Uniqueness versus non-uniqueness for the 2D incompressible Euler equations with vorticity in $L^p$, $p<\infty$ (the Yudovich gap)

For the two-dimensional incompressible Euler equations, Yudovich (1963) proved global existence and uniqueness of weak solutions with bounded vorticity, and DiPerna–Majda (1987) proved global existence for initial vorticity in $L^1\cap L^p$ for every finite $p$; yet for no unbounded vorticity class between these two regimes is either uniqueness or non-uniqueness known for the unforced equation. As of August 2026, the forced analogue is fully resolved (Vishik: non-uniqueness for every finite $p$), and the unforced problem has partial non-uniqueness results in adjacent classes — admissible solutions whose vorticity is only a distribution for $t>0$ (Mengual), and genuine $L^p$ vorticity only for some small $p>1$ in a non-renormalized, non-conservative class (Bruè–Colombo–Kumar) — while the expectation in the literature is that non-uniqueness holds for every $p<\infty$. The precise open core survives: for any specified finite $p$ (most prominently $2<p<\infty$, where the velocity is bounded and Hölder but not Osgood), determine whether the Cauchy problem is unique or non-unique among weak solutions with vorticity in $L^\infty_t L^p_x$, at least in the admissible subclass, with the renormalized and vanishing-viscosity subclasses as desirable further refinements.

Background

The two-dimensional incompressible Euler equations describe an ideal (inviscid) fluid. In the velocity formulation one seeks a divergence-free field $v$ and a pressure $p$ with $\partial_t v + \operatorname{div}(v\otimes v) + \nabla p = 0$. Taking the curl eliminates the pressure and transports the vorticity $\omega = \operatorname{curl}\,v$, giving the vorticity formulation

$$\partial_t \omega + v\cdot\nabla\omega = 0, \qquad v = \nabla^\perp \Delta^{-1}\omega,$$

where the second relation is the Biot–Savart law (the velocity is recovered from the vorticity by a singular integral operator, roughly a derivative of the Newtonian potential of $\omega$). A weak solution here is a distributional solution with finite kinetic energy, $v\in C([0,T];L^2)$; it is called admissible if its kinetic energy does not increase in time.

The cornerstone result is Yudovich's 1963 theorem: for bounded initial vorticity, $\omega^\circ\in L^1\cap L^\infty$, there is a global weak solution, unique within the class of bounded admissible solutions. The proof exploits that Biot–Savart maps $L^\infty$ vorticity to a log-Lipschitz velocity, whose modulus of continuity decays like $1/|\log r|$ and satisfies the Osgood criterion (a modulus of continuity $\kappa$ with $\int_0^1 \mathrm{d}s/\kappa(s)=\infty$, the classical condition guaranteeing uniqueness of solutions of ordinary differential equations), so the Lagrangian flow map is uniquely defined and an energy estimate for the difference of two solutions closes. Uniqueness was later extended to the Yudovich spaces $Y^\Theta$ of unbounded vorticities lying in every $L^p$ with $\|\omega\|_{L^p}\le\Theta(p)$ where $\Theta$ grows slowly enough for an Osgood-type condition to hold (Yudovich's energy method, Taniuchi's variant for vorticity in $\mathrm{bmo}$, the space of locally integrable functions with locally bounded mean oscillation, and the localized treatment of Crippa–Stefani). Below these classes nothing is known on the uniqueness side: for a single exponent $p>2$ the Biot–Savart velocity is merely Hölder continuous (exponent $1-2/p$), a non-Osgood modulus for which ordinary differential equations already have non-unique solutions; for $p\le 2$ the velocity is not even bounded.

On the existence side, DiPerna and Majda (1987) proved that every $\omega^\circ\in L^1\cap L^p$, $1<p<\infty$, generates a global weak solution with $\omega\in L^\infty_t(L^1\cap L^p)$ (the DiPerna–Majda class). Whether this solution is unique is the classical open problem considered in this task. Ceci and Seis, studying convergence of Euler flows with unbounded vorticity ($p>2$) to point-vortex dynamics, state the status verbatim: existence holds for every $p\ge 1$, 'however still nothing is known about uniqueness if $p<\infty$'.

