Global Existence and Singularities for the Euler–Nordström System

Definitions and mathematical statement

The source asks about the Euler--Nordström system, a coupled fluid/scalar-gravity evolution, without supplying its equations, equation of state, spatial setting, or physical parameters. Let $U$ denote all unknown fields and let $X$ be the intended function or distribution space, with an admissible initial-data subset $\mathcal D\subseteq X$. Its first question is local well-posedness: for data $U_0\in\mathcal D$, does there exist a positive time and a unique solution that remains in $X$ and depends continuously on $U_0$? In a usual strong-topology formulation this would mean a solution $U\in C([0,T];X)$ with $U(0)=U_0$, uniqueness in a specified solution class, and continuity of the solution map on an initial-data neighborhood with a common positive time. This is a conditional explanation of well-posedness, not a theorem for every unspecified distribution space. The choice of topology and any higher time regularity remain missing.

Its second question is to determine which initial data give global solutions and which give finite-time singularities, and to determine the nature of any singularity. Once a local theory is fixed, write $T_*(U_0)$ for maximal forward existence time. The question is a classification of the alternatives $T_*(U_0)=\infty$ and $T_*(U_0)<\infty$, not simply the logical observation that a maximal time is finite or infinite. A finite maximal time must be connected to a stated continuation criterion before it implies a particular norm blow-up, shock, or loss of physical admissibility. No such criterion or singularity type is supplied. The 2026-08-21 triage labels the general item open and reports that existing well-posedness and small-data results do not settle this broad global-versus-singular behavior question.

Short English statement

Ask for local well-posedness in the intended space and for a classification of global evolution versus finite-time breakdown in the Euler--Nordström system. The equations, data class, topology, continuation criterion, and requested singularity description are missing.

Sources

• Local UnsolvedMath registry, permanent ID 30000981, source problem catalogue entry, Global Existence and Singularities for the Euler--Nordström System, proposed year 2008; no individual proposer supplied.
• Supplied citation: Nonlinear Evolution Equations (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/27, pp. 1475--1536: https://doi.org/10.4171/owr/2008/27. Supplied literature label: original Oberwolfach report.
• Supplied literature label: Speck, Euler--Nordstrom well-posedness, https://arxiv.org/abs/0802.2090.