Finite-time Properties of the Navier-Stokes Equations Under Lebesque Space Disturbances

A complete understanding of the stability characteristics of the Navier-Stokes equations involve understanding both the transient response and the steady state response. The steady state (or infinite-time) response of the Navier-Stokes equations is characterized by the point spectrum and has been well studied. In this work, we study the transient (or finite-time) response of the unsteady Navier-Stokes equations linearized about plane Couette base flow under spatial and temporal varying disturbance forcing. The forcing and response are assumed to belong to infinite-dimensional Lebesque function spaces, $L_2$ and $L_{\infty}$. An analytical characterization is given for the induced norms that characterize the response. It is shown that the $L_2$ induced norm is tightly bounded by the $H_{\infty}$ norm of the transfer function operator and the $L_{\infty}$ induced norm is upper bounded by the $L_1$ norm of the impulse response operator. The structure of the worst case disturbances and their amplification rates are computed using spectral methods---with Fourier modes in homogeneous direction and Chebyshev collocation in non-homogeneous direction. The relevance of the present results to the channel flow laminar-turbulent transition experiments will be discussed.