Subsequent work has delimited the gap without closing it. Vishik (2018) resolved the forced analogue: with a suitable radially symmetric force, non-uniqueness holds in the class $\omega\in L^\infty_t(L^1\cap L^p)$ for every finite $p$; his tour-de-force proof was verified and exposited by Albritton, Bruè, Colombo, De Lellis, Giri, Janisch and Kwon, and rederived by shorter routes (Dolce–Mescolini; Castro–Faraco–Mengual–Solera). For the unforced equation, Mengual constructed, for any $2<p<\infty$, a steady power-law vortex with $\omega\in L^1\cap L^p$ admitting infinitely many bounded admissible solutions — but their vorticity is only a distribution at positive times, so the $L^p$-class question is untouched. Bruè–Colombo proved non-uniqueness with vorticity in the Lorentz space $L^{1,\infty}$ (weak-$L^1$, slightly below $L^1$ in scaling), and Buck–Modena with vorticity in Hardy spaces $H^p$, $0<p<1$. Bruè, Colombo and Kumar finally overcame the long-standing $L^1$-vorticity obstruction of two-dimensional convex integration (the method behind Onsager-type flexibility results — Onsager's conjecture posits a critical Hölder regularity for the velocity, above which Euler flows must conserve energy — based on iteratively adding highly oscillatory perturbations), producing non-unique solutions with vorticity in $C_tL^p(\mathbb{T}^2)$ — but only for some small, unspecified $p>1$, and the solutions are non-conservative and non-renormalized. Renormalized here means that $\beta(\omega)$ solves the transport equation for every test function $\beta$; by the DiPerna–Lions theory renormalization is automatic for $p\ge 2$, which structurally confines all current unforced constructions to subcritical or weaker classes and leaves the range $p\ge 2$ (in particular $2<p<\infty$) with no known non-uniqueness example. On the energy side, Cheskidov, Lopes Filho, Nussenzveig Lopes and Shvydkoy proved that every solution with $\omega\in L^\infty_tL^p_x$, $p\ge 3/2$, conserves kinetic energy, so any non-uniqueness above that threshold must occur among energy-conserving solutions — a two-dimensional, vorticity-based analogue of the Onsager threshold.

The problem also propagates into approximation theory. Vanishing-viscosity solutions (limits of Navier–Stokes solutions as the viscosity $\nu\to0$) with $L^p$ vorticity, $p>1$, are renormalized (Crippa–Spirito) and conserve energy; within the Yudovich class the inviscid limit is unique (Constantin–Drivas–Elgindi; Ciampa–Crippa–Spirito). Whether the vanishing-viscosity or vortex-method solution for a given unbounded $L^p$ vorticity is unique is open; in the forced case, Albritton–Colombo–Mescolini showed vanishing viscosity resolves Vishik's non-uniqueness only below a perturbation threshold $\varepsilon\sim\nu^{\kappa_c}$, so selection is genuinely scale-dependent there. Adding a rough Kraichnan-type transport noise (a spatially singular stochastic perturbation used in turbulence modeling) restores pathwise uniqueness for $p>3/2$ (Coghi–Maurelli), a regularization unavailable deterministically; and Galeati showed that a Baire-generic set of $L^2$ initial data is well-posed, the genericity counterpart to the dense 'wild' data produced by convex integration.

Problem Statement

For the unforced two-dimensional incompressible Euler equations on the torus $\mathbb{T}^2$ (the same question on the plane $\mathbb{R}^2$ is equally open, and an answer in either setting is acceptable), determine on which side of the Yudovich threshold the Cauchy problem falls in each vorticity integrability class $L^p$ with finite $p$. Concretely, for initial data $v^\circ\in L^2$ divergence-free with vorticity $\omega^\circ\in L^1\cap L^p$, $1<p<\infty$:

(a) Non-uniqueness: exhibit, for any specified finite $p$ — most prominently any $2<p<\infty$ as in Mengual's Conjecture 2.1, where the velocity is bounded and Hölder continuous but not Osgood — a datum admitting at least two distinct global weak solutions $v\in C([0,T];L^2)$ with vorticity $\omega\in L^\infty((0,T);L^p)$, required at least to be admissible (satisfying the energy inequality); realising the same datum also inside the renormalized or vanishing-viscosity subclasses (limits of Navier–Stokes solutions or of vortex-method approximations) is desirable but not required;

(b) Uniqueness: prove, if true, that all such solutions coincide for some finite $p$ or for any vorticity class strictly larger than the Osgood-type Yudovich spaces, thereby locating the sharp uniqueness boundary for unbounded vorticity.

Deliverables (a) and (b) are the two sides of a single dichotomy: a definite answer for a single specified exponent $p$ — either two distinct solutions with vorticity in $L^\infty_t(L^1\cap L^p)$, or uniqueness in that class — resolves the Cauchy problem in the class $L^p$ and completes the task, while an answer valid uniformly for every finite $p$ settles well-posedness of the whole DiPerna–Majda class $\omega\in L^\infty_t(L^1\cap L^p)$ and is the strongest form of a solution. Note the structural constraints any answer must respect: for $p\ge3/2$ every solution conserves energy, and for $p\ge2$ every solution is renormalized.

