Well-posedness of the 2D Euler equations with bounded, non-decaying planar vorticity: uniqueness of Yudovich-type limits beyond symmetry and far-field prescriptions

Whether the two-dimensional incompressible Euler equations on the whole plane are well posed for initial vorticity that is merely bounded, with no decay at infinity, is an open problem sitting at the sharp boundary of Yudovich's classical $L^1\cap L^\infty$ uniqueness theory. Elgindi and Jeong showed that the natural approximation scheme is genuinely underdetermined: for the half-plane indicator vorticity $\chi_{\{y\ge 0\}}$, limits of the unique Yudovich solutions for smooth, compactly supported approximating data are non-unique — stationary disk approximations and rotating Kirchhoff-ellipse approximations converge to different evolutions, and infinitely many distinct limits arise — while bounded $m$-fold rotational symmetry with $m\ge 3$ restores a unique global evolution. Later work has circumscribed the question without entering it: global well-posedness is known for bounded vorticity with sublinear velocity growth $O(|x|^{1/2-\varepsilon})$ (Cobb–Koch) and almost surely for special random non-decaying data (Iyer–Lopes Filho–Nussenzveig Lopes), while non-uniqueness is known for vorticity in every $L^p$, $p<\infty$, and in Lorentz classes strictly below $L^\infty$. The linear-growth regime containing the half-plane indicator, with no symmetry hypothesis and no hand-prescribed far-field behavior, remains open. The problem is to determine whether the 2D Euler equations admit a notion of solution that restores a unique global evolution for bounded non-decaying planar vorticity without symmetry and without hand-prescribed far-field behavior — or whether non-uniqueness is inherent to this regime.

Background

The two-dimensional incompressible Euler equations describe an ideal (inviscid), volume-preserving fluid. In vorticity form the unknown is the scalar vorticity $\omega=\partial_1 u_2-\partial_2 u_1$ (the curl of the velocity field $u$), transported by the flow it generates: $\partial_t\omega+u\cdot\nabla\omega=0,\qquad u=\nabla^\perp\Delta^{-1}\omega,$ where the second relation is the Biot–Savart law ($\nabla^\perp=(\partial_2,-\partial_1)$), recovering the divergence-free velocity from the vorticity. A classical theorem of Yudovich (1963) gives global existence and uniqueness of weak solutions when $\omega_0\in L^1\cap L^\infty(\mathbb{R}^2)$, and equally for $\omega_0\in L^p\cap L^\infty$ for any finite $p$: the vorticity is bounded and decays at infinity in the sense of integrability. The velocity is then log-Lipschitz (Hölder-continuous with a logarithmic modulus, one step below Lipschitz), which is enough for uniqueness of particle trajectories by Osgood's criterion for ordinary differential equations (the classical test under which a velocity modulus of continuity that is integrable at zero forces unique trajectories), and uniqueness of the solution follows by transporting the difference of two solutions. Some decay of $\omega_0$ has always been an essential hypothesis: it makes the Biot–Savart integral converge and fixes the velocity uniquely.

When the vorticity is merely bounded and does not decay, both pillars fail. For the model datum $\omega_0=\chi_{\{y\ge 0\}}$, the indicator of a half-plane, every associated velocity grows linearly in $|x|$, and the stream function $\Psi$ solving $\Delta\Psi=-\omega$ is determined only up to harmonic functions: adding rotation-type fields built from the harmonic polynomials $x_1x_2$ and $x_1^2-x_2^2$, or Galilean constant velocities, again yields a divergence-free velocity with the same vorticity. Elgindi and Jeong (2020) proved that this ambiguity is not a notational artifact but a genuine failure of the solution concept. Their Proposition 1.2 approximates $\chi_{\{y\ge 0\}}$ in $L^2_{\mathrm{loc}}$ by smooth, compactly supported data and passes to the limit in the corresponding unique Yudovich solutions — the Yudovich scheme. Approximating disks are stationary vortex patches, while approximating ellipses are Kirchhoff vortices (elliptical vortex patches rotating rigidly at a size-independent angular speed, a classical exact solution), and at a fixed positive time the two sequences converge in $L^2_{\mathrm{loc}}$ to different functions; in fact infinitely many distinct limiting evolutions arise. On the positive side, the same authors proved their main theorem: for initial vorticity that is bounded and $m$-fold rotationally symmetric with $m\ge 3$, there is a unique global solution on the whole plane — symmetry replaces the $L^1$ decay assumption, because the $m$-fold symmetrization of the Biot–Savart kernel acquires enough cancellation at infinity ($|x|^{m-1}/|y|^m$ decay, integrable only for $m\ge 3$). Their argument barely fails at $m=2$, precisely where the harmonic-ambiguity mechanism persists.

