Structure-Preserving Variational Closures of Kinetic Equations

Definitions and mathematical statement

Write $x$ for the spatial variable, $t$ for time, $c$ for the kinetic variable, and $W(x,t)$ for the reduced state. The prescribed form of the closure ansatz is \[ f(x,t,c)=f^{\star}(W(x,t),c). \] The kinetic equation, the closure map $f^{\star}$, the dimensions and domains of these variables, and the admissible reduced states are not specified in the supplied record. Let $\phi_{\alpha}(x,t,c)$ denote the test functions sought for the variational formulation; their index set and admissible function spaces are likewise unspecified.

Statement of the modeling problem. For a kinetic equation reduced by this closure ansatz, determine test functions $\phi_{\alpha}$ such that the reduced equations preserve the physical conservation laws of the kinetic model. Determine the resulting evolution equations for $W(x,t)$, and investigate whether they are symmetric hyperbolic balance laws admitting an entropy. The conserved quantities and the sense in which their conservation is to be retained must be specified for the particular kinetic model. The record does not distinguish local balance identities from global conservation under boundary conditions.

For terminology, a first-order system is symmetric hyperbolic on an admissible state region if, in suitable state coordinates $U$, it has the form \[ A^0(U)\partial_t U+\sum_{j=1}^{d}A^j(U)\partial_{x_j}U=B(U), \] where $A^0(U)$ is symmetric positive definite and every $A^j(U)$ is symmetric throughout that region. Here $d$, the coordinate change from $W$ to $U$, and all coefficients would have to be determined from the specified model and reduction; this display is a definition of the structural property under investigation, not an asserted reduced equation.

If a reduced system admits a conservative balance-law representation \[ \partial_t W+\sum_{j=1}^{d}\partial_{x_j}F^j(W)=S(W), \] a differentiable entropy pair $(\eta,q^1,\ldots,q^d)$ satisfies \[ Dq^j(W)=D\eta(W)\,DF^j(W),\qquad j=1,\ldots,d. \] For smooth solutions this compatibility yields the entropy balance \[ \partial_t\eta(W)+\sum_{j=1}^{d}\partial_{x_j}q^j(W) =D\eta(W)S(W). \] The supplied record does not specify whether the intended entropy must be convex or strictly convex, what sign condition its production should satisfy, or whether an entropy inequality for weak solutions is required. Neither the conservative representation nor any of these additional properties is assumed to hold.

Unresolved scope. This is an underspecified modeling question. The source does not state a universal assertion over all kinetic equations or all closures, and it does not supply the data needed to formulate one particular reduced initial or boundary value problem. A precise model-specific existence or characterization statement requires those missing data.

Short English statement

For the closure $f^{\star}(W(x,t),c)$, find variational test functions that preserve the kinetic model's physical conservation laws, derive the equations for $W$, and determine whether they form symmetric hyperbolic balance laws with an entropy. The kinetic model, closure, conserved quantities, domains, boundary conditions, and intended entropy requirements are unspecified, so the supplied question remains a modeling target requiring further hypotheses.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003162$, source problem code catalogue entry, titled Structure-Preserving Variational Closures of Kinetic Equations, in the collection Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2016.
• Supplied report citation: Hyperbolic Techniques in Modelling, Analysis and Numerics (2017), Oberwolfach Reports, pp. 1683--1751, DOI 10.4171/owr/2016/30. https://doi.org/10.4171/owr/2016/30. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page or named contributor is supplied.
• Supplied literature source: Oberwolfach Report 30. https://ems.press/content/serial-article-files/46634?nt=1.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the entry requests test functions and an evolution equation for $W(x,t)$ without specifying a universal closure, and that the report gives no general solution to this modeling problem. This is supplied dated curation, not an independently verified literature assessment; its status and cited claims require independent verification.
• The registry labels the statement corrected_verified, with the qualification that only safe notation, typography, and TeX-delimiter cleanup was applied and that hypotheses, quantifiers, and subquestions were preserved. This supplied label does not establish mathematical completeness or independent source verification of this draft.