Kinra and Mohan (SIAM Journal on Mathematical Analysis, 2024) established global weak (analytic) solutions, uniqueness under bounded-vorticity assumptions, and pullback stochastic weak attractors for the two-dimensional incompressible Euler equations on a bounded domain perturbed by linear multiplicative Ito noise, and explicitly deferred the existence of invariant measures — stationary probability measures describing the long-time statistics of the flow — to future work. All known invariant-measure results for Euler-type stochastic models require damping or fractional dissipation (Bessaih–Ferrario 2020; Constantin–Glatt-Holtz–Vicol 2014; Bai–Feng–Zhao 2026), concern the deterministic Euler equations with noise used only as an approximation device (Brzeźniak–Ferrari 2026), or construct non-unique stationary solutions only in low Hölder regularity for the 3D additive-noise problem (Kinra–Koley 2026). For the inviscid 2D stochastic Euler equations with linear multiplicative noise on a bounded domain, the existence of invariant measures in the natural energy-based phase spaces remains open.

Background

The incompressible Euler equations model an ideal (inviscid, zero-viscosity) incompressible fluid: the velocity field uu satisfies ∂tu+(u⋅∇)u+∇p=f\partial_t u + (u\cdot\nabla)u + \nabla p = f with ∇⋅u=0\nabla\cdot u = 0. In two space dimensions the vorticity ω=∇∧u\omega = \nabla\wedge u (the curl of the velocity, measuring local rotation) is transported by the flow, and Yudovich (1963) proved global existence and uniqueness of weak solutions when the initial vorticity is bounded in L∞L^\infty. Uniqueness of weak solutions below this bounded-vorticity class — in particular for data merely in L2L^2 or H1H^1 — is a classical open problem that predates the source paper and is mentioned there only as background context.

Several works have perturbed the 2D Euler equations by noise: Brzeźniak and Peszat (2001) proved existence of martingale solutions (solutions in a weak probabilistic sense, defined on a possibly enlarged probability space), and Glatt-Holtz and Vicol (2014) proved local and global existence of smooth solutions for stochastic Euler equations with multiplicative noise, including the case of linear multiplicative noise of Ito type (noise of the form σu dW\sigma u\,dW, where WW is a Wiener process and the Ito interpretation fixes the stochastic calculus convention). Kinra and Mohan (2024) considered the non-autonomous 2D stochastic Euler equations on a bounded domain with C2C^2 boundary, driven by a one-dimensional linear multiplicative Ito noise:

du+(u⋅∇)u dt+∇p dt=f dt+σu dW,∇⋅u=0,u⋅n=0 on ∂O,du + (u\cdot\nabla)u\,dt + \nabla p\,dt = f\,dt + \sigma u\,dW,\qquad \nabla\cdot u = 0,\qquad u\cdot n = 0 \text{ on } \partial\mathcal{O},

with σ>0\sigma > 0 and u(τ)=uτu(\tau) = u_\tau. They proved existence of global weak (analytic) solutions for divergence-free data uτ∈H1(O)u_\tau \in H^1(\mathcal{O}) and forcing f∈Lloc2(R;H1(O))f \in L^2_{\mathrm{loc}}(\mathbb{R};H^1(\mathcal{O})), uniqueness under the stronger assumptions ∇∧uτ∈L∞(O)\nabla\wedge u_\tau \in L^\infty(\mathcal{O}) and ∇∧f∈Lloc∞\nabla\wedge f \in L^\infty_{\mathrm{loc}}, and they developed an abstract theory of weak asymptotic autonomy for pullback stochastic weak attractors (random sets attracting trajectories when the initial time is pulled to −∞-\infty). They state that for these equations the existence of invariant measures "will be addressed in a future work".

An invariant measure is a probability measure μ\mu on the phase space that is stationary for the Markov transition semigroup PtP_t of the dynamics, (Pt)∗μ=μ(P_t)^*\mu = \mu; physically it describes equilibrium long-time statistics of the randomly forced flow. For dissipative stochastic PDEs, such as the 2D stochastic Navier–Stokes equations, invariant measures are constructed by the Krylov–Bogoliubov procedure: dissipation yields absorbing sets that make time averages of the process tight, and weak limits of those averages are invariant. The Euler equations contain no dissipative term, so this mechanism is unavailable, and for linear multiplicative Ito noise the formal Ito energy balance d E∥u∥2/dt=2 E⟨u,f⟩+σ2 E∥u∥2d\,\mathbb{E}\|u\|^2/dt = 2\,\mathbb{E}\langle u,f\rangle + \sigma^2\,\mathbb{E}\|u\|^2 shows the noise injects energy at rate σ2∥u∥2\sigma^2\|u\|^2, making even uniform-in-time moment bounds unclear. Accordingly, every known invariant-measure result near this problem adds dissipation: Constantin, Glatt-Holtz and Vicol (2014) proved unique ergodicity for fractionally dissipated stochastically forced 2D Euler; Bessaih and Ferrario (2020) constructed invariant measures for stochastically damped 2D Euler on the torus; Bai, Feng and Zhao (2026) proved exponential mixing for the 2D stochastic damped Euler equation under large damping. In the inviscid setting, Brzeźniak and Ferrari (2026) constructed invariant measures for the deterministic 2D Euler equations on R2\mathbb{R}^2 as inviscid limits of stochastically forced hyperviscous Navier–Stokes equations (equations augmented by a higher-order dissipative term) — there the noise serves only as an approximation tool and the invariant measure governs the deterministic dynamics — and Kinra and Koley (2026) constructed infinitely many non-unique stationary (in law) Hölder-continuous solutions of the 3D stochastic Euler equations with additive noise on the torus via stochastic convex integration (an iterative oscillatory-perturbation method producing wildly non-unique weak solutions), a much weaker regularity class than the energy-based spaces of the source paper. Whether the 2D stochastic Euler equations with linear multiplicative noise on a bounded domain admit invariant measures in their natural phase spaces is not settled by any of these works.

