Blowup of smooth solutions to the compressible Euler equations with radial symmetry on bounded domains
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Blowup of smooth solutions to the compressible Euler equations with radial symmetry on bounded domains Jianwei Dong and Manwai Yuen Abstract. In this paper, we study the blowup of smooth solutions to the compressible Euler equations with radial symmetry on some fixed bounded domains (BR = {x ∈ RN : |x| ≤ R}, N = 1, 2, . . .) by introducing some new averaged quantities. We consider two types of flows: initially move inward and initially move outward on average. For the flow initially moving inward on average, we show that the smooth solutions will blow up in a finite time if the density vanishes at the origin only (ρ(t, 0) = 0, ρ(t, r) > 0, 0 < r ≤ R) for N ≥ 1 or the density vanishes at the origin and the velocity field vanishes on the two endpoints (ρ(t, 0) = 0, v(t, R) = 0) for N = 1. For the flow initially moving outward, we prove that the smooth solutions will break down in a finite time if the density vanishes on the two endpoints (ρ(t, R) = 0) for N = 1. The blowup mechanisms of the compressible Euler equations with constant damping or time-depending damping are obtained as corollaries. Mathematics Subject Classification. 35B44, 35L67. Keywords. Compressible Euler equations, Radial symmetry, Blowup.
1. Introduction In the last several decades, the blowup phenomena of solutions to the compressible Euler and Euler– Poisson equations have attracted many researchers’ attention. It is well known that the method of characteristics is often used to analyze the formation of singularities of solutions to the one-dimensional Euler equations [8,9,11,16,23], but it is difficult to be applied to the multi-dimensional case. For this, Sideris [21] first introduced some averaged quantities to study the blowup mechanism of solutions to the compressible Euler equations. The advantage of using these averaged quantities is that they can avoid the part of the local analysis of solutions. Under the conditions that the initial velocity field has a compact support (u0 = 0, |x| ≥ R) and initial density is strictly positive in the whole space and is equal to a positive constant outside the support of the initial velocity field (ρ0 (x) > 0, x ∈ R3 , ρ0 (x) = ρ > 0, |x| ≥ R), Sideris [21] proved two interesting results. The first result in [21] is that the solutions will blow up if the radial component of initial momentum ( xρ0 u0 dx) is sufficiently large. The second one is to R3
show that a fluid will develop singularities if it is compressed and outgoing on average near the wave front. The proof was based on the finite propagation of compactly supported disturbances of the solution. After Sideris’ classical work [21], many researchers have investigated the blowup phenomena of the compressible Euler equations by using the property of the finite propagation speed, see [1,4,22,25,27,36–38] and the references therein. When the initial density is surrounded by vacuum, there are also some blowup works on the compressible Euler and Euler–Poisson equations which have been published. Makino et al. [19
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