Existence of energy-admissible global weak solutions to the incompressible Euler equations for arbitrary $L^2$ initial data

For the incompressible Euler equations on the periodic box $Q=[0,2\pi]^d$, $d\ge 2$, weak solutions issuing from every divergence-free $v_0\in L^2$ exist, but the known constructions violate the energy inequality. The problem, posed explicitly by Wiedemann (2011) and Székelyhidi–Wiedemann (2012), asks whether every such initial datum admits a global weak solution whose kinetic energy never exceeds its initial value ($E(t)\le E(0)$ for all $t\ge 0$, i.e. an admissible solution). As of 2026 this is open: admissible solutions are known only on $L^2$-dense (in 2D: Baire-generic) classes of data, and no counterexample datum is known either.

Background

The incompressible Euler equations $\partial_t v+\mathrm{div}(v\otimes v)+\nabla p=0$, $\mathrm{div}\,v=0$ describe an ideal (inviscid) fluid. On the periodic box $Q=[0,2\pi]^d$ with $d\ge 2$, write $H(Q)$ for the space of $Q$-periodic, mean-zero, divergence-free vector fields in $L^2$ (the natural energy space for the problem; the mean-zero condition is harmless by Galilean invariance). A weak solution with initial datum $v_0\in H(Q)$ is a field $v\in L^\infty((0,\infty);H(Q))$ satisfying the equations in the sense of distributions, with the initial datum included in the distributional identity through test functions on $[0,\infty)$ (as in Wiedemann's formulation, in which the solutions lie in $C([0,\infty);H_w(Q))$ and attain $v(0)=v_0$), so that $E(0)=\frac12\|v_0\|_{L^2}^2$; its kinetic energy is $E(t)=\frac12\int_Q|v(x,t)|^2\,dx$. For the viscous Navier–Stokes equations, Leray (1934) constructed global weak solutions satisfying the energy inequality $E(t)\le E(0)$ for arbitrary $L^2$ data, and this remains the foundational existence theory of mathematical fluid dynamics. For Euler the analogous statement is open. A weak Euler solution is called admissible (or energy admissible) if $E(t)\le E(0)$ for all $t\ge 0$. Admissibility carries a strong uniqueness consequence: by weak–strong uniqueness, any admissible solution coincides with the classical solution for as long as the latter exists, and in two dimensions the classical Yudovich theory (1963) provides a unique global admissible solution whenever the vorticity $\omega=\mathrm{curl}\,v$ is bounded. Since the 2000s, convex integration (De Lellis–Székelyhidi, 'On admissibility criteria for weak solutions of the Euler equations', 2010) has produced 'wild' Euler solutions — one starts from a strict subsolution, a smoother relaxed field whose defect relative to the kinetic energy is pointwise positive, and adds oscillations to obtain infinitely many exact weak solutions with prescribed energy profiles. This mechanism, however, always produces infinitely many solutions from a given construction, so it can never work for data on which uniqueness holds; Székelyhidi–Wiedemann ('Young measures generated by ideal incompressible fluid flows', 2012) deduced from the DiPerna–Majda theory of measure-valued solutions (objects recording the oscillation and concentration behavior of weakly convergent approximate-solution sequences, which exist and are admissible for all $L^2$ data) that an $L^2$-dense set of initial data carries infinitely many admissible weak solutions, and stated that improving this to a larger class is 'a very delicate issue'. Wiedemann (2011) closed the admissibility-free question: for every $v_0\in H(Q)$ there exist infinitely many global weak solutions with bounded energy (even $E(t)\to 0$), but necessarily with a discontinuous energy increase at $t=0$, so no energy inequality. Recent 2D progress connects admissibility with anomalous dissipation (nonvanishing energy dissipation in the vanishing-viscosity limit): De Rosa–Park (2025) showed that for 2D data whose vorticity is a measure with positive singular part (e.g. vortex-sheet type data), vanishing-viscosity limits converge to an admissible weak solution; Galeati (2026) proved that a residual set in the sense of Baire (a countable intersection of open dense sets, i.e. a generic set in the topological sense) of $L^2$ divergence-free 2D data carries a unique global energy-conserving weak solution recovered by vanishing viscosity — yet for a given arbitrary datum even $L^2$-compactness of vanishing-viscosity sequences is not known. In $d\ge 3$ there is no analogue: only dense classes of data (smooth, Hölder, vortex-sheet, or $L^p$-vorticity type) are covered, and even for smooth data a global-in-time admissible continuation past a potential finite-time singularity is not constructed, since that is entangled with the open blow-up problem.

Problem Statement

On the periodic box $Q=[0,2\pi]^d$ with $d\ge 2$, let $H(Q)$ denote the space of $Q$-periodic, mean-zero, divergence-free $L^2$ vector fields. Does every initial datum $v_0\in H(Q)$ admit a global-in-time weak solution $v\in L^\infty((0,\infty);H(Q))$ of the incompressible Euler equations with $v(0)=v_0$ that is admissible in the energy sense, i.e. whose kinetic energy $E(t)=\frac12\int_Q|v(x,t)|^2\,dx$ satisfies $E(t)\le E(0)$ for all $t\ge 0$? Any answer must be consistent with the known constraints: for smooth data the admissible solution must coincide with the classical solution by weak–strong uniqueness (globally in 2D for bounded vorticity by Yudovich's theorem), while infinitely many admissible solutions exist for dense classes of rough data such as $L^p$-vorticity or vortex-sheet data. Both the two-dimensional case (where currently only a Baire-generic set of data is covered) and the case $d\ge 3$ are open.

References

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