Partial Differential Equations and Applications

First order non-linear equations. The Cauchy problem for quasi-linear equations. Burgers’ equation and the shock condition (Rankine-Hugoniot condition). Shock waves and rarefaction waves. Buckley-Leverett equations. Second order hyperbolic equations. Propagation of singularity. The wave equation. Shallow water equations. The Cauchy-Kowalevski theorem, Green identities and the Holmgren uniqueness theorem. Weak solutions; Distributions. Elliptic equations. The Laplace equation. The Poison equation for pressure or stream function. The wave equation in several spatial variables. The method of spherical means, the Duhamel principle and methods of energy. Parabolic equations. The maximum principle. Uniqueness and regularity analysis. 

References:
COURANT, R. and HILBERT, D. – Methods of Mathematical Physics, vol. II, Partial Differential Equations, Intersciense Publisher, 1953.
EVANS, L.C. – Partial Differential Equations (Graduate Studies in Mathematics, v. 19) GSM/19, AMS, 1998.
JOHN, F. – Partial Differential Equations, Springer-Verlag, 1982.
WHITHAM, G. B. – Linear and Nonlinear Waves, Wiley-Interscience, 1974.
ZAUDERER, E. – Partial Differential Equations of Applied Mathematics, 2^nd ed., John Wiley, 2000.

* Standard program. The teacher has the autonomy to make any changes