  We propose a framework to understand input-output amplification properties of
non- linear partial differential equation (PDE) models of wall-bounded shear
flows, which are spatially invariant in one coordinate (e.g.,
streamwise-constant plane Couette flow). Our methodology is based on the notion
of dissipation inequalities in control theory. In particular, we consider flows
with body and other forcings, for which we study the input- to-output
properties, including energy growth, worst-case disturbance amplification, and
stability to persistent disturbances. The proposed method can be applied to a
large class of flow configurations as long as the base flow is described by a
polynomial. This includes many examples in both channel flows and pipe flows,
e.g., plane Couette flow, and Hagen-Poiseuille flow. The methodology we use is
numerically implemented as the solution of a (convex) optimization problem. We
use the framework to study input-output amplification mechanisms in rotating
Couette flow, plane Couette flow, plane Poiseuille flow, and Hagen-Poiseuille
flow. In addition to showing that the application of the proposed framework
leads to results that are consistent with theoretical and experimental
amplification scalings obtained in the literature through linearization around
the base flow, we demonstrate that the stability bounds to persistent forcings
can be used as a means to predict transition to turbulence in wall-bounded
shear flows.
