Fractional Stochastic Navier-Stokes Equations: Local Well-Posedness and Enstrophy Balance

Abstract

In this paper, we investigate a stochastic incompressible fluid on the three-dimensional torus in which power-law temporal memory, represented by a Caputo derivative of order $\beta \in (1/2,1)$, is coupled to non-local dissipation, represented by a fractional Laplacian of order $\alpha \in (1,2)$, under spatially smooth trace-class noise with a superlinear coefficient. We first derive the equation from constrained Hamiltonian mechanics through a Mori-Zwanzig elimination of fast degrees of freedom, so that the memory kernel and the random force emerge together rather than being postulated; a Tauberian limit identifies the Caputo operator as the exact analytic image of a scale-free harmonic heat bath. Representing vorticity as a differential two-form, we prove that the vortex-stretching functional coincides with the sign-indefinite quadratic form of the strain tensor, and that the velocity is recovered from the vorticity with an exact one-derivative gain. Using a subordination identity for the Mittag-Leffler propagator, we show that the linear kernel is a probability density with an algebraic far-field tail. Within a Gelfand triple adapted to the Stokes operator we establish local existence and uniqueness in $H^s$ for $s>3/2$, with the data and the forcing transported by two distinct propagators, and we prove that the memory threshold $\beta >1/2$ is sharp for the existence of function-valued solutions under white-in-time forcing. We then derive an enstrophy balance, governed by a fractional differential inequality, read off the critical exponent $\sigma =\beta (1+3/\alpha )$ separating the dissipation-dominated regime from possible stretching-driven growth, and close with a Beale-Kato-Majda continuation criterion together with a geometric condition for global existence.

24 pages, 4 figures