Global solutions to the relativistic Euler equation with spherical symmetry

Abstract

The relativistic Euler equation inR ³ is given by

$$\begin{gathered} \frac{\partial }{{\partial t}} \left( {\frac{{\rho c^2 + P}}{{c^2 }} \frac{1}{{1 - \frac{{\upsilon ^2 }}{{c^2 }}}} - \frac{P}{{c^2 }}} \right) + \sum\limits_{j = 1}^3 {\frac{\partial }{{\partial x_j }}} \left( {\frac{{\rho c^2 + P}}{{c^2 }} \frac{{\upsilon _j }}{{1 - \frac{{\upsilon ^2 }}{{c^2 }}}}} \right) = 0, \hfill \\ \frac{\partial }{{\partial t}} \left( {\frac{{\rho c^2 + P}}{{c^2 }} \frac{{\upsilon _i }}{{1 - \frac{{\upsilon ^2 }}{{c^2 }}}}} \right) + \sum\limits_{j = 1}^3 {\frac{\partial }{{\partial x_j }}} \left( {\frac{{\rho c^2 + P}}{{c^2 }} \frac{{\upsilon _i \upsilon _j }}{{1 - \frac{{\upsilon ^2 }}{{c^2 }}}} + \delta _{ij} P} \right) = 0, i = 1,2,3. \hfill \\ \end{gathered} $$

. In 1993, Smoller and Temple have constructed global weak solutions to this equation for 1 dimensional case. In this article we succeed, to show the existence of global weak solutions with spherical symmetry. Suppose that solutions are spherically symmetric. Then the equation becomes

$$\begin{gathered} \frac{\partial }{{\partial t}} \left( {\frac{{\rho c^2 + P}}{{c^2 }} \frac{1}{{1 - \frac{{\upsilon ^2 }}{{c^2 }}}} - \frac{P}{{c^2 }}} \right) + \frac{\partial }{{\partial r}} \left( {\frac{{\rho c^2 + P}}{{c^2 }} \frac{\upsilon }{{1 - \frac{{\upsilon ^2 }}{{c^2 }}}}} \right) + \frac{{\rho c^2 + P}}{{c^2 }} \frac{1}{{1 - \frac{{\upsilon ^2 }}{{c^2 }}}}\frac{{2\upsilon }}{r} = 0, \hfill \\ \frac{\partial }{{\partial t}} \left( {\frac{{\rho c^2 + P}}{{c^2 }} \frac{\upsilon }{{1 - \frac{{\upsilon ^2 }}{{c^2 }}}}} \right) + \frac{\partial }{{\partial r}} \left( {\frac{{\rho c^2 + P}}{{c^2 }} \frac{{\upsilon ^2 }}{{1 - \frac{{\upsilon ^2 }}{{c^2 }}}} + P} \right) + \frac{{\rho c^2 + P}}{{c^2 }} \frac{\upsilon }{{1 - \frac{{\upsilon ^2 }}{{c^2 }}}}\frac{{2\upsilon ^2 }}{r} = 0. \hfill \\ \end{gathered} $$

. To obtain our desired uniform estimates, we use the transformation

$$\upsilon = c tanh u = c\frac{{e^u - e^{ - u} }}{{e^u + e^{ - u} }}.$$

. This transformation is well-known among physicians and also plays very important role in our paper. We firmly believe that this transformation is also very useful for analyzing this equation mathematically.