Blowup phenomenon for the initial-boundary value problem of the non-isentropic compressible Euler equations

Abstract

The blowup phenomenon for the initial-boundary value problem of the non-isentropic compressible Euler equations is investigated. More precisely, we consider a functional F(t) associated with the momentum and weighted by a general test function f and show that if F(0) is sufficiently large, then the finite time blowup of the solutions of the non-isentropic compressible Euler equations occurs. As the test function f is a general function with only mild conditions imposed, a class of blowup conditions is established.

Formation of Singularity for Isentropic Irrotational Compressible Euler Equations

“…As observed in [36], the functional (15) has potential applications in exploring threedimensional non-isentropic rotational solutions of Euler equations. The formation of singularity of irrotational solutions for compressible Euler equations with general timedependent damping in R n ρ t + ▽ • (ρu) = 0, ρ[u t + (u • ▽)u] + ▽P = −a(t)ρu, (167) could be analyzed using a similar approach, where a(t) > 0.…”

“…Their findings indicated that singular solutions with radial symmetry must occur in finite time when F(0) reaches a sufficient value. Additionally, Cheung, Wong and Yuen con-structed a test function f = f (r) that represents an increasing C 1 property on [0, +∞) and vanishes at r = 1 in [36]. They used the functional…”

“…We have G 5 (F 1 (t)) ≥ 0 if F 1 (0) ≥ µ 2A and G 5 (F 1 (0)) ≥ 0. Thus, by (36), we obtain G 4 (t) ≥ G 5 (F 1 (t)) ≥ 0 for t ∈ [0, T]. From condition (37) and inequality (158), we have…”