The global smooth symmetric solution to 2-D full compressible Euler system of Chaplygin gases

Abstract

For one dimensional or multidimensional compressible Euler system of polytropic gases, it is well known that the smooth solution will generally develop singularities in finite time. However, for three dimensional Chaplygin gases, due to the crucial role of “null condition” in the potential equation which is derived by the irrotational and isentropic flow, P. Godin in [9] has proved the global existence of a smooth 3-D spherically symmetric flow with variable entropy when the initial data are of small smooth perturbations with compact supports to a constant state. It is noted that there are some clear differences for the global solution or blowup problems between 2-D and 3-D hyperbolic equations or systems. In this paper, we will focus on the global symmetric solution problem of 2-D full compressible Euler system of Chaplygin gases. Through carrying out involved analysis and finding an appropriate weight we can derive some uniform weighted energy estimates on the small symmetric solution to 2-D compressible Euler system of Chaplygin gases and further establish the global existence of the smooth solution by the continuous induction method.