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 general variable viscosity coefficient, under a scaling-invariant smallness condition on the initial data. The viscosity coefficient μ may depend on the freely transported density function in a general nonlinear manner and may exhibit large jumps across W$^{2,2+ϵ}$-interfaces. The Lipschitz estimate for the velocity field, even in the presence of significant discontinuities in μ, can be achieved by a novel analysis of the viscous stress tensor μSu. Specifically, the fourth-order elliptic operator for the stream function, which is hidden in the viscosity term div (μSu), is analyzed both globally- and locallyin-space via two newly introduced ``good unknowns''. The ``global good unknown'' (R$^⊥$ ⊗ R):(μSu), where R denotes the Riesz operator, is shown to satisfy timeweighted H$^1$-energy estimates. Combined with tangential regularity, this leads to the W$^{1,2+ϵ}$-regularity of the ``local good unknown'', ($\overline{τ}$ ⊗ n):(μSu), where $\overline{τ}$ and n denote the unit tangential and normal vectors of some free interfaces, respectively. ... mehrThis regularity implies finally the uniform boundedness of the discontinuous normal derivative ∂$_n$u across interfaces.
As applications, we investigate the Boussinesq equations without heat conduction and establish the global-in-time well-posedness for the density-dependent incompressible Navier-Stokes equations in two spatial dimensions.