Stability of Inflow Problem for Hyperbolic Systems

Inflow BC plays a critical role in the study of hyperbolic PDE in a bounded domain. We establish $W^{1,\mathrm{\infty }}$ stability for 1D hyperbolic conservation laws with inflow data in a bounded interval, and $W^{2,3+}$ stability of a large class of shear flows for the 3D incompressible Euler system with inflow BC in finite square or circular pipes.