For the two-dimensional incompressible Euler equations in a planar domain DD with impermeable boundary, the velocity is recovered from the vorticity through the Dirichlet Green function GG of the domain. In bounded domains, Yudovich theory gives global existence and uniqueness under the single assumption that the initial vorticity ω0\omega_0 is bounded, but in unbounded domains boundedness alone no longer guarantees that the Biot–Savart-type velocity integral is finite, because vorticity far away can make the velocity diverge. Caprino and Marchioro proved global well-posedness for initial vorticity in L∞L_\infty without any decay assumption in narrow sectors (opening below π/2\pi/2) and in strip-like domains of comparably narrow far-field, and they left open the corresponding question for wider sectors and general unbounded domains: whether a sufficient condition is the finiteness, at one single reference point x∗∈Dx^*\in D, of the far-field velocity contribution ∫D∣∇x⊥G(x∗,y)∣ ∣ω0(y)∣ dy\int_D |\nabla_x^\perp G(x^*,y)|\,|\omega_0(y)|\,dy. The boundaryless case D=R2D=\mathbb{R}^2 of this conjecture was settled by Brunelli and substantially generalized since (Cozzi, Cozzi–Kelliher, Cobb–Koch), but the case of sectors of opening at least π/2\pi/2 and of general unbounded domains with boundary, where the Green-function decay exponent satisfies γ≤2\gamma\le2 and the kernel estimates behind the narrow-sector proof are not available, remains open, as does the general question of sharp sufficient conditions on the vorticity and the geometry.

Background

The two-dimensional incompressible Euler equations describe the motion of an ideal (inviscid) fluid. Written for the vorticity ω=∂1u2−∂2u1\omega=\partial_1 u_2-\partial_2 u_1 (the curl of the velocity field uu), they read ∂tω+u⋅∇ω=0\partial_t\omega+u\cdot\nabla\omega=0: vorticity is transported by the flow, so its L∞L_\infty norm is conserved. On a simply connected domain DD with impermeable boundary (u⋅n=0u\cdot n=0 on ∂D\partial D), the velocity is reconstructed from the vorticity by the Biot–Savart law u=∇⊥ψu=\nabla^\perp\psi, where ψ\psi solves the Dirichlet problem −Δψ=ω-\Delta\psi=\omega in DD, ψ=0\psi=0 on ∂D\partial D; equivalently u(x)=∫D∇x⊥G(x,y) ω(y) dyu(x)=\int_D \nabla_x^\perp G(x,y)\,\omega(y)\,dy, where GG is the Dirichlet Green function of the Laplacian in DD (the function of xx vanishing on ∂D\partial D whose Laplacian is a point mass at yy). In bounded domains, Yudovich's classical theorem gives global existence and uniqueness of the solution for every ω0∈L1∩L∞\omega_0\in L_1\cap L_\infty; uniqueness relies on the velocity being log-Lipschitz, the borderline regularity at which the Osgood uniqueness criterion for the flow map still applies (the Osgood criterion: trajectories of x˙=b(x)\dot x=b(x) are unique whenever the modulus of continuity η\eta of bb satisfies ∫01ds/η(s)=∞\int_0^1 ds/\eta(s)=\infty; logarithmic moduli are exactly borderline). In unbounded domains the integrability of ω0\omega_0 can be dropped from the hypotheses only at a price: the Green-function kernel decays slowly at infinity, so for a merely bounded vorticity that does not decay the velocity integral may diverge, and the equations can cease to be meaningful. Serfati treated the whole plane with bounded velocity and bounded vorticity by a renormalized identity (the Serfati identity, which splits the Biot–Savart kernel into near and far fields), and Benedetto–Marchioro–Pulvirenti treated velocities growing like ∣x∣α|x|^\alpha with additional integrability of ω\omega. Caprino and Marchioro (J. Math. Fluid Mech. 12 (2010) 151–169) proved, for the sector D={x=(x1,x2):x1≥0, ∣x2∣≤αx1}D=\{x=(x_1,x_2): x_1\ge0,\ |x_2|\le\alpha x_1\} with 0<α<10<\alpha<1, i.e. opening angle 2θ0=2arctan⁡α<π/22\theta_0=2\arctan\alpha<\pi/2, global existence and uniqueness for every ω0∈L∞(D)\omega_0\in L_\infty(D) with no decay assumption, the velocity satisfying u/(1+∣x∣)∈L∞u/(1+|x|)\in L_\infty; they construct the solution as the limit of the flows generated by compactly supported truncations ω0,M\omega_{0,M}, which expresses that far-away fluid has negligible influence on the local evolution. They extended the result to finite unions of strip-like domains {−gi(x1)≤x2≤fi(x1)}\{-g_i(x_1)\le x_2\le f_i(x_1)\} with fi(x1)≤α∣x1∣f_i(x_1)\le\alpha|x_1|, gi(x1)≤α∣x1∣g_i(x_1)\le\alpha|x_1|, α<1\alpha<1. The proofs rest on estimates for ∂1G\partial_1 G that are established precisely in the regime α<1\alpha<1: in a sector of opening 2θ02\theta_0 the Green function decays at infinity like a power r−γr^{-\gamma} with γ=π/(2θ0)\gamma=\pi/(2\theta_0), and γ>2\gamma>2 exactly when α<1\alpha<1. For wider sectors (θ0≥π/4\theta_0\ge\pi/4, equivalently α≥1\alpha\ge1, γ≤2\gamma\le2) these estimates are not available, and the authors left the problem open, formulating the conjecture studied in this record.

