Formation of Singularity for Isentropic Irrotational Compressible Euler Equations

Abstract

The domain of science and engineering relies heavily on an in-depth comprehension of fluid dynamics, given the prevalence of fluids such as water, air, and interstellar gas in the universe. Euler equations form the basis for the study of fluid motion. This paper is concerned with the Cauchy problem of isentropic compressible Euler equations away from the vacuum. We use the integration method with the general test function f=f(r), proving that there exist the corresponding blowup results of C1 irrotational solutions for Euler equations and Euler equations with time-dependent damping in Rn (n≥2), provided the density-independent initial functional is sufficiently large. We also provide two simple and explicit test functions f(r)=r and f(r)=1+r, to demonstrate the blowup phenomenon in the one-dimensional case. In particular, our results are applicable to the non-radial system.

Keywords

• Euler equations
• blowup
• irrotational solutions
• time-dependent damping
• initial value problem
• test function