On Stochastic N-S equations via Gevrey Class Normal Form of Lie Algebra Frameworks in Reflexive Banach Spaces in Infinite Dimensions

  • Authors

    • Isamu Ohnishi Faculty of Mathematical Science, Graduate School of Integrated Sciences for Life, Hiroshima University, Kagamiyama 1-3-1, Higashi-Hiroshima, Hiroshima-Pref., JAPAN 739-8526
    https://doi.org/10.14419/s6g86k45
  • Lie algebra decoupling, normal forms, infinite-dimensional dynamical systems, reflexive Banach spaces, quadratic nonlinearities, resonant conditions, (AMS Classification NO.s:35Q55, 37L10, 17B66, 46G20, 81V70)

  • Abstract

    We extend the Lie algebra decoupling framework for first-order evolution equations from separable Hilbert spaces to reflexive Banach spaces with a countable Schauder basis, incorporating stochastic perturbations via Stochastic Advection by Lie Transport (SALT). We consider equations of the form

    du = [Au+B(u,u)]dt +  i P(ξi ・∇u) ◦dWit ,

    where A is a (possibly unbounded) linear operator, B is a quadratic bilinear form, and the noise models transport uncertainties with divergence-free fields ξi. Leveraging reflexivity for well-defined adjoints and weak compactness, we establish resonant conditions using adjoint representations and prove the solvability of the stochastic homological equation under non-resonance assumptions. This yields normal forms that eliminate non-resonant quadratic terms, addressing domain issues in non-Hilbert settings like Lp spaces for 1 < p <∞. The extension to higher-order normal forms achieves convergence in Gevrey classes through involutive PDE theory and the Cartan-K”ahler theorem, mitigating small divisors via spectral gaps and stochastic regularization. We derive explicit resonant conditions for basis triples and demonstrate solvability under Diophantine-type non-resonance. Applications include stochastic quantum many-body systems (e.g., Hartree and Hartree-Fock equations in Sobolev embeddings) and fluid dynamics in reflexive spaces. For the stochastic 3D Navier-Stokes equations under SALT noise, we prove global well-posedness by constructing solutions as deviations from Gevrey-class normal form solutions using Banach fixed-point arguments. Numerical validations on truncated models, such as the Bose-Hubbard system and 1D stochastic Burgers analogs, underscore reduced computational complexity, mode decoupling, and preservation of invariants like reversibility. This work broadens finite-dimensional Lie theory to unbounded operators in reflexive Banach spaces, offering insights into resonances, stability, and emergent behaviors in complex infinite-dimensional stochastic systems.

