Damped-driven Navier-Stokes equation

A damped-driven Navier-Stokes equation adds linear damping and external forcing. In two-dimensional steady vorticity form it reads

For a steady damped vorticity equation with $\gamma >0$, divergence-free transport, and bounded forcing, multiplication by $\omega |\omega {|}^{q-2}$ gives $\gamma \Vert \omega {\Vert }_q\le \Vert g{\Vert }_q$. Sending $q$ to infinity yields

A vanishing-viscosity limit studies solutions as the viscosity coefficient tends to zero. Uniform estimates and compactness can make the viscous term disappear weakly while retaining enough strong convergence to pass nonlinear transport terms to an inviscid equation.