Blow-up of viscous heat-conducting compressible flows

Abstract

We show the blow-up of strong solution of viscous heat-conducting flow when the initial density is compactly supported. This is an extension of Z. Xin's result[5] to the case of positive heat conduction coefficient but we do not need any information for the time decay of total pressure nor the lower bound of the entropy. We control the lower bound of second moment by total energy and obtain the exact relationship between the size of support of initial density and the existence time. We also provide a sufficient condition for the blow-up in case that the initial density is positive but has a decay at infinity

Global Solutions to the Three-Dimensional Full Compressible Navier--Stokes Equations with Vacuum at Infinity in Some Classes of Large Data

“…[2] which shows that there is no global strong solution in R 3 if the initial density is of nontrivial compactly support. While the authors in [2] essentially impose on the strong solution an assumption for the entropy function S (i.e., S = S(x, t) < ∞ for all (x, t) ∈ R 3 × [0, T ]) in order that the temperature function vanishes in the vacuum region. Thus here we find a global strong solution in a bigger space than that in [2].…”

“…While the authors in [2] essentially impose on the strong solution an assumption for the entropy function S (i.e., S = S(x, t) < ∞ for all (x, t) ∈ R 3 × [0, T ]) in order that the temperature function vanishes in the vacuum region. Thus here we find a global strong solution in a bigger space than that in [2]. As a byproduct, when the initial mass is small in some sense, our result shows that the entropy function S of the strong solution is not always less than infinity in R 3 × [0, T ] even if it is initially.…”

Compressible Navier–Stokes Equations with hyperbolic heat conduction

“…Because of its physical importance and mathematical challenging, the well-posed theory has been widely studied for the system (1.1), (1.2) combined with Fourier's law (1.6), see [1,2,3,4,6,8,11,12,14,15,16,17,18,21,23]. In particular, the local existence and uniqueness of smooth solutions was established by Serrin [21] and Nash [18] for initial data far away from vacuum.…”

“…Later, Matsumura and Nishida [16] got global smooth solutions for small initial data without vacuum. For large data, Xin [23], Cho and Jin [1] showed that smooth solutions must blow up in finite time if the initial data has a vacuum state. The existence of global non-vacuum smooth solutions for large data is a famous open problem in fluid dynamics.…”

Blow-Up of Smooth Solutions to the Navier–stokes Equations of Compressible Viscous Heat-Conducting Fluids

“…The key points of his proof are the lower bound of the entropy and the time decay of total pressure, but his proof seems hard to apply for the case k > 0. For the case k > 0, but in a different way by estimating the quantity R n ρ(t, x)|x| 2 d x, Cho and Jin [1] also showed that with the initial data compactly supported, the life span of smooth solutions is finite. In this paper, we shall give a simpler and refined proof, by estimating the time evolution of the quantity R n ρ(t, x)x i d x, to obtain the blow-up phenomena for smooth solutions to (NS) with k ≥ 0 and the initial data compactly supported; the precise results are stated in Theorem 1.1.…”