Classic PDE systems
Andrew Gibiansky, Fluid Dynamics: The Navier-Stokes Equations is a great and compact introduction.
TBD. Operator adjoints.
Basis function methods
for fluids specifically
Bluman, George W. 1983. “On Mapping Linear Partial Differential Equations to Constant Coefficient Equations.” SIAM Journal on Applied Mathematics 43 (6): 1259–73.
Borthwick, David. 2016. Introduction to Partial Differential Equations. 1st ed. 2016. Universitext. Cham: Springer International Publishing : Imprint: Springer.
Borzì, Alfio, and Volker Schulz. 2012. Computational Optimization of Systems Governed by Partial Differential Equations. Computational Science and Engineering Series. Philadelphia: Society for Industrial and Applied Mathematics.
Brezis, Haim. 2010. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer Science & Business Media.
Dynkin, E. B. 2004. Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations. University Lecture Series, v. 34. Providence, R.I: American Mathematical Society.
Evans, Lawrence C. 1998. Partial Differential Equations. Amer Mathematical Society.
Kovacic, Jerald J. 1986. “An Algorithm for Solving Second Order Linear Homogeneous Differential Equations.” Journal of Symbolic Computation 2 (1): 3–43.
Kuzmin, Dmitri. 2010. A Guide to Numerical Methods for Transport Equations.
Le Gall, Jean-François. 1999. Spatial Branching Processes, Random Snakes and Partial Differential Equations. Basel: Birkhäuser Basel.
Reid, Homer. 2015. Advanced Analytic 18.305 Methods in Science and Engineering.