Beyond the Yudovich class for the 2D Euler equations: propagating localized instability to full non-uniqueness of unforced $L^p$ weak solutions, Gevrey-type well-posedness, and the self-similar structure of the algebraic singularity

For the unforced two-dimensional incompressible Euler equations, uniqueness of weak solutions is known only for bounded vorticity (Yudovich's theorem); for unbounded vorticity in $L^p$ the problem is open. Elgindi, Murray and Said constructed a local well-posedness class for $m$ -fold symmetric vorticities with algebraic point singularities $(r+\theta)^{-\alpha}$ , showed that solutions in this class blow up and exit it in finite time, and proved a finite-time instability localized at the singular point, which suggests — but does not establish — non-uniqueness of $L^p$ weak solutions after the blow-up time. The three open questions left by that work are: (i) whether the localized instability can be propagated into the bulk to produce genuinely distinct weak solutions of the unforced equation with vorticity in $L^p$ for large $p$ (in the strong form, two solutions coinciding up to the blow-up time and separating after), a question independently formalized as an explicit conjecture by Mengual; (ii) whether local well-posedness extends from analytic (Gevrey-1) regularity to Gevrey classes of exponent $\sigma \le 1/\alpha$ ; (iii) whether the finite-time singularity is asymptotically self-similar. All three questions remain open.

Background

The two-dimensional incompressible Euler equations describe an ideal (inviscid, incompressible) fluid. Written in vorticity form, the unknown is the scalar vorticity $\omega=\operatorname{curl}u$ , which is transported by the velocity field it generates via the Biot–Savart law: $\partial_t\omega+u\cdot\nabla\omega=0$ , $u=\nabla^\perp\Delta^{-1}\omega$ , where $\nabla^\perp=(-\partial_y,\partial_x)$ . Because $\omega$ is transported, its size is conserved: whatever singularity the initial vorticity carries cannot be smoothed, only moved and deformed. Yudovich proved in 1963 that for initial vorticity in $L^1\cap L^\infty$ the Cauchy problem is globally well-posed; the velocity is then log-Lipschitz (Lipschitz up to a logarithmic correction), which is exactly regular enough for uniqueness of the Lagrangian flow. The class of bounded vorticities is accordingly called the Yudovich class. For unbounded vorticity in $L^p$ , existence of global weak solutions was proved by DiPerna and Majda, but uniqueness is open: for $2<p<\infty$ the velocity is Hölder-continuous yet its modulus of continuity $\pi$ fails the Osgood criterion (the classical condition $\int_0^1 ds/\pi(s)=\infty$ that guarantees unique trajectories for the particle ordinary differential equation $\dot X=u(X,t)$ , on which uniqueness of the Lagrangian flow — the trajectory map of the velocity field — rests), so both the Yudovich uniqueness argument and standard Lagrangian tools break down. Whether weak solutions of the unforced equation with vorticity in $L^p$ are unique is one of the basic open problems of the field. The strongest related results are: (a) Vishik proved non-uniqueness in the vorticity class $L^1\cap L^p$ for any $2<p<\infty$ for the forced equation, where an external body force sustains an unstable self-similar eigenmode of a singular vortex; simpler proofs of this forced result were later given by Dolce–Mescolini and by Castro–Faraco–Mengual–Solera; (b) Mengual removed the force but only at the price of leaving the $L^p$ vorticity class, obtaining non-uniqueness among admissible $L^2$ -velocity solutions whose vorticity is merely a distribution; (c) Bruè–Colombo–Kumar, via a new convex integration scheme (convex integration is the h-principle-style method, initiated by De Lellis and Székelyhidi, that builds wildly oscillating weak solutions), proved non-uniqueness for the unforced equation with vorticity in $C_tL^p_x$ for some small $p>1$ , their solutions satisfying only the symmetrized (Delort–Schochet) vorticity formulation — a weaker weak formulation, meaningful even for measure-valued vorticity, in which the vorticity equation is tested through a symmetric double integral against the Biot–Savart interaction kernel — rather than the distributional transport equation, which requires $p\ge 4/3$ ; the large- $p$ regime relevant here is beyond all current convex-integration constructions. Against this background, Elgindi, Murray and Said introduced a local well-posedness theory for vorticities with algebraic point singularities, without assuming self-similarity. Their data are $m$ -fold symmetric (invariant under rotation by $2\pi/m$ ) and odd in the polar angle $\theta$ , with $m\ge 3$ , and carry a singularity of order $(r+\theta)^{-\alpha}$ , $0<\alpha<1$ , near the origin; such vorticities lie in $L^p$ exactly when $p<3/\alpha$ , so the interesting regime is $2<p<3/\alpha$ . Their Theorem 1.2 proves local existence, uniqueness, and continuous dependence in an analytic-type norm that weights derivatives by powers of $(r+\theta)$ ; the analytic (Gevrey-1, i.e. roughly $k!$ -weighted derivative) regularity compensates a derivative loss in closing the estimates. Their Theorem 1.4 shows that for the explicit datum $\omega_0=-(r+\theta)^{-\alpha}$ the solution blows up in finite time $T^*$ and exits this class, and that this yields a finite-time nonlinear instability among weak solutions in $L^p_{loc}$ for $p<1/\alpha$ (in particular, for some $p>1$ ), localized at the singular point. The dynamics at the origin reduce to the 0-homogeneous (angle-only) Euler equation $\partial_t g+2G\partial_\theta g=0$ , $(4+\partial_{\theta\theta})G=g$ , studied by the same authors and by Jeong–Said; the long-time relaxation theory for that reduced equation was developed by Elgindi–Murray–Said for $m\ge 4$ and recently completed for $m=3$ by Cao–Fan–Qin, while Shao–Wei–Zhang constructed self-similar algebraic spiral solutions near power-law vorticities.

