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

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.