Problem Statement

For the two-dimensional incompressible Euler equations on a bounded domain O⊂R2\mathcal{O}\subset\mathbb{R}^2 with C2C^2 boundary, perturbed by linear multiplicative Ito noise,

du+(u⋅∇)u dt+∇p dt=f dt+σu dW,du + (u\cdot\nabla)u\,dt + \nabla p\,dt = f\,dt + \sigma u\,dW,

with ∇⋅u=0\nabla\cdot u = 0 in O\mathcal{O}, u⋅n=0u\cdot n = 0 on ∂O\partial\mathcal{O}, WW a standard one-dimensional two-sided Wiener process, σ>0\sigma > 0, and a fixed but arbitrary time-independent forcing f∈Lloc2(R;H1(O))f \in L^2_{\mathrm{loc}}(\mathbb{R};H^1(\mathcal{O})) with nonzero solenoidal (divergence-free) part — so that ff is not pressure-equivalent to zero — the class of the source paper's existence theory — the setting in which Kinra and Mohan proved global weak (analytic) solutions, uniqueness under L∞L^\infty-vorticity assumptions, and pullback stochastic weak attractors — does the associated Markovian dynamics admit an invariant probability measure in a natural energy-based phase space (such as H1(O)H^1(\mathcal{O}) or the vorticity class used for uniqueness)?

The invariant probability measure of interest is a probability measure μ\mu with (Pt)∗μ=μ(P_t)^*\mu = \mu and finite energy moments in such a phase space, for the Markov semigroup well defined under the relevant solution concept (Markov or martingale solutions). The two directions of the question are the construction of such a measure and the proof that no invariant probability measure with finite energy moments exists in these phase spaces. The degenerate forcing cases pressure-equivalent to zero are not the target: f≡0f \equiv 0 makes the Dirac measure at the zero solution stationary, and more generally a pure-gradient forcing f=∇qf = \nabla q is absorbed into the pressure (take u≡0u \equiv 0 and ∇p=f\nabla p = f, the noise term σu dW\sigma u\,dW vanishing), so that the Dirac measure at the zero solution is again an invariant probability measure with finite energy moments; the question is the one deferred by the source authors, namely the existence of invariant measures for the 2D stochastic Euler equations driven by linear multiplicative noise. The core formulation fixes time-independent forcing because a strict invariant measure, (Pt)∗μ=μ(P_t)^*\mu = \mu, is defined for autonomous dynamics; extensions to time-dependent (e.g. periodic) forcing, matching the non-autonomous framework of the source, fall within the same question.

References

  1. Kush Kinra, Manil T. Mohan, Theory of Weak Asymptotic Autonomy of Pullback Stochastic Weak Attractors and Its Applications to 2D Stochastic Euler Equations Driven by Multiplicative Noise, SIAM Journal on Mathematical Analysis 56(5):6268–6301 (2024). DOI: 10.1137/24M1637878; arXiv:2302.06342. https://doi.org/10.1137/24M1637878
  2. Hakima Bessaih, Benedetta Ferrario, Invariant Measures for Stochastic Damped 2D Euler Equations, Communications in Mathematical Physics 377(1):531–549 (2020). DOI: 10.1007/s00220-020-03714-3. https://doi.org/10.1007/s00220-020-03714-3
  3. Peter Constantin, Nathan Glatt-Holtz, Vlad Vicol, Unique Ergodicity for Fractionally Dissipated, Stochastically Forced 2D Euler Equations, Communications in Mathematical Physics 330(2):819–857 (2014). DOI: 10.1007/s00220-014-2003-3. https://doi.org/10.1007/s00220-014-2003-3
  4. Nathan E. Glatt-Holtz, Vlad C. Vicol, Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise, The Annals of Probability 42(1):80–145 (2014). DOI: 10.1214/12-AOP773. https://doi.org/10.1214/12-AOP773
  5. Zdzisław Brzeźniak, Szymon Peszat, Stochastic two dimensional Euler equations, The Annals of Probability 29(4):1796–1832 (2001). DOI: 10.1214/aop/1015345773. https://doi.org/10.1214/aop/1015345773
  6. 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
  7. Kush Kinra, Ujjwal Koley, Nonuniqueness of Hölder Continuous Solutions for Stochastic Euler and Hypodissipative Navier–Stokes Equations, SIAM Journal on Mathematical Analysis 58(2):1436–1484 (2026). DOI: 10.1137/24M1680027; arXiv:2407.20270. https://doi.org/10.1137/24M1680027
  8. Zdzisław Brzeźniak, Matteo Ferrari, Inviscid limit of the stochastic hyperviscous Navier–Stokes equations and invariant measures for the Euler equations in R2\mathbb{R}^2, Nonlinearity 39(1):015004 (2026). DOI: 10.1088/1361-6544/ae2a9c. https://doi.org/10.1088/1361-6544/ae2a9c
  9. Rui Bai, Chunrong Feng, Huaizhong Zhao, Exponential Mixing for 2D Stochastic Damped Euler Equation Driven by Bounded Noise, arXiv:2606.29352 (2026). https://arxiv.org/abs/2606.29352