Existence of global smooth solution to the relativistic Euler equations
The authors study the Cauchy problem for the one-dimensional relativistic Euler equations \[ \begin{aligned} &{\partial\over\partial t} \Biggl[{\rho+ Pu^2/c^4\over 1- u^2/c^2}\Biggr]+ {\partial\over\partial x} \Biggl[{(\rho+ P/c^2)u\over 1- u^2/c^2}\Biggr]= 0,\\ & {\partial\over\partial t}\Biggl[{(\rho+ P/c^2)u\over 1- u^2/c^2}\Biggr]+ {\partial\over\partial x}\Biggl[{P+\rho u^2\over 1- u^2/c^2}\Biggr]= 0,\end{aligned} \] with initial conditions \[ \rho(0,x)= \rho_0(x),\quad u(x,0)= u_0(x), \] where \(c\) is the speed of light and \(P\) is a function of \(\rho\) satisfying certain conditions. These equations govern the one-dimensional motion of a perfect gas in Minkowski space-time. The authors prove that, under certain specified conditions, these equations admit a unique global smooth solution on \(t\geq 0\).
scientific article; zbMATH DE number 2136982