Subsequent work has mapped the surroundings without entering this core. Uniqueness theories that drop decay all re-impose structure at infinity: Serfati (1995) treats bounded velocity and vorticity, uniqueness holding only after a far-field term $U_\infty(t)$ (the velocity contribution from infinity) is prescribed; Benedetto–Marchioro–Pulvirenti (1993) and later Cozzi (2015, logarithmic velocity growth) and Cozzi–Kelliher (2019, local well-posedness without integrability at nearly square-root growth) keep decay-type or growth restrictions; Kelliher (2015) characterized what the far field must determine for bounded vorticity and velocity. Chae–Wolf (2021) reach linear velocity growth in a critical Campanato space (a scale of function classes measuring the local mean oscillation of the velocity, adjacent to the Morrey scale), but by prescribing the asymptotic behavior of the pressure force; McOwen–Topalov (2022) prove well-posedness in weighted Sobolev spaces with growth/decay conditions at infinity. Most decisively, Cobb–Koch (2026) prove global existence and uniqueness for bounded vorticity with no decay at all, but only for velocity growth $O(|x|^{1/2-\varepsilon})$, and explicitly state that linear growth — the regime of the half-plane indicator — is beyond their method, with a follow-up announced. Iyer–Lopes Filho–Nussenzveig Lopes (2025) obtain almost-sure global well-posedness for a random, non-decaying array of vortex blobs by constructing initial velocities with sub-polynomial growth and invoking Cobb–Koch, and record that, to the best of their knowledge, even local well-posedness is open once velocity growth is square-root or faster. From below, non-uniqueness is known for forced solutions with $\omega_0\in L^p$, $p<\infty$ (Vishik, with a detailed exposition by Albritton–Brué–Colombo–De Lellis–Giri–Janisch–Kwon), for vorticity in the Lorentz space $L^{1,\infty}$ — the weak-$L^1$ scale, refining the Lebesgue integrability hierarchy (Brué–Colombo) — and, by convex integration — a scheme that superposes high-frequency oscillatory perturbations to build wild solutions — for unforced solutions with vorticity in $L^p$, $p>1$ (Bruè–Colombo–Kumar; Mengual for admissible $L^2$ solutions with $L^p$ vorticity). None of these reach bounded vorticity: $L^\infty$ remains the sharp, still-open threshold. Linear-growth classes also host genuine pathologies — for data such as $u_0=(x_1,-x_2)$ the pressure blows up in finite time, and Miller (2021) exhibited finite-time blow-up with linear growth at infinity for Navier–Stokes — so any well-posed class must exclude such far fields using conditions on the initial data alone, which is the conceptual core of the difficulty. Complementary directions (unbounded rather than non-decaying vorticity) include Elgindi–Murray–Said (2023), who build a local well-posedness class beyond Yudovich's from which solutions can exit in finite time, and Drivas–La (2025), who reprove Yudovich theory Lagrangianly but retain the $L^1$ decay.

Problem Statement

Determine whether the initial-value problem for the two-dimensional incompressible Euler equations on the whole plane is well posed when the initial vorticity is merely bounded, $\omega_0\in L^\infty(\mathbb{R}^2)$, with no decay, integrability, or symmetry assumption — the regime containing the half-plane indicator $\omega_0=\chi_{\{y\ge 0\}}$, whose velocities grow linearly and for which Elgindi–Jeong's Proposition 1.2 exhibits infinitely many distinct limits of the Yudovich scheme.

Precisely: what is the correct notion of solution for such data, and does well-posedness hold? On the positive side, find a notion of weak solution — imposing no symmetry and no prescribed time evolution of the far field (no imposed pressure or velocity asymptotics, no externally given far-field term in the sense of Serfati) — for which a global solution with $\omega\in L^\infty$ exists for every $\omega_0\in L^\infty(\mathbb{R}^2)$ and is unique, as $m$-fold symmetry with $m\ge 3$ achieves in the Elgindi–Jeong main theorem. Proposition 1.2 shows that the Yudovich scheme alone — limits of the unique solutions for smooth compactly supported approximations of $\omega_0$ — cannot provide such a selection: for $\chi_{\{y\ge 0\}}$ its limits realize infinitely many distinct evolutions. On the negative side, prove that non-uniqueness is inherent to the merely-bounded-vorticity regime, so that no such refinement of the solution concept restores uniqueness without re-imposing symmetry-type or far-field prescriptions.

Sharp open instances contained in this generality (indicative sub-problems, each open in its own right; each is contained in the general determination posed above for arbitrary bounded non-decaying data): uniqueness for 2-fold rotationally symmetric bounded vorticity (the case $m=2$, where the Elgindi–Jeong cancellation argument barely fails and the harmonic ambiguity persists); local or global well-posedness at exactly square-root or linear velocity growth (Cobb–Koch reach $O(|x|^{1/2-\varepsilon})$; Iyer–Lopes Filho–Nussenzveig Lopes record square-root-or-faster as open); and the classification of which evolutions the Yudovich scheme realizes for a given non-decaying datum.

References

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