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Convergence of a fully discrete approximation for the stochastic 2D Euler equations with Kraichnan transport noise

Abhishek Chaudhary, Ujjwal Koley, Andreas Prohl

arXiv:2609.31369v1math.PRcs.NAmath.APmath.NA

TL;DR

A fully discrete approximation theory for stochastic 2D Euler with Kraichnan transport noise has been lacking. The paper combines torus coercivity with continuous interpolants and localized estimates, and proves subsequential convergence to a weak martingale solution.

  • Problem

    A fully discrete, implementable approximation yielding weak martingale solutions for stochastic 2D Euler equations had not been available.

    Source: §1 · p. 2

  • Method

    The paper combines a torus coercivity estimate with continuous equation-based interpolants and localized stopping-time estimates for compactness.

    Source: §6.6 · pp. 51–52

  • Results

    Along subsequences, the fully discrete approximations converge to a weak martingale solution of the limiting equation.

    Source: §6.6 · p. 51

  • Takeaways & Limitations

    The scheme provides a constructive numerical existence theory for rough weak martingale solutions on the torus with H−1-valued initial vorticity.

    Source: §1 · p. 2, §1.1 · p. 2

    The regularization estimate requires γ > 1; for 0 < γ ≤ 1, no positive dissipative contribution is obtained.

    Source: §4.1 · p. 13

Abstract

from arXiv · show

We study a fully discrete approximation theory for the two-dimensional incompressible Euler equations on $\mathbb{T}^{2}$, in vorticity form, driven by Kraichnan-type transport noise and with $\mathbb{H}^{-1}(\mathbb{T}^2)$-valued initial vorticity. The main new ingredient is a finite-dimensional coercivity estimate for the torus Kraichnan model, which captures at the discrete level the regularizing mechanism induced by the rough transport noise. Together with a new compactness strategy based on continuous equation-based interpolants and localized discrete estimates, this coercivity estimate gives tightness of the approximations and shows that subsequential limit is a weak martingale solution of the Euler equations.

1 Introduction

The paper develops a fully discrete approximation for stochastic 2D Euler equations with Kraichnan transport noise and rough vorticity. It combines a torus coercivity estimate with a localized compactness strategy to prove subsequential convergence to a weak martingale solution.

1 Introduction

  • Prior rough-vorticity work identifies compactness and weak-limit identification as central issues, while existing Kraichnan results are primarily analytical and continuous.A fully discrete approximation theory producing weak martingale solutions had not been available.Source: §1 · p. 1, §1.1 · p. 2
  • Kraichnan transport noise models unresolved incompressible transport fluctuations coupled to the resolved Euler velocity, rather than an additive external force.The Stratonovich formulation preserves the geometric character of transport by a random velocity field.Source: §1 · p. 2

1.1 Our aims and contributions in this work

  • The torus is chosen because explicit implementable noise formulas are available there, unlike on R2, making a finite-dimensional approximation scheme feasible.The scheme combines Fourier truncation, a finite noise cutoff, mixed implicit–explicit time discretization, and an implicit Laplacian correction.Source: §1.1 · pp. 2–3
  • The torus model yields a finite-dimensional uniform coercivity estimate under scale separation, with high noise modes supplying discrete negative-Sobolev dissipation.The estimate is new for the torus and is the key input to the discrete H−1 energy estimates.Source: §1.1 · p. 3
  • Along subsequences, the fully discrete scheme converges to a weak martingale solution, while the discrete velocities converge strongly to identify the nonlinear transport term.The Biot–Savart law supplies velocity compactness even though vorticities are compact only in weak negative-Sobolev topologies.Source: §1.1 · pp. 3–4
  • A continuous equation-based interpolant and localized stopping-time estimates establish tightness, and all three approximating objects converge to the same limit process.The interpolant agrees with the discrete approximation at grid points and has the temporal regularity required of the limit.Source: §1.1 · pp. 3–4

2 Mathematical setup

The setup fixes Kraichnan transport noise on the two-dimensional torus using a spectral density and divergence-free modes, then establishes notation for Sobolev and path spaces.

