A Study On Existence Of BV Weak Solutions To A Boundary Value Problem Of Time-discretized 2-D Isentropic Euler Equations

The Euler system is an important model describing the motion of gases,which is of great value to the applications of physics and engineering. A well-established mathematical theory for 1-D Euler equations has been available now. However, for multi-dimensional cases, we are still far away from such a satisfactory theory. In this paper, we study the 2-D isentropic Euler system by discretizing the time-derivatives,so that they become nonhomogeneous terms to steady Euler equations. The deduced equations are called time-discretized 2-D Euler equations and we are going to investigate the existence of BV weak solutions for given boundary conditions. We assume that the initial data for the original non-steady Euler equations are BV functions, and their values at the given boundary are small BV perturbation of a steady supersonic shock solution to the Euler equations. Under these assumptions, we applied a modified Glimm scheme and constructed a Glimm-type functional to prove the existence of local BV weak solutions to the boundary value problem. The difficulties for the proof are the existence of the nonhomogeneous terms and a strong shock wave. To overcome the difficulties, we modify the method of Glimm’s scheme to construct the approximate solutions, referring to Dafermos-Hsiao[9], and establish estimates for the interactions not only between weak waves but also between the strong shock and weak waves. These estimates are then employed to establish the convergence of the approximate solutions which implies the local existence of the BV weak solutions of the system.