Local existence and non-relativistic limits of shock solutions to a multidimensional piston problem for the relativistic Euler equations
The authors of this interesting paper consider a multidimensional piston problem with initial-boundary value data. The main goal is the local existence of shock front solutions to the spherically symmetric piston problem for relativistic Euler equations. The basic equations are \[ \partial_t\biggl(\frac{(\rho +p\epsilon )u}{1-u^2\epsilon }\biggr)+\partial_r\biggl(\frac{(\rho +p\epsilon )u^2}{1-u^2\epsilon }+p\biggr)+\frac{(d-1)(\rho +p\epsilon )u^2}{(1-u^2\epsilon )r}=0, \] \[ \partial_t\biggl(\frac{(\rho +p\epsilon )u^2\epsilon }{1-u^2\epsilon }+\rho \biggr)+\partial_r\biggl(\frac{(\rho +p\epsilon )u}{1-u^2\epsilon }\biggr)+\frac{(d-1)(\rho +p\epsilon )u}{(1-u^2\epsilon )r}=0, \] where \(r=|\vec{x}|\), \(u=|\vec{u}|\), \(\epsilon = 1/c^2\). Obviously this system tends to classical non-relativistic case when accept \(c\to\infty \) (\(\epsilon\to 0\)). The path of the piston is described as \(r=b(t)\), and the boundary condition is \(u=b'(t)\), \(r=b(t)\). The state of the gas of the expanding body at the initial time is \(U_0=(\rho_a,0)\). The main result is that for \(b\in C^{\infty }\), \(b(0)=0\), \(b'(0)=b_0>0\), \(b^{(k)}(0)=0\) (\(2\leq k\leq K\)), where \(K\) is a large integer, there is a constant \(T_0>0\), such that there exists a shock front solution \((\rho^{c},u^{c})\) to the problem stated above for \(t\leq T_0\), \(b(t)<r<s(t)\). Moreover, there exists a subsequence \(c_n\to \infty \), as \(n\to\infty \), such that \((\rho^{c_n},u^{c_n})\to (\rho ,u)\) a.e., and the limit \((\rho ,u)\) is the corresponding solution of the classical non-relativistic system; this solution is piecewise smooth.
scientific article; zbMATH DE number 6140988
scientific article