a|(F^\prime F_*^\prime-FF_*)\,dv_*\,d\omega$. One datum $F_0\ge0$ is claimed to have bounded velocity support, finite mass, energy and absolute entropy, yet admit two global solutions differing on a positive-measure set. Both belong to $C([0,\infty);L^1(\Omega))$, obey the renormalized equation for every $\beta\in C^1([0,\infty))$ with $\beta(0)=0$ and $|\beta^\prime(z)|\le C/(1+z)$ relative to the Maxwellian, and satisfy both the global energy bound and the full relative-entropy dissipation inequality at every $t\ge0$. Each solution is nonnegative, takes initial value $F_0$, and for every finite $T$ has \[\sup_{0\le t\le T}\int_\Omega F(t)(1+|v|^2+|\log F(t)|)\,dx\,dv<\infty,\qquad\int_0^T\int_\Omega(1+|v|^3)F\,dx\,dv\,dt<\infty.\] The second condition ensures that the local fluxes below are integrable. In addition \[\partial_t\int F\psi\,dv+\nabla_x\cdot\int vF\psi\,dv=0\quad(\psi=1,v_1,v_2,v_3,|v|^2/2)\] distributionally with the prescribed initial traces. For every finite $T$ and $k\ge0$, \[\int_0^T\int_\Omega(1+|v|^2)^{k/2}(Q^+(F,F)+Q^-(F,F))\,dx\,dv\,dt<\infty.\] Thus the stronger nonuniqueness claim retains exact local mass, momentum and energy conservation.

Outline of the argument

Periodic Boltzmann nonuniqueness: Two cutoff approximation families separate a dormant hot component from one activated by a seed of vanishing mass. Velocity-weight caps provide uniform integrability and permit removal of the annular and collision cutoffs. A rescaled branching argument prevents an apparent cap hit from disappearing into negligible mass, proving that the admissible limits differ despite sharing the same initial datum.

Boltzmann nonuniqueness with local conservation: Kernel estimates and carefully arranged cold jets select short-time hot-component approximations. A global continuation bound and weighted compactness preserve collision terms and each local conservation law. Dilation and branching produce a nonzero rescaled limit with a finite horizon, which separates the two limiting solutions rather than merely their approximations.

Formal verification

Family 363 describes Lean coverage for the initial periodic nonuniqueness result with its stated conservation and integrability properties; the stronger local-conservation refinement is separate.

Periodic Boltzmann nonuniqueness: The manuscript links BoltzmannNonuniqueness.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

Boltzmann nonuniqueness with local conservation: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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