Global smooth solutions of some quasilinear hyperbolic systems with large data (Q5928961)

This work is concerned with the Cauchy problem for quasilinear hyperbolic system NEWLINE\[NEWLINE\begin{aligned} & \frac{\partial u}{\partial t} u(t,x)+\sum^n_{i=1} c_i(u(t,x)) \frac{\partial}{\partial x_i} u(t,x)=0,\quad t>0,\;x\in\mathbb{R},\\ & u(0,x)=u_0(x).\end{aligned}NEWLINE\]NEWLINE This system is the very particular case where all the eigenvalues are equal. The author proved that this system admits smooth solutions, which are global in time for large Cauchy data, which satisfy a particular criterion. It is in fact a generalization of the famous property of the Burgers equation in 1D to be smooth when the initial data is not decreasing. The existence theorem is based on a property of a matrix-valued Riccati equation. Also, the existence of a global smooth solution for some Hamilton-Jacobi equations is proved.