The hyperbolic system forms an important class
of partial differential equations. It is involved in many scientific fields
such as gas dynamics, fluid mechanics and kinetic theory, etc. There is a
strong relation between the solution of the system and wave propagations
with finite speed. A fundamental property of the quasilinear hyperbolic
system is the formation of singularities in finite time. However,
the dissipative structure of the system may prevent the formation of
the singularity and lead to global smooth solutions in a neighborhood
of equilibrium states.
In this course, I will introduce basic concepts of the system, discuss
nonlinearity and formation of singularities, study dissipation conditions,
energy estimates and global smooth solutions near constant states
in Sobolev spaces.