Global-in-time well-posedness for the two-dimensional incompressible Navier-Stokes equations with freely transported viscosity coefficient

We establish the global-in-time well-posedness of the two-dimensional incompressible Navier-Stokes equations with freely transported viscosity coefficient, under a scaling-invariant smallness condition on the initial data. The viscosity coefficient is allowed to exhibit large jumps across $W^{2,2+\epsilon}$-interfaces. The viscous stress tensor $\mu Su$ is carefully analyzed. Specifically, $(R^\perp\otimes R):(\mu Su)$, where $R$ denotes the Riesz operator, defines a ``good unknown'' that satisfies time-weighted $H^1$-energy estimates. Combined with tangential regularity, this leads to the $W^{1,2+\epsilon}$-regularity of another ``good unknown'', $(\bar{\tau}\otimes n):(\mu Su)$, where $\bar{\tau}$ and $n$ denote the unit tangential and normal vectors of the interfaces, respectively. These results collectively provide a Lipschitz estimate for the velocity field, even in the presence of significant discontinuities in $\mu$. As applications, we investigate the well-posedness of the Boussinesq equations without heat conduction and the density-dependent incompressible Navier-Stokes equations in two spatial dimensions.