  We propose a modification to the nonlinear term of the three-dimensional
incompressible Navier-Stokes equations (NSE) in either advective or rotational
form which "calms" the system in the sense that the algebraic degree of the
nonlinearity is effectively reduced. This system, the calmed Navier-Stokes
Equations (calmed NSE), utilizes a "calming function" in the nonlinear term to
locally constrain large advective velocities. Notably, this approach avoids the
direct smoothing or filtering of derivatives, thus we make no modifications to
the boundary conditions. Under suitable conditions on the calming function, we
are able to prove global well-posedness of calmed NSE and show the convergence
of calmed NSE solutions to NSE solutions on the time interval of existence for
the latter. In addition, we prove that the dynamical system generated by the
calmed NSE in the rotational form possesses both an energy identity and a
global attractor. Moreover, we show that strong solutions to the calmed
equations converge to strong solutions of the NSE without assuming their
existence, providing a new proof of the existence of strong solutions to the 3D
Navier-Stokes equations.
