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Abstract We investigate the well-posedness of a class of stochastic second-order in time damped evolution equations in Hilbert spaces, subject to the constraint that the solution lies within the unitary sphere. Then, we focus on a specific example, the stochastic damped wave equation in a bounded domain of ad-dimensional Euclidean space, endowed with the Dirichlet boundary condition, with the added constraint that the$$L^2$$ $L^2$-norm of the solution is equal to one. We introduce a small mass$$\mu >0$$ $\mu >0$in front of the second-order derivative in time and examine the validity of a Smoluchowski–Kramers diffusion approximation. We demonstrate that, in the small mass limit, the solution converges to the solution of a stochastic parabolic equation subject to the same constraint. We further show that an extra noise-induced drift emerges, which in fact does not account for the Stratonovich-to-Itô correction term. 

The Voigt regularization is a technique used to model turbulent flows, offering advantages such as sharing steady states with the Navier-Stokes equations and requiring no modification of boundary conditions; however, the parabolic dissipative character of the equation is lost. In this work we propose and study a generalization of the Voigt regularization technique by introducing a fractional power $$r$$ in the Helmholtz operator, which allows for dissipation in the system, at least in the viscous case. We examine the resulting fractional Navier-Stokes-Voigt (fNSV) and fractional Euler-Voigt (fEV) and show that global well-posedness holds in the 3D periodic case for fNSV when the fractional power $$r \geq \frac{1}{2}$$ and for fEV when $$r>\frac{5}{6}$$. Moreover, we show that the solutions of these fractional Voigt-regularized systems converge to solutions of the original equations, on the corresponding time interval of existence and uniqueness of the latter, as the regularization parameter $$\alpha \to 0$$. Additionally, we prove convergence of solutions of fNSV to solutions of fEV as the viscosity $$\nu \to 0$$ as well as the convergence of solutions of fNSV to solutions of the 3D Euler equations as both $$\alpha, \nu \to 0$$. Furthermore, we derive a criterion for finite-time blow-up for each system based on this regularization. These results may be of use to researchers in both pure and applied fluid dynamics, particularly in terms of approximate models for turbulence and as tools to investigate potential blow-up of solutions.