Qualitative analysis of solution for the full compressible Euler equations in R

The present article is devoted to the study of the well-posedness, blow-up criterion and lower semicontinuity of the existence time for the full compressible Euler equations in R , N ≥ 1. First, let π = p(γ−1)/2γ and B 2,r ↪→ C, we show that the solution v = (π, u, S) is well-posed in B 2,r to the full compressible Euler equations, provided there exists a c > 0 such that the initial data entropy c ≤ S0 ≤ c−1. Next, we prove that the solution v is stable. i.e. the solution to Euler equations is continuously dependent of the initial data v0 in C([0, T [;B 2,r) and state the blow-up criterion of the solution. Thirdly, with the Osgood modulus of continuity, we derive a refined blow-up criterion of the solution by Theorem 4.1. Finally, we investigate the lower bound of maximal existence time and the lower semicontinuity of the existence time.