conventions in analysis of pde

Tao has a book on nonlinear dispersive PDE's which ought to be of help. It should be easily obtainable as a PDF online somewhere. I would recommend control+f'ing for the terms you want.
I encountered the same frustrations with these terms in my own reading. The best explanations I have found come from Tao's lecture notes as well as Section 3.1 of his book: Nonlinear Dispersive Equations. This describes most of what you are looking for. The Sobolev ladder and control of norms probably reference inequalities between Sobolev norms (like from the embedding theorem) or other quantities like initial energy, etc. It is not as formal as you are probably hoping for but "control" really just refers to one quantity being bounded by another over time. This mostly refers to bounds of some norm of the solution in terms of initial data and/or forcing functions.

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