Physically Admissible Weak Solutions of Incompressible Euler

Definitions and mathematical statement

The supplied question concerns the incompressible Euler equations and the meaning of physical admissibility for their weak solutions. The record does not specify a spatial domain, dimension, time interval, initial or boundary data, regularity class, or admissibility criterion. Consequently, it does not determine a unique mathematical problem.

For reference only, a parameterized interior distributional formulation is as follows. Let $d$ be a positive integer, let $\Omega\subseteq{\mathbb{R}}^d$ be open, and let $I\subseteq{\mathbb{R}}$ be an open interval. Write $u=(u_1,\ldots,u_d)$ for velocity and $p$ for pressure. The equations are \[ \partial_tu+\operatorname{div}_x(u\otimes u)+\nabla_xp=0, \qquad \operatorname{div}_xu=0 \quad\text{in }\Omega\times I, \] where $(u\otimes u)_{ij}=u_i u_j$. One possible interior weak-solution class consists of pairs \[ u\in L^2_{\mathrm{loc}}(\Omega\times I;{\mathbb{R}}^d), \qquad p\in\mathcal D'(\Omega\times I), \] such that, for every $\varphi\in C_c^\infty(\Omega\times I;{\mathbb{R}}^d)$, \[ \int_I\!\int_\Omega \left(\sum_{i=1}^d u_i\partial_t\varphi_i +\sum_{i,j=1}^d u_i u_j\partial_{x_j}\varphi_i\right)\,{\,\mathrm{d}} x\,{\,\mathrm{d}} t +\langle p,\operatorname{div}_x\varphi\rangle=0, \] and, for every $\psi\in C_c^\infty(\Omega\times I)$, \[ \int_I\!\int_\Omega u\cdot\nabla_x\psi\,{\,\mathrm{d}} x\,{\,\mathrm{d}} t=0. \] Here $\mathcal D'$ denotes distributions and $\langle\cdot,\cdot\rangle$ denotes their action on test functions. Compactly supported interior tests impose no initial or boundary conditions. This reference definition is not asserted to be the solution class intended by the source.

Source question. What definition of a weak solution of the incompressible Euler equations, together with what additional admissibility requirements, captures solutions that are physically reasonable?

The phrase "physically reasonable" is not defined in the record. No particular energy inequality, regularity threshold, approximation procedure, or other selection criterion is specified. Thus the intended admissible class and a precise property to establish about that class remain unresolved; the question cannot faithfully be converted into a single quantified conjecture from the supplied information.

Short English statement

The source asks how to define physically reasonable weak solutions of the incompressible Euler equations. An interior distributional formulation is provided as a reference, but the intended domain, data, solution class, and physical admissibility criterion are unspecified. The source therefore records a formulation question requiring further mathematical clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001308$, title "Physically Admissible Weak Solutions of Incompressible Euler," source problem code catalogue entry, in the collection "Oberwolfach Reports Open Problems." No individual proposer is supplied; the proposed year is recorded as 2009.
• Supplied report citation: Mathematical Aspects of Hydrodynamics (2010), Oberwolfach Reports, DOI 10.4171/owr/2009/34, pp. 1921--1942. https://doi.org/10.4171/owr/2009/34. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled "Hydrodynamics report": https://doi.org/10.4171/owr/2009/34.
• Supplied literature source, labeled "De Lellis--Szekelyhidi existence theorem": https://arxiv.org/abs/1102.3614.
• Supplied literature source, labeled "Admissibility nonuniqueness": https://arxiv.org/abs/0712.3288.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open, describes the prompt as underspecified because weak solutions depend on regularity and admissibility classes, and reports active existence and uniqueness questions with strong nonuniqueness results already known. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The record expressly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-status field is "exact," and its verification note reports manual confirmation of a self-contained, mathematically coherent extraction without LaTeX or report-layout artifacts. That supplied extraction assessment does not resolve the missing formulation choices identified here.