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October 5, 2025Open Access

On the vanishing viscosity limit for incompressible flows with inflow/outflow boundary conditions

Key Points

The convergence of Navier-Stokes equations to Euler equations is rigorously established as viscosity vanishes.
Interior convergence in L2 and Sobolev H1 norms is achieved at rates similar to injection/suction normal to the boundary.
The study allows for a general injection and suction angle, enhancing the application of boundary layer correctors.
The method involves constructing boundary layer correctors using Prandtl-type equations to improve convergence rates.

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 L² and the Sobolev H¹ 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.

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