On the vanishing viscosity limit for incompressible flows with inflow/outflow boundary conditions
Anna L. Mazzucato, Dehua Wang, Wei Wei
TL;DR
This work establishes the vanishing viscosity limit for incompressible flows in bounded domains with inflow/outflow and oblique injection, proving convergence to Euler solutions in the energy norm and achieving interior $L^2$ and $H^1$ convergence with rates matching the normal-injection case. The authors develop a unified framework based on curvilinear boundary-layer coordinates, Prandtl-type boundary-layer correctors, and higher-order asymptotics to obtain interior convergence, including scenarios with umbilical boundary points. They derive explicit leading-order and higher-order boundary-layer corrections, achieving optimal rates $\|v^\epsilon - v^0 - \varphi^0\|_{L^\infty L^2} \le C\epsilon$ and $\|v^\epsilon - v^0 - \varphi^0\|_{L^2 H^1} \le C\epsilon^{1/2}$, and improving to $\|v^\epsilon - v^0 - \varphi^0 - \varepsilon(v^1+\varphi^1)\|_{L^\infty L^2} \le C\varepsilon^2$ and $\|\cdot\|_{L^2 H^1} \le C\varepsilon^{3/2}$ with higher-order expansions. The methodology enables arbitrary-order convergence under suitable compatibility conditions, providing a robust analytic justification for the influence of injection/suction on boundary-layer behavior and vorticity production in the zero-viscosity limit.
Abstract
We study the vanishing viscosity limit for the incompressible Navier-Stokes equations (NSE) in a general bounded domain with inflow-outflow boundary conditions. Extending the work of Gie, Hamouda, and Temam ( Netw. Heterog. Media 7, 2012) and also of Lombardo and Sammartino (SIAM J. Math. Anal. 33, 2001), we allow for a general injection and suction angle, as long as it is bounded away from zero. We rigorously establish the convergence of NSE solutions to those of the Euler equations (EE) as viscosity vanishes in the energy norm. We prove interior convergence in both the $L^2$ and the Sobolev $H^1$ norms at the same rates as in the case of injection/suction normal to the boundary. The proof relies on the construction of boundary layer correctors via Prandtl-type equations and a higher-order asymptotic expansion that improves the convergence rate.