Uniqueness for the two-dimensional incompressible Euler equations with unbounded $L^p$ vorticity

For the two-dimensional incompressible Euler equations, uniqueness of weak solutions is known only for bounded vorticity (Yudovich, 1963); for vorticities in $L^p$ with finite $p$ the uniqueness question has been open for decades. Ciampa, Crippa and Spirito (Arch. Ration. Mech. Anal. 240 (2021) 295-326), in their study of the vanishing viscosity limit of the 2D Navier-Stokes equations, prove that the vorticities converge strongly to a renormalized Euler solution and explicitly flag uniqueness for unbounded vorticity as the central unresolved issue of their framework, since their convergence result identifies a canonical limit exactly in the class where uniqueness is unknown. The current status is as follows: Vishik settled the forced problem by proving nonuniqueness for every $p>2$ with a suitable force; convex integration has produced nonunique unforced solutions only outside the $L^p$ scale, in the admissible class, or among non-renormalized solutions; uniqueness beyond Yudovich-type classes and nonuniqueness within the renormalized class both remain open for every finite $p$. The surviving open core is: are distributional/renormalized solutions of the unforced 2D Euler equations with initial vorticity in $L^1\cap L^p$, $1\le p<\infty$, unique or nonunique?

Background

The two-dimensional incompressible Euler equations describe the velocity field $u$ of an ideal (inviscid, incompressible) fluid; in the vorticity formulation one transports the scalar vorticity $\omega=\operatorname{curl}u$ along the flow: $\partial_t\omega+u\cdot\nabla\omega=0$ with $u=K*\omega$ recovered from $\omega$ by the Biot-Savart law (a singular integral which in 2D gains one derivative). Classical existence theory (DiPerna and Majda, 1987) yields global weak solutions for initial vorticity $\omega_0\in L^1\cap L^p$, $p>1$. Uniqueness is a different matter: Yudovich (1963) proved that for bounded vorticity ($\omega_0\in L^\infty$) the solution is unique in the class $\omega\in L^\infty(L^\infty)$, the key point being that the Biot-Savart velocity is then almost Lipschitz so an Osgood/Grönwall argument closes. For any finite $p$ the velocity is merely Sobolev, Yudovich's argument breaks down, and uniqueness has remained a famous open problem. A key structural notion is that of renormalized solutions in the sense of DiPerna and Lions (1989): solutions for which $\beta(\omega)$ solves the transport equation for every smooth compactly supported $\beta$; by their theory any solution with vorticity in $L^p$, $p\ge 2$, is automatically renormalized, and Crippa and Spirito (2015) showed that solutions obtained as vanishing viscosity limits of the Navier-Stokes equations are renormalized for every $p\ge 1$. Ciampa, Crippa and Spirito (2021) proved, jointly with an independent simultaneous result of Nussenzveig Lopes, Seis and Wiedemann, that in the inviscid limit $\nu\to 0$ the Navier-Stokes vorticities $\omega^\nu$ converge strongly in $C(L^q)$, $1\le q\le p$, to the renormalized Euler solution with initial data $\omega_0\in L^1\cap L^p$, $1\le p<\infty$, extending the earlier bounded-vorticity result of Constantin, Drivas and Elgindi (2022) for the Yudovich class; on the whole space this yields energy conservation of the limit. Convergence alone, however, does not fix the dynamics uniquely unless the renormalized solution itself is unique, which is open for $p<\infty$; the authors call uniqueness for unbounded vorticities 'an old and outstanding open problem'. Two modern research programs border the problem. First, Vishik (2018) proved via spectral instability of radial vortices that with a suitable (smooth, compactly supported in time away from $t=0$) external force the forced Euler equations admit uncountably many distributional solutions with vorticity in $L^\infty(L^1\cap L^p)$ for every $p\in(2,\infty)$, even from initial vorticity $\omega_0\equiv 0$; the detailed exposition by Albritton, Brué, Colombo, De Lellis, Giri, Janisch and Kwon (2021, also a Princeton University Press monograph) makes this precise. This shows Yudovich's uniqueness cannot extend to the $L^p$ scale in the forced case, but leaves the unforced equations untouched. Second, convex integration (a Nash-Moser-type iteration producing infinitely many weak solutions, pioneered by De Lellis and Székelyhidi in the Onsager program) has produced nonunique unforced solutions in neighbouring classes: vorticity in the Lorentz space $L^{1,\infty}$ (Brué and Colombo, 2023; $L^{1,\infty}$ is strictly larger than every $L^p$), in Hardy spaces $H^p$ with $2/3<p<1$ (Buck and Modena, 2023), admissible solutions (an energy-admissibility class in the convex-integration framework) with $L^p$ vorticity data for $2<p<\infty$, extended to $1<p<\infty$ (Mengual, 2024), and most recently nonconservative weak solutions with vorticity in $C(L^p)$ for some $p>1$ (Brué, Colombo and Kumar, 2026) — but the latter are non-renormalized, satisfy only the symmetrized (Delort) vorticity formulation, and provably cannot be vanishing viscosity limits. In the opposite, rigidity direction: every weak solution with vorticity in $L^p$, $p\ge 3/2$, conserves kinetic energy (Cheskidov, Lopes Filho, Nussenzveig Lopes and Shvydkoy, 2016); Elgindi, Murray and Said (2023) exhibited a local-in-time uniqueness class strictly beyond the Yudovich class from which solutions can nevertheless exit in finite time; Galeati (2026) proved generic (Baire-residual in $L^2$) global well-posedness; and Albritton, Colombo and Mescolini (2025) showed in Vishik's forced scenario that vanishing viscosity acts as a selection principle with a sharp uniqueness threshold for perturbation size versus viscosity.

Problem Statement

For the unforced two-dimensional incompressible Euler equations on the torus $\mathbb{T}^2$ or the whole plane $\mathbb{R}^2$, written in vorticity form $\partial_t\omega+u\cdot\nabla\omega=0$ with $u=K*\omega$ the Biot-Savart velocity, consider initial vorticities $\omega_0\in L^1\cap L^p$ with finite $p$, $1\le p<\infty$, and the natural weak-solution class $\omega\in L^\infty((0,T);L^1\cap L^p)$ solving the vorticity equation in the sense of distributions (the distributional formulation is well defined for $p\ge 4/3$; for lower $p$ one uses the renormalized formulation; the still weaker symmetrized (Delort) formulation, in which only a symmetrized version of the vorticity equation is imposed, is mentioned only for context — the known solutions in that class are non-renormalized — and is not the target class of this question), in particular within the class of renormalized solutions, which for $p\ge 2$ coincides with the distributional class and for every $p\ge 1$ contains all solutions selected by the vanishing viscosity limit. Determine whether solutions in this class are unique: either prove that for some range of finite $p$ any two such solutions with the same initial vorticity coincide, thereby extending Yudovich's $L^\infty$ uniqueness theorem to the $L^p$ scale, or construct for some finite $p$ two distinct distributional/renormalized solutions of the unforced equations sharing the same initial vorticity in $L^1\cap L^p$. The sharp $p^*$ separating renormalized from non-renormalized weak solutions with $L^p$ vorticity remains open (posed as Problem 1.5 by Brué-Colombo-Kumar, with $p^*=3/2$ speculated). The question concerns the unforced equations and the distributional/renormalized solution class; nonuniqueness for the forced equations, in the admissible class of convex integration, or among non-renormalized solutions is already known.

References

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