This paper studies the dynamics of two incompressible immiscible fluids in 2D modeled by the inhomogeneous Navier-Stokes equations. We prove that if initially the viscosity contrast is small then there is global-in-time regularity. This result has been proved recently in [32] for $H^{ 5/2}$ Sobolev regularity of the interface. Here we provide a new approach which allows to obtain preservation of the natural $C^{ 1+\gamma}$ H\"older regularity of the interface for all $0<\gamma<1$. Our proof is direct and allows for low Sobolev regularity of the initial velocity without any extra technicality. It uses new quantitative harmonic analysis bounds for $C^{ \gamma}$ norms of even singular integral operators on characteristic functions of $C^{ 1+\gamma}$ domains [21].
Categories: math.AP