2 Mathematical setup

  • On the standard two-dimensional torus, the setup fixes 0 < α < 1 and defines a spectral density and two real divergence-free noise modes.Source: §2 · pp. 4–5
  • The noise modes are grouped into cosine–sine families and driven by independent standard Brownian motions, giving compact and expanded forms of the stochastic transport term.Source: §2 · p. 5

2.1 Functional setting

  • For s > 0, H^-s(T2) is paired with H^s(T2) through the L2 pairing, using duality when the pairing is not directly meaningful.Source: §2.1 · p. 5
  • For reflexive spaces, the specified L2-in-time space has the weak topology, and Brownian path spaces use the product topology.Source: §2.1 · p. 5

3 Fully discrete spectral scheme on T2

The section defines a Fourier-truncated spatial and temporal framework for an adapted, implementable spectral scheme, with mixed-time transport and an implicit Laplacian supporting its energy estimate.

3 Fully discrete spectral scheme on T2

  • The time grid uses tn = n∆t through NT = ⌊T/∆t⌋, alongside fixed Fourier and noise cutoffs N and K.Source: §3 · p. 6

3.1 Spectral space, projector, and discrete filtration

  • The resolved space contains mean-zero trigonometric polynomials with Fourier modes restricted to the box |m|∞ ≤ N, and PN removes the zero mode and modes outside that box.Source: §3.1 · p. 6
  • The noise cutoff retains nonzero modes with |m|∞ ≤ K; the stated spectral-density decay makes cK finite for each K in dimension two when α > 0.Source: §3.1 · p. 6

3.2 Fully discrete scheme

  • At each discrete step, the scheme seeks an adapted VN-valued iterate driven by Brownian increments and is presented as an implementable algorithm.Source: §3.2 · p. 7
  • The scheme's compact sum over noise indices is equivalent to an explicit cosine–sine decomposition of the stochastic term.Source: §3.2 · p. 7
  • The transport term uses subsequent-time-level iterates to support the H^-1 energy estimate, while the implicit Laplacian supplies its coercive contribution.Source: §3.2 · p. 7

4 Coercivity estimate

This section develops a torus coercivity estimate using the Kraichnan noise’s Fourier representation and continuum and discrete ingredients. Its main discrete estimate yields negative-Sobolev dissipation when sufficiently many high-frequency noise modes are retained, with scale separation between resolved vorticity and noise modes.

4.1 Uniform Riemann-sum estimate and positive continuum defect

  • The boundary-layer contribution is bounded by Cr uniformly in e, completing the claimed uniform Riemann-sum estimate.Source: §4.1 · pp. 9–10
  • For 0 < α < 1, the continuum defect is positive when γ > 1; for 0 < γ ≤ 1, no positive dissipative contribution is obtained.Thus γ > 1 is necessary to detect the regularizing effect of the high Kraichnan modes.Source: §4.1 · pp. 10, 12, 13

4.2 Discrete noise quadratic form

  • A self-contained torus analogue of the discrete noise inequality establishes constants c0 > 0 and C0 ≥ 0, depending only on α, for every nonzero frequency in KN.The proof is new in the torus setting and differs from the Euclidean argument, whose structural features are unavailable here.Source: §4.2 · p. 13
  • Theorem 4.6 establishes the main coercivity estimate for 0 < α < 1 under its stated cutoff condition.Its proof uses the preceding lemma as a key structural input together with further Fourier-analytic arguments.Source: §4.2 · p. 24
  • Fourier expansion, exact cancellation of cosine–sine cross terms, and a diagonal representation reduce the quadratic form to the estimate supplied by Lemma 4.5.The proof first reduces to the unprojected form; its conclusion is the coercivity estimate (4.58).Source: §4.2 · pp. 24, 27, 28
  • The cutoff condition enforces scale separation: the coercive effect comes from noise modes above the resolved spatial frequencies, and γ > 1 is intrinsic.For 0 < γ ≤ 1, the continuum defect j_α,γ is zero, so the corresponding noise modes yield no positive dissipative contribution.Source: §4.2 · p. 28
  • Separation of resolved modes and retained noise modes keeps their interactions in the coercive regime, allowing control of the remainder term in (4.25) in Lemma 4.5.Source: §4.2 · p. 28
  • Sufficiently many high-frequency Kraichnan modes relative to the spatial discretization make the discrete noise quadratic form produce negative-Sobolev dissipation and capture regularization in the fully discrete scheme.This dissipation is quantified by the coercivity estimate (4.58).Source: §4.2 · p. 28

5 Uniform discrete energy estimate

The fully discrete scheme admits uniform H−1 energy bounds, with stability relying on its implicit–explicit structure. Testing at the implicit time level combines transport cancellation, Laplacian coercivity, and the discrete Kraichnan coercivity estimate.

