Travelling Wave-Like Solutions of the Navier]Stokes and the Related Equations

We present a new family of travelling wave-like solutions to the Navier]Stokes equations of incompressible fluid flows, and other regularized equations of the Euler equations, obtain their trend to the solutions of the Euler equations as the viscosity tends to zero, and estimate the rate of convergence. We also find a ‘‘singularizing effect’’ of the viscosity term in the Navier]Stokes equations, i.e., we have a local moving blow-up of unbounded solutions with the blow-up’s speed depending on viscosity. We demonstrate that if the initial function is the Beltrami flow then the solution of the Navier]Stokes equations conserves the Beltrami flow property for all time.