*L. C. Evans, Partial Differential Equations, Second Edition. Graduate Studies in Mathematics, volume 19, American Mathematical Society, Providence, RI, 2010*
1. Chap. 8: Calculus of variations.
2. Chap. 9: Nonvariational techniques.
3. Chap. 3.4 & 11: Conservation laws.
4. Additionally, the appendices in the book; they cover relevant background material from real analysis, functional analysis, and measure theory.
*Benoit Perthame, Kinetic formulation of conservation laws, Oxford University Press, 2002*
1. Chap. 2: Preliminary material (kinetic tools).
2. Chap. 3: Kinetic formulation of conservation laws.
3. Chap. 5: Regularizing effects (averaging lemmas).
If time permits, a quick introduction to stochastic partial differential equations , e.g. parabolic and hyperbolic PDEs driven by temporal Ito noise.