5 Uniform discrete energy estimate

  • The H−1 test cancels the mixed-time convection term exactly, so nonlinear transport does not contribute to the energy balance.The cancellation is recorded in Lemma 5.1.Source: §5 · pp. 28, 34
  • Theorem 5.3 establishes a uniform strong H−1(T2) estimate for the fully discrete scheme.Its proof uses the estimate of Theorem 5.2.Source: §5 · pp. 31, 33
  • Testing with the new-time unknown lets a stochastic increment term be absorbed by a positive discrete energy term.The implicit Laplacian also supplies a positive time-integrated L2 term, which combines with stochastic quadratic variation through coercivity estimate (4.58).Source: §5 · pp. 33–34
  • These stability features depend on the scheme’s implicit–explicit structure and are not available in the same form for fully explicit or incompatible time discretizations.The structure supports the pathwise uniform bounds used later to prove tightness.Source: §5 · pp. 31, 34

6 Convergence of the fully discrete scheme to a weak martingale solution

This section proves that the fully discrete scheme converges along a subsequence, after a change of probability space, to a weak martingale solution. It establishes tightness using continuous interpolants and localized estimates, then passes the discrete weak formulation to the limit.

6 Convergence of the fully discrete scheme to a weak martingale solution

  • Along a subsequence and after changing probability space, the fully discrete approximations converge to a weak martingale solution of the stochastic Euler equations.The theorem assumes mean-zero initial vorticity in H^-1(T2) and 0 < α < 1/2.Source: §6 · p. 35

6.1 Discrete interpolants

  • Uniform energy estimates provide L2(0, T; H^-α) bounds for the left- and right-endpoint interpolants needed in the compactness argument.The section also defines a continuous equation-based interpolant alongside the piecewise-constant interpolants.Source: §6.1 · p. 36

6.2 Localization and time regularity in H−4(T2)

  • A discrete stopping-time localization is essential for tightness; the section establishes high probability of no stopping and an expected bound on the localized drift.The drift estimate treats its nonlinear and Laplacian parts separately.Source: §6.2 · pp. 38–40
  • Localization yields time regularity in H−4 and convergence in probability of the continuous interpolant to the left interpolant and of the left and right interpolants to each other.These estimates control the discrepancies among the interpolants as the discretization is refined.Source: §6.2 · pp. 41–45
  • In the localized H−4 time-regularity argument, both error terms on the right converge to zero in probability as j tends to infinity.Source: §6.2 · p. 45

6.3 Canonical path space and tightness

  • The laws of the path-space coordinates are tight on X, completing the tightness argument for the approximations.Tightness is proved coordinate by coordinate, and combining the four coordinates yields tightness on X.Source: §6.3 · pp. 46–47

6.4 Skorokhod–Jakubowski representation

  • After subsequence extraction, the represented approximations converge almost surely, and the limiting process is adapted with an independent Brownian family.The Brownian motions are independent with respect to the usual augmentation of the filtration generated by the path variables.Source: §6.4 · pp. 47–48

6.5 Passage to the limit

  • The discrete weak formulation passes to the limit as the interpolants converge strongly in L2([0, T]; H−β(T2)) and the stochastic coefficients converge.The resulting limit satisfies the weak identity for every smooth test function and every t in [0, T].Source: §6.5 · pp. 49–51

6.6 Conclusion

  • The limiting process satisfies the weak formulation and, with the independent Brownian family, is a weak martingale solution of the limiting equation.The compactness strategy uses the continuous interpolant with localized stopping-time estimates and the Kraichnan noise’s negative-Sobolev coercivity structure.Source: §6.6 · pp. 51–52

A.1 Noise estimate

The noise estimate controls the finite-dimensional transport contribution in a negative Sobolev norm. Its proof uses Fourier projection properties and reduces the relevant bound to Fourier and velocity estimates.

A.1 Noise estimate

  • Lemma A.1 gives a noise estimate in H−4(T2), uniformly in the discretization index for mean-zero H−1(T2) inputs.The stated constant depends only on α.Source: §A.1 · p. 52
  • Because the Fourier projection is orthogonal on Sobolev spaces, the proof reduces the estimate to the case without that projection.The argument then uses Fourier-mode estimates and a lattice bound.Source: §A.1 · pp. 52–53
  • The proof bounds the Fourier sum by estimating the velocity tensor u⊗u in H−α and using the Biot–Savart relation between velocity and vorticity.The passage combines the estimates from these steps to conclude the bound.Source: §A.1 · pp. 55–56
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