Global solutions to the relativistic Euler equation with spherical symmetry
The relativistic Euler equation in \(\mathbb{R}^3\) is given by \[ {\partial\over\partial t} \Biggl({\rho c^2+P\over c^2} {1\over 1-{v^2\over c^2}}-{P\over c^2}\Biggr)+ \sum^3_{j=1} {\partial\over\partial x_j} \Biggl({\rho c^2+P\over c^2} {v_j\over 1-{v^2\over c^2}}\Biggr)=0, \] \[ {\partial\over\partial t} \Biggl({\rho c^2+P\over c^2} {v_i\over 1-{v^2\over c^2}}\Biggr)+ \sum^3_{j=1} {\partial\over\partial x_j} \Biggl({\rho c^2+P\over c^2} {v_iv_j\over 1-{v^2\over c^2}}+\delta_{ij}P\Biggr)= 0,\;i=1,2,3. \] In 1993, Smoller and Temple have constructed global weak solutions to this equation in the one-dimensional case. In this article we show the existence of global weak solutions with spherical symmetry.
existence of global weak solutions with spherical symmetry
scientific article; zbMATH DE number 1054333