  • References

    1. A Agresti and M Luongo. On the navier-stokes equations with stochastic lie transport. arXiv preprint arXiv:2211.01265, 2022.
    2. M. Berti and P. Bolle. The renormalization of non-commutative field theories in the limit of large non-commutativity. Nuclear
    3. Physics B, 664:371–400, 2003.
    4. Leonardo Blanco et al. Observing ground-state properties of the fermi-hubbard model using a scalable algorithm on a quantum
    5. computer. Nature Communications, 13(1):1–10, 2022.
    6. Robert L. Bryant. An Introduction to Lie Groups and Symplectic Geometry. Duke University Press, 2001.
    7. Keaton J Burns et al. Dedalus: A spectral dynamo code with applications to astrophysical dynamos. Astrophysical Journal
    8. Supplement Series, 257(2):63, 2021.
    9. C Canuto, MY Hussaini, A Quarteroni, and TA Zang. Spectral methods in fluid dynamics. Springer, 1988.
    10. K. Chen. Bounded lie algebras in banach spaces. Advances in Mathematics, 115:1–30, 1995.
    11. H. Chiba. Lie algebra decoupling in banach spaces. Journal of Mathematical Physics, 64:012345, 2023.
    12. CJ Cotter, D Crisan, DD Holm, Wei Pan, and Igor Shevchenko. A particle filter for stochastic advection by lie transport (salt): A
    13. case study for the damped and forced incompressible 2d euler equation. arXiv preprint arXiv:1907.11884, 2019.
    14. Colin J Cotter, Dan Crisan, Darryl D Holm, Wei Pan, and Igor Shevchenko. A particle filter for stochastic advection by lie
    15. transport (salt): A case study for the damped and forced incompressible 2d euler equation. International Journal for Numerical
    16. Methods in Fluids, 93(4):1123–1143, 2021.
    17. Theodore D Drivas and Darryl D Holm. Lagrangian averaged stochastic advection by lie transport for fluids. arXiv preprint
    18. arXiv:1908.11481, 2019.
    19. Theodore D Drivas and Darryl D Holm. Lagrangian averaged stochastic advection by lie transport for fluids. Journal of Statistical
    20. Physics, 179:417–447, 2020.
    21. Ivar Ekeland. Some applications of the cartan-k”ahler theorem to economic theory. Mathematical Models and Methods in Applied
    22. Sciences, 5:1–16, 1995.
    23. D Goodair. Navier-stokes equations with navier boundary conditions and stochastic lie transport: Well-posedness and inviscid
    24. limit. arXiv preprint arXiv:2310.06335, 2023.
    25. Darryl D Holm. Stochastic advection by lie transport (salt). In CliMathParis Notes, 2015.
    26. Darryl D Holm. Stochastic geometric mechanics of thermal ocean dynamics. arXiv or Spiral repository, 2015.
    27. Darryl D Holm. Variational principles for stochastic fluid dynamics. Proceedings of the Royal Society A: Mathematical, Physical
    28. and Engineering Sciences, 471(2176):20140963, 2015.
    29. Darryl D Holm et al. Stochastic transport in upper ocean dynamics ii. 2024.
    30. J Jordan, Roman Or’us, and Guifre Vidal. Numerical study of the hard-core bose-hubbard model on an infinite square lattice.
    31. Physical Review B, 77(24):245110, 2008.
    32. Peter E Kloden and Eckhard Platen. Numerical solution of stochastic differential equations. Springer, 1992.
    33. Korbinian Kottmann. Lie-algebraic classical simulations for variational quantum computing. https://pennylane.ai/qml/demos/
    34. tutorial_liesim, 2024.
    35. Z. Li and Y. Wang. Well-posedness for stochastic navier-stokes equations in critical spaces. Journal of Differential Equations,
    36. :1–45, 2023.
    37. Elliott H Lieb. Two theorems on the hubbard model. Physical Review Letters, 62(4):397, 1988.
    38. Qinlong Luo. Numerical study of hubbard model for iron-based superconductors. University of Tennessee, Knoxville, 2013.
    39. Mikael Mortensen and Hans Petter Langtangen. Spectral-dns in python. ACM Transactions on Mathematical Software, 45(3):1–26,
    40. Marios-Andreas Nikolaidis. A vectorized navier-stokes ensemble direct numerical simulation code for plane parallel flows. arXiv
    41. preprint arXiv:2306.12739, 2023.
    42. Isamu Ohnishi. Lie algebra decoupling and normal forms for differential equations defined in infinite-dimensional hilbert space
    43. with quadratic nonlinearities. 2025. Preprint in Submittion.
    44. Peter Prelovˇsek and Lev Vidmar. Dynamical mean-field theory for real-time dynamics. Rep. Prog. Phys., 81(10):105701, 2018.
    45. M Razaghi et al. Spectral collocation method for stochastic burgers equation driven by additive noise. Mathematics and Computers
    46. in Simulation, 82(12):2245–2255, 2012.
    47. Valentin Resseguier, Wei Pan, and Baylor Fox-Kemper. Data-driven stochastic lie transport modeling of the 2d euler equations.
    48. Journal of Advances in Modeling Earth Systems, 15(1):e2022MS003268, 2023.
    49. Anders W Sandvik. Computational studies of quantum spin liquids. Physical Review Letters, 104(12):121501, 2010.
    50. Christopher Subich. Simulation of the navier-stokes equations in three dimensions with a spectral collocation method. International
    51. Journal for Numerical Methods in Fluids, 73(4):321–340, 2013.
    52. Leticia Tarruell and Laurent Sanchez-Palencia. Quantum simulation of the hubbard model with ultracold fermions in optical
    53. lattices. Cold Atoms and Molecules, pages 383–494, 2012.
    54. Yu Wang et al. Data-driven model reduction for stochastic burgers equations. SIAM Journal on Scientific Computing, 42(5):A3172–
    55. A3196, 2020.
    56. Dongbin Xiu and George Em Karniadakis. Efficient stochastic galerkin methods for random diffusion equations. Journal of
    57. Computational Physics, 217(2):662–688, 2007.

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  • How to Cite

    Ohnishi, I. (2026). On Stochastic N-S equations via Gevrey Class Normal Form of Lie Algebra Frameworks in Reflexive Banach Spaces in Infinite Dimensions. International Journal of Advanced Mathematical Sciences, 12(1), 16-30. https://doi.org/10.14419/s6g86k45