Existence of global energy-admissible weak solutions of the incompressible Euler equations for arbitrary initial data in the energy space

For the incompressible Euler equations on the periodic box $Q=[0,2\pi]^d$ , $d\ge 2$ , global distributional weak solutions are known to exist for every divergence-free initial datum $v_0\in H(Q)$ (the energy space of solenoidal zero-mean $L^2$ vector fields), but the known constructions force the kinetic energy to jump upward at $t=0$ . A weak solution is called admissible if its total kinetic energy $E(t)$ never exceeds the initial energy $E(0)$ for all $t\ge 0$ . Admissible global weak solutions are currently constructed only for an $L^2$ -dense subset of initial data, and convex integration cannot plainly be extended to all data, since for smooth data any admissible solution must coincide with the classical solution by weak-strong uniqueness. As of 2026 the problem remains open in every dimension: does every $v_0\in H(Q)$ admit a global admissible weak solution, and if not, which data do? Recent partial results (generic 2D well-posedness with energy balance, prescribed-energy-profile constructions, and topological smallness results for decaying solutions) sharpen the landscape on both sides without settling the core question.

Background

The incompressible Euler equations $\partial_t v+\operatorname{div}(v\otimes v)+\nabla p=0$ , $\operatorname{div}v=0$ model an ideal (inviscid, incompressible) fluid. On the periodic box $Q=[0,2\pi]^d$ , $d\ge 2$ , the natural finite-energy phase space is $H(Q)$ : the space of $Q$ -periodic, square-integrable, divergence-free vector fields with zero mean (the zero-mean condition is harmless because of Galilean invariance). A weak solution with initial datum $v_0\in H(Q)$ is a map $v\in L^\infty((0,\infty);H(Q))$ satisfying the equations in the sense of distributions against smooth periodic divergence-free test fields and attaining $v(\cdot,0)=v_0$ weakly in $L^2$ (such solutions are weakly continuous in time, so the initial condition is attained in the weak $L^2$ sense, the standard convention in the source paper). Its kinetic energy is $E(t)=\frac12\int_Q|v(x,t)|^2\,dx$ , which for smooth solutions is conserved. For the viscous Navier–Stokes equations, global weak solutions satisfying the energy inequality $E(t)\le E(0)$ have been known since the 1930s, but the inviscid case offers no parabolic smoothing and hence no comparable compactness argument. Following the terminology of De Lellis and Székelyhidi (2010), a weak solution of Euler is called admissible if $E(t)\le E(0)$ for all $t\ge 0$ ; admissibility is the minimal physical energy constraint, and it implies weak-strong uniqueness: any admissible solution whose initial datum admits a (Lipschitz) classical solution must coincide with that classical solution for as long as the latter exists. Modern existence theory for non-smooth data rests on convex integration, a flexible-construction technique in which one first builds a smooth subsolution — a triple solving the linear parts of the equations whose associated generalized energy strictly undershoots a prescribed energy profile — and then corrects it to infinitely many exact weak solutions; the solutions produced this way are often called wild solutions. De Lellis and Székelyhidi (2010) showed that admissibility criteria (energy conservation or dissipation) do not restore uniqueness of such wild solutions, and Buckmaster, De Lellis, Székelyhidi, and Vicol (2019) proved Onsager's conjecture for admissible weak solutions on the three-dimensional torus (Onsager's conjecture asserts that Hölder-continuous weak solutions with exponent above $1/3$ conserve energy while below $1/3$ dissipative solutions exist). Wiedemann (2011) proved that every $v_0\in H(Q)$ admits infinitely many global weak solutions, but with a discontinuous energy increase at $t=0$ , so no energy inequality holds; and Székelyhidi and Wiedemann (2012) proved, via generation of admissible measure-valued solutions (solution concepts relaxed to Young measures carrying oscillation and concentration effects) by exact weak solutions, that an $L^2$ -dense set of initial data each admits infinitely many admissible weak solutions. Whether the dense set can be enlarged to all of $H(Q)$ is stated in both papers to be a delicate open issue: the initial-time energy jump is intrinsic to the subsolution construction for rough data, while for smooth data weak-strong uniqueness forces any admissible solution to be the classical one, so any all-data construction must be rigid exactly where data are smooth and flexible where they are rough.

