Global propagation of regular nonlinear hyperbolic waves / Li Tatsien, Wang Libin

The book studies the global propagation of the regular nonlinear waves described by first-order quasilinear hyperbolic equations in one spatial dimension. The focus is the global existence and the blow-up for Cauchy problems, the one-sided mixed problem, generalized Riemann problems, inverse Riemann and piston problems.

In the introduction many examples that appear in applications are described, and the problems to be studied in the sequel are posed. Then the main tools of the investigation is described: the concept of weak linear degeneracy and the method of normalized coordinates. Two chapters are devoted to a deep study of the Cauchy problem. Necessary conditions and sufficient conditions of global existence and uniqueness of the smooth solution and a priori estimates are obtained, and a mechanism of formation of singularities and blow-ups is described. Some examples from gas dynamics and elasticity are given. Other chapters describe (from the same point of view) the Cauchy problem on a semi-bounded axis, one-sided mixed problems, generalized linear, nonlinear, and inverse Riemann problem and the piston problem. Most chapters contain examples from gas dynamics and the theory of elasticity. (Zentralblatt)