References

1. V. I. Yudovich, Non-stationary flow of an ideal incompressible liquid, USSR Computational Mathematics and Mathematical Physics 3(6):1407–1456 (1963). DOI: 10.1016/0041-5553(63)90247-7, https://doi.org/10.1016/0041-5553(63)90247-7
2. R. J. DiPerna and A. J. Majda, Concentrations in regularizations for 2-D incompressible flow, Communications on Pure and Applied Mathematics 40(3):301–345 (1987). DOI: 10.1002/cpa.3160400304, https://doi.org/10.1002/cpa.3160400304
3. Y. Taniuchi, Uniformly local Lp estimate for 2-D vorticity equation and its application to Euler equations with initial vorticity in bmo, Communications in Mathematical Physics 248(1):169–186 (2004). DOI: 10.1007/s00220-004-1095-6, https://doi.org/10.1007/s00220-004-1095-6
4. G. Crippa and G. Stefani, An elementary proof of existence and uniqueness for the Euler flow in localized Yudovich spaces, arXiv:2110.15648 (2021), https://arxiv.org/abs/2110.15648
5. S. Ceci and C. Seis, Vortex dynamics for 2D Euler flows with unbounded vorticity, Revista Matemática Iberoamericana 37(2) (2021). DOI: 10.4171/rmi/1255, https://doi.org/10.4171/rmi/1255; preprint arXiv:1907.10923, https://arxiv.org/abs/1907.10923
6. M. Vishik, Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part I, arXiv:1805.09426 (2018), https://arxiv.org/abs/1805.09426
7. M. Vishik, Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part II, arXiv:1805.09440 (2018), https://arxiv.org/abs/1805.09440
8. D. Albritton, E. Bruè, M. Colombo, C. De Lellis, V. Giri, M. Janisch and H. Kwon, Instability and nonuniqueness for the 2d Euler equations in vorticity form, after M. Vishik, arXiv:2112.04943 (2021), https://arxiv.org/abs/2112.04943
9. M. Dolce and G. Mescolini, Self-similar instability and forced nonuniqueness: an application to the 2D Euler equations, Journal of the London Mathematical Society (2025). DOI: 10.1112/jlms.70274, https://doi.org/10.1112/jlms.70274; preprint arXiv:2411.18452, https://arxiv.org/abs/2411.18452
10. Á. Castro, D. Faraco, F. Mengual and M. Solera, A proof of Vishik's nonuniqueness theorem for the forced 2D Euler equation, Journal für die reine und angewandte Mathematik (Crelles Journal) (2025). DOI: 10.1515/crelle-2025-0025, https://doi.org/10.1515/crelle-2025-0025
11. F. Mengual, Non-uniqueness of admissible solutions for the 2D Euler equation with $L^p$ vortex data, Communications in Mathematical Physics 405 (2024). DOI: 10.1007/s00220-024-05065-9, https://doi.org/10.1007/s00220-024-05065-9; preprint arXiv:2304.09578, https://arxiv.org/abs/2304.09578
12. E. Bruè and M. Colombo, Nonuniqueness of solutions to the Euler equations with vorticity in a Lorentz space, Communications in Mathematical Physics 403(2):1171–1192 (2023). DOI: 10.1007/s00220-023-04816-4, https://doi.org/10.1007/s00220-023-04816-4
13. M. Buck and S. Modena, Non-uniqueness and energy dissipation for 2D Euler equations with vorticity in Hardy spaces, Journal of Mathematical Fluid Mechanics 26 (2024). DOI: 10.1007/s00021-024-00860-9, https://doi.org/10.1007/s00021-024-00860-9
14. E. Bruè, M. Colombo and A. Kumar, Flexibility of two-dimensional Euler flows with integrable vorticity, Duke Mathematical Journal (2026). DOI: 10.1215/00127094-2025-0056, https://doi.org/10.1215/00127094-2025-0056; preprint arXiv:2408.07934, https://arxiv.org/abs/2408.07934
15. A. Cheskidov, M. C. Lopes Filho, H. J. Nussenzveig Lopes and R. Shvydkoy, Energy conservation in two-dimensional incompressible ideal fluids, Communications in Mathematical Physics 348(1):129–143 (2016). DOI: 10.1007/s00220-016-2730-8, https://doi.org/10.1007/s00220-016-2730-8
16. G. Crippa and S. Spirito, Renormalized solutions of the 2D Euler equations, Communications in Mathematical Physics 336 (2015). DOI: 10.1007/s00220-015-2411-z, https://doi.org/10.1007/s00220-015-2411-z
17. G. Ciampa, G. Crippa and S. Spirito, Strong convergence of the vorticity for the 2D Euler equations in the inviscid limit, Archive for Rational Mechanics and Analysis 240:295–326 (2021). DOI: 10.1007/s00205-021-01612-z, https://doi.org/10.1007/s00205-021-01612-z
18. P. Constantin, T. D. Drivas and T. M. Elgindi, Inviscid limit of vorticity distributions in the Yudovich class, Communications on Pure and Applied Mathematics 75(1):60–82 (2022). DOI: 10.1002/cpa.21940, https://doi.org/10.1002/cpa.21940
19. H. J. Nussenzveig Lopes, C. Seis and E. Wiedemann, On the vanishing viscosity limit for 2D incompressible flows with unbounded vorticity, Nonlinearity 34 (2021). DOI: 10.1088/1361-6544/abe51f, https://doi.org/10.1088/1361-6544/abe51f
20. D. Albritton, M. Colombo and G. Mescolini, Vanishing viscosity non-unique solutions to the forced 2D Euler equations, arXiv:2507.19257 (2025), https://arxiv.org/abs/2507.19257
21. M. Coghi and M. Maurelli, Existence and uniqueness by Kraichnan noise for 2D Euler equations with unbounded vorticity, Nonlinearity (2026). DOI: 10.1088/1361-6544/ae621b, https://doi.org/10.1088/1361-6544/ae621b; preprint arXiv:2308.03216, https://arxiv.org/abs/2308.03216
22. L. 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