On the instability of incompressible Navier-Stokes equations with unbounded shear layer

We considered the construction of the linear instability of the Navier-Stokes equations around an unbounded shear layer. We employed the Rayleigh-Airy iteration scheme, developed by Grenier, Guo, and Nguyen [14], to construct unstable long waves and provide a qualitative description of the instability. Moreover, the growth rate of the unstable solution sheds light on the appropriate functional space for studying the inviscid limit problem of the incompressible Navier-Stokes equations with vortex sheet initial data.