Problem Statement

The problem comprises the three open questions left by Elgindi–Murray–Said's construction of a well-posedness class beyond the Yudovich class, tied together by one scientific objective: determine the fate of the Cauchy problem for the unforced 2D Euler equations when the vorticity has an algebraic point singularity — in particular, whether uniqueness fails there. Throughout, the equation is the unforced system $\partial_t\omega+u\cdot\nabla\omega=0$ , $u=\nabla^\perp\Delta^{-1}\omega$ on $\mathbb{R}^2$ , with $m$ -fold symmetric, odd-in- $\theta$ data, $m\ge 3$ , and singularity of order $(r+\theta)^{-\alpha}$ , $0<\alpha<1$ , so that $\omega\in L^p$ for $p<3/\alpha$ . The questions are: (1) [core, non-uniqueness] Can the instability localized at the singular point be propagated into the bulk of the fluid to establish full non-uniqueness of weak solutions of the unforced equation in the $L^p$ vorticity class for large $p$ (the strong form: two distinct weak solutions, both with vorticity in $L^p$ for some $2<p<3/\alpha$ , that coincide up to the finite blow-up time $T^*$ of Theorem 1.4 — in particular coincide with the unique well-posedness-class solution on $[0,T^*)$ — and differ for later times)? The weaker form of the question drops the coincidence up to $T^*$ and concerns distinct $L^p$ weak solutions. A positive answer in the non-uniqueness direction would provide an instance, for some $2<p<3/\alpha$ , of the general conjecture, stated openly by Mengual, that for any $2<p<\infty$ there is $L^1\cap L^p$ vorticity data admitting more than one weak solution with vorticity in $L^\infty_t(L^1\cap L^p)$ . (2) [well-posedness threshold] Does the local existence, uniqueness, and continuous-dependence result of Theorem 1.2 extend from the analytic-type (Gevrey-1) class to Gevrey classes of exponent $\sigma\le 1/\alpha$ within the weighted framework of Theorem 1.2 (cf. Remark 1.3 of the source), i.e. can the estimate be closed with the expected sharp loss of $\alpha$ derivatives rather than a full derivative? (3) [structure of the singularity] Determine whether the singularity formed in Theorem 1.4 is asymptotically self-similar, and if so describe the profile — as announced by the authors for a future paper. The core question (1) is two-sided — a non-uniqueness construction or a uniqueness theorem in the class — and (2) and (3) are the companion questions named by the same source, each standing independently of the others.

References

Tarek M. Elgindi, Ryan W. Murray, Ayman R. Said, Wellposedness and singularity formation beyond the Yudovich class, 2023, arXiv:2312.17610, DOI 10.48550/arXiv.2312.17610, https://arxiv.org/abs/2312.17610
Tarek M. Elgindi, Ryan W. Murray, Ayman R. Said, On the long-time behavior of scale-invariant solutions to the 2d Euler equation and applications, 2022, arXiv:2211.08418, https://arxiv.org/abs/2211.08418
V. I. Yudovich, Non-stationary flow of an ideal incompressible liquid, 1963, USSR Computational Mathematics and Mathematical Physics 3(6):1407-1456 (English translation of the Russian original), DOI 10.1016/0041-5553(63)90247-7, https://doi.org/10.1016/0041-5553(63)90247-7
R. J. DiPerna, A. J. Majda, Concentrations in regularizations for 2-D incompressible flow, 1987, Communications on Pure and Applied Mathematics 40(3):301-345, DOI 10.1002/cpa.3160400304, https://doi.org/10.1002/cpa.3160400304
Misha Vishik, Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part I, 2018, arXiv:1805.09426, https://arxiv.org/abs/1805.09426
Misha Vishik, Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part II, 2018, arXiv:1805.09440, https://arxiv.org/abs/1805.09440
Francisco Mengual, Non-uniqueness of admissible solutions for the 2D Euler equation with $L^p$ vortex data, Communications in Mathematical Physics 405(9), 2024 (preprint 2023, arXiv:2304.09578), DOI 10.1007/s00220-024-05065-9, https://doi.org/10.1007/s00220-024-05065-9
Elia Bruè, Maria Colombo, Anuj Kumar, Flexibility of Two-Dimensional Euler Flows with Integrable Vorticity, 2024, arXiv:2408.07934, https://arxiv.org/abs/2408.07934
In-Jee Jeong, Ayman R. Said, Logarithmic spirals in 2d perfect fluids, Journal de l'École polytechnique — Mathématiques 11:655-682, 2024 (preprint 2023, arXiv:2302.09447), DOI 10.5802/jep.262, https://doi.org/10.5802/jep.262
Feng Shao, Dongyi Wei, Zhifei Zhang, Self-similar algebraic spiral solution of 2-D incompressible Euler equations, 2023, arXiv:2305.05182, https://arxiv.org/abs/2305.05182
Michele Dolce, Giulia Mescolini, Self-similar instability and forced nonuniqueness: an application to the 2D Euler equations, 2024, arXiv:2411.18452, https://arxiv.org/abs/2411.18452
Ángel Castro, Daniel Faraco, Francisco Mengual, Marcos Solera, A proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation, 2024, arXiv:2404.15995, https://arxiv.org/abs/2404.15995
Daomin Cao, Junhong Fan, Guolin Qin, Gradient growth and relaxation to jump profiles for 3-fold symmetric scale-invariant Euler flows, 2026, arXiv:2608.16755, https://arxiv.org/abs/2608.16755