Problem Statement

Let DD be an unbounded planar domain with boundary, starting with the sector Dα={x=(x1,x2)∈R2:x1≥0, ∣x2∣≤αx1}D_\alpha=\{x=(x_1,x_2)\in\mathbb{R}^2: x_1\ge0,\ |x_2|\le\alpha x_1\} with α≥1\alpha\ge1 (opening angle 2arctan⁡α≥π/22\arctan\alpha\ge\pi/2; equivalently θ0=arctan⁡α≥π/4\theta_0=\arctan\alpha\ge\pi/4), the regime in which the exponent γ=π/(2θ0)\gamma=\pi/(2\theta_0) governing the power-law decay r−γr^{-\gamma} of the Dirichlet Green function at infinity satisfies γ≤2\gamma\le2 and the kernel estimates of Caprino–Marchioro are not available. Consider the initial value problem for the two-dimensional incompressible Euler equations in DD with impermeability boundary condition and initial vorticity ω0∈L∞(D)\omega_0\in L_\infty(D), the velocity being defined through the Dirichlet Green function GG of DD. Prove or disprove the conjecture of Caprino and Marchioro: the existence of a single point x∗∈Dx^*\in D at which the velocity integral is absolutely convergent, ∫D∣∇x⊥G(x∗,y)∣ ∣ω0(y)∣ dy<∞,\int_D |\nabla_x^\perp G(x^*,y)|\,|\omega_0(y)|\,dy<\infty, is, together with ω0∈L∞(D)\omega_0\in L_\infty(D), a sufficient condition for global-in-time existence and uniqueness of the solution (in the Yudovich class, with velocity finite and obtained from the Green function); that is, finiteness of the far-field vorticity contribution at one reference point propagates to a well-defined velocity field and to a unique dynamics. More generally, determine the sharp sufficient conditions on the initial vorticity and/or on the far-field geometry of the domain — for instance on the Green-function decay exponent γ\gamma relative to the sector opening, or on suitable geometric quantities for general unbounded domains — under which the Euler initial value problem with merely bounded vorticity admits global-in-time existence and uniqueness of the solution in the Yudovich class.

References

  1. S. Caprino, C. Marchioro, On the Euler Equation in an Unbounded Domain of the Plane, Journal of Mathematical Fluid Mechanics 12 (2010) 151–169 (published online 2008). DOI: 10.1007/s00021-008-0279-9, https://doi.org/10.1007/s00021-008-0279-9
  2. E. Brunelli, On the Euler Equation in the Plane, Communications in Partial Differential Equations 35 (2010) 480–495. DOI: 10.1080/03605300903393954, https://doi.org/10.1080/03605300903393954
  3. V. I. Yudovich, Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid, Mathematical Research Letters 2 (1995), no. 1, 27–38. DOI: 10.4310/MRL.1995.v2.n1.a4, https://doi.org/10.4310/MRL.1995.v2.n1.a4
  4. P. Serfati, Solutions C∞C^\infty en temps, nn-log Lipschitz bornées en espace et équation d'Euler [English title: C∞C^\infty-solutions in time, nn-log Lipschitz bounded in space and Euler equation], Comptes Rendus de l'Académie des Sciences. Série I 320 (1995) 555–558. zbMATH: 0835.76012, https://zbmath.org/0835.76012
  5. D. M. Ambrose, J. P. Kelliher, M. C. Lopes Filho, H. J. Nussenzveig Lopes, Serfati solutions to the 2D Euler equations on exterior domains, Journal of Differential Equations 259 (2015) 4509–4560. DOI: 10.1016/j.jde.2015.06.001, https://doi.org/10.1016/j.jde.2015.06.001
  6. E. Cozzi, Solutions to the 2D Euler equations with velocity unbounded at infinity, Journal of Mathematical Analysis and Applications 423 (2015) 144–161. DOI: 10.1016/j.jmaa.2014.09.053, https://doi.org/10.1016/j.jmaa.2014.09.053
  7. E. Cozzi, J. Kelliher, Well-posedness of the 2D Euler equations when velocity grows at infinity, Discrete and Continuous Dynamical Systems 39 (2019) 2361–2392. DOI: 10.3934/dcds.2019100, https://doi.org/10.3934/dcds.2019100
  8. C. Lacave, A. Zlatoš, The Euler Equations in Planar Domains with Corners, Archive for Rational Mechanics and Analysis 234 (2019) 57–79. DOI: 10.1007/s00205-019-01384-7, https://doi.org/10.1007/s00205-019-01384-7
  9. Z. Han, A. Zlatoš, Euler Equations on General Planar Domains, Annals of PDE 7 (2021), article 20. DOI: 10.1007/s40818-021-00107-0, https://doi.org/10.1007/s40818-021-00107-0
  10. S. Agrawal, A. R. Nahmod, Uniqueness of the 2D Euler equation on a corner domain with non-constant vorticity around the corner, Nonlinearity 35 (2022) 2767–2808. DOI: 10.1088/1361-6544/ac586a, arXiv:2009.14816, https://doi.org/10.1088/1361-6544/ac586a
  11. D. Cobb, Bounded solutions in incompressible hydrodynamics, Journal of Functional Analysis 286 (2024) 110290. DOI: 10.1016/j.jfa.2023.110290, https://doi.org/10.1016/j.jfa.2023.110290
  12. D. Cobb, H. Koch, Unbounded Yudovich Solutions of the Euler Equations, Archive for Rational Mechanics and Analysis 250 (2026), no. 3. DOI: 10.1007/s00205-026-02193-5, arXiv:2410.05054, https://doi.org/10.1007/s00205-026-02193-5
  13. G. Iyer, M. C. Lopes Filho, H. J. Nussenzveig Lopes, Incompressible 2D Euler equations with non-decaying random initial vorticity (2025), arXiv:2512.07096, https://arxiv.org/abs/2512.07096