Classical solutions to relativistic Burgers equations in FLRW space-times

Abstract

We are interested in the classical solutions to the Cauchy problem of relativistic Burgers equations evolving in Friedmann-Lemat^itre-Robertson-Walker (FLRW) space-times, which are spatially homogeneous, isotropic expanding or contracting universes. In such kind of space-times, we first derive the relativistic Burgers equations from the relativistic Euler equations by letting the pressure be zero. Then we can show the global existence of the classical solution to the derived equation in the accelerated expanding space-times with small initial data by the method of characteristics when the spacial dimension $n=1$ and the energy estimate when $n\geq2$, respectively. Furthermore, we can also show the lifespan of the classical solution by similar methods when the expansion rate of the space-times is not so fast.