I collect here topics that (in my opinion!) forms solid background for further studies in PDEs. This should be considered to be "under construction".

Analysis

• bacis analysis : H. Amman, Analysis I, II, III
• Lebesgue integral : G. B. Folland, Real Analysis: Modern Techniques and Their Applications
• weak differentiability, function spaces: L.C. Evans, R. Gariepy, Measure Theory and Fine Properties of Functions
• functional analysis: H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations
• semigroups: K. Engel, R. Nagel, A Short Course on Operator Semigroups
• harmonic analysis: J. Duoandikoetxa, Fourier Analysis

PDEs and related topics

• introductory PDEs: L.C. Evans, Partial Differential Equations
• introductory PDEs: L.C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations
• elliptic PDEs: D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order
• parabolic PDEs: G. M. Lieberman, Second Order Parabolic Differential Equations
• porous media: J.L. Vazquez, The Porous Medium Equation: Mathematical Theory
• conservation laws: C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics (say, chapters 4-6 and 15-16)
• renormalization theory: C. de Lellis, ODEs with Rough Coefficients and the Renormalization Theorem of Ambrosio
• introductory kinetic theory: R. T. Glassey, The Cauchy Problem in Kinetic Theory
• introductory fluid mechanics: P.-L. Lions, Mathematical Topics in Fluid Mechanics (vol. 1)
• Boltzmann: C. Cercignani, R. Illner, M. Pulvirenti, The Mathematical Theory of Dilute Gases
• Navier-Stokes: J. Robinson, J. Rodrigo, W. Sadowski, The Three-Dimensional Navier–Stokes Equations
• convex integration: L. Szekelyhidi Jr, From Isometric Embeddings to Turbulence (+ references therein + this paper )
• calculus of variations: F. Rindler, Calculus of Variations
• gradient flows, optimal transport: F. Santambrogio, Optimal Transport for Applied Mathematicians

Stochastic PDEs

• readable stochastic analysis: B. Øksendal, Stochastic Differential Equations: An Introduction with Applications
• stochastic PDEs: W. Liu, M. Röckner, Stochastic Partial Differential Equations: An Introduction
• stochastic flows: H. Kunita, Stochastic Flows and Stochastic Differential Equations
• Markov processes: T. M. Ligett, Continuous Time Markov Processes: An Introduction (skip Chapter 1, it's terrible!)