Problem Statement

Let $Q=[0,2\pi]^d$ with $d\ge2$ , and let $H(Q)$ be the space of $Q$ -periodic, square-integrable, divergence-free vector fields with zero mean. Consider the incompressible Euler equations $\partial_t v+\operatorname{div}(v\otimes v)+\nabla p=0$ , $\operatorname{div}v=0$ on $Q\times[0,\infty)$ , with weak solutions understood distributionally against smooth $Q$ -periodic divergence-free test fields and kinetic energy $E(t)=\frac12\int_Q|v(x,t)|^2\,dx$ . The problem, stated as open in the source paper and still open as of 2026, has one coherent objective — an admissible-existence theory over the whole energy space — with two facets: (i) does every initial datum $v_0\in H(Q)$ admit a global weak solution $v\in L^\infty((0,\infty);H(Q))$ with $v(\cdot,0)=v_0$ (attained weakly in $L^2$ ) and $E(t)\le E(0)$ for all $t\ge0$ (an admissible solution)? (ii) if not, characterize the class of initial data in $H(Q)$ for which admissible global weak solutions exist. Any resolution must be consistent with weak-strong uniqueness (admissible solutions with data admitting a local classical solution coincide with that classical solution on its lifespan, so non-unique admissible solutions can occur only for highly irregular data) and must handle the initial time: the inequality is required from $t=0$ onward, which excludes the upward energy jumps at $t=0$ inherent in the known all-data weak-existence constructions. The question is open in every dimension $d\ge2$ on the periodic box, and analogously on smooth bounded domains with the corresponding energy space.

References

Emil Wiedemann, Existence of weak solutions for the incompressible Euler equations, Ann. Inst. Henri Poincaré C, Anal. Non Linéaire 28 (2011) 727–730, DOI 10.1016/j.anihpc.2011.05.002, arXiv:1102.3614, https://arxiv.org/abs/1102.3614
Camillo De Lellis and László Székelyhidi Jr., On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal. 195 (2010) 225–260, DOI 10.1007/s00205-008-0201-x, https://doi.org/10.1007/s00205-008-0201-x
László Székelyhidi Jr. and Emil Wiedemann, Young measures generated by ideal incompressible fluid flows, Arch. Ration. Mech. Anal. 206 (2012) 333–366, DOI 10.1007/s00205-012-0540-5, arXiv:1101.3499, https://arxiv.org/abs/1101.3499
S. Daneri, Cauchy problem for dissipative Hölder solutions to the incompressible Euler equations, Comm. Math. Phys. 329 (2014) 745–786, DOI 10.1007/s00220-014-1973-5, https://doi.org/10.1007/s00220-014-1973-5
Tristan Buckmaster, Camillo De Lellis, László Székelyhidi Jr., and Vlad Vicol, Onsager's conjecture for admissible weak solutions, Comm. Pure Appl. Math. 72 (2019) 229–274, DOI 10.1002/cpa.21781, https://doi.org/10.1002/cpa.21781
Tristan Buckmaster and Vlad Vicol, Convex integration constructions in hydrodynamics, Bull. Amer. Math. Soc. 58 (2020) 1–44, DOI 10.1090/bull/1713, https://doi.org/10.1090/bull/1713
Calvin Khor and Changxing Miao, Construction of solutions to the 3D Euler equations with initial data in $H^\beta$ for $\beta>0$ , Commun. Inf. Syst. 25 (2025) 389–425, DOI 10.4310/cis.250823005729, https://doi.org/10.4310/cis.250823005729
Calvin Khor, Changxing Miao, and Weikui Ye, Infinitely many non-conservative solutions for the three-dimensional Euler equations with arbitrary initial data in $C^{1/3-\varepsilon}$ , arXiv:2204.03344 (2022), https://arxiv.org/abs/2204.03344
André Guerra, Lukas Koch, and Sauli Lindberg, Nonlinear open mapping principles, with applications to the Jacobian equation and other scale-invariant PDEs, Adv. Math. 415 (2023) 108869, DOI 10.1016/j.aim.2023.108869, arXiv:2010.10497, https://arxiv.org/abs/2010.10497
Yann Brenier, The initial value problem for the Euler equations of incompressible fluids viewed as a concave maximization problem, Comm. Math. Phys. 364 (2018) 579–605, DOI 10.1007/s00220-018-3240-7, arXiv:1706.04180, https://arxiv.org/abs/1706.04180
Francisco Mengual, Non-uniqueness of admissible solutions for the 2D Euler equation with $L^p$ vortex data, Comm. Math. Phys. 405 (2024), article 207, DOI 10.1007/s00220-024-05065-9, https://doi.org/10.1007/s00220-024-05065-9
Miriam Buck and Stefano Modena, Non-uniqueness and energy dissipation for 2D Euler equations with vorticity in Hardy spaces, arXiv:2306.05948 (2023), https://arxiv.org/abs/2306.05948
Lucio Galeati, The 2D Euler equations are well-posed for generic initial data in $L^2$ , arXiv:2604.14100 (2026), https://arxiv.org/abs/2604.14100
Jule Schindler and Emil Wiedemann, The nonlocal-to-local limit for the inviscid Leray- $\alpha$ equations, arXiv:2601.14813 (2026), https://arxiv.org/abs/2601.14813
Daniel W. Boutros and Simon Markfelder, Non-uniqueness of global-in-time admissible weak solutions to the isentropic compressible Euler equations for a dense set of initial data, arXiv:2606.14308 (2026), https://arxiv.org/abs/2606.14308