Blow‐up of the smooth solutions to the compressible Navier–Stokes equations

Abstract

In this paper, we will firstly extend the results about Jiu, Wang, and Xin (JDE, 2015, 259, 2981–3003). We prove that any smooth solution of compressible fluid will blow up without any restriction about the specific heat ratio γ. Then we prove the blow‐up of smooth solution of compressible Navier–Stokes equations in half space with Navier‐slip boundary. The main ideal is constructing the differential inequality. Copyright © 2017 John Wiley & Sons, Ltd.

Blow-up for the compressible isentropic Navier-Stokes-Poisson equations

“…It should be remarked that conditions (1.13) and (1.14) guarantee that integration by parts in our calculations makes sense (see also [8], [10], [13]).…”

“…For example, motivated by [16], under the assumption that the initial density has compact support, Xie in [15] showed blow-up result of smooth solutions to the full compressible N-S-P system in R 3 , Jiang and Tan in [7] obtained blow-up result of the compressible reactive self-gravitating gas with chemical kinetics equations in R 3 , Tang and Zhang in [12] established the blow-up result for both isentropic and isothermal N-S-P system in R 2 . The aim of this paper is to extend the work (see [13]) to the compressible unipolar isentropic N-S-P system with attractive forcing and compressible bipolar isentropic N-S-P system. To our best knowledge, this paper is the first work about the blow-up of smooth solutions to the compressible bipolar N-S-P system.…”

“…Recently, Jiu, Wang and Xin in [8] showed the blow-up of smooth solutions to the Cauchy problem for the full compressible Navier-Stokes equations and isentropic compressible Navier-Stokes equations with constant and degenerate viscosities in arbitrary dimensions under some restrictions on the initial data, but they do not require that the initial data has compact support or contains vacuum in any finite regions. For further generalization of the blow-up results of [8] about the full compressible Navier-Stokes equations and isentropic compressible Navier-Stokes equations with constant viscosities, we can refer to [13].…”

Remarks on blowup of solutions for one‐dimensional compressible Navier–Stokes equations with Maxwell's law

“…As mentioned by Sideris [18], the maximum propagation speed of a front into a constant state is a priori determined by the background state, whereas one cannot expect the property of finite propagation speed when the solutions decay to zero at far fields. For the compressible Navier-Stokes equations with solutions decay at far fields, Jiu et al [6] obtained some blowup results by establishing some relations among some physical quantities such as mass, momentum, momentum of inertia, internal energy, potential energy, total energy, and some combined functionals of these quantities, see the extended results in [19]. So, the second aim of this note is to generalize the blowup results of [6,19] to the compressible Navier-Stokes equations with Maxwell's law.…”

“…For the compressible Navier-Stokes equations with solutions decay at far fields, Jiu et al [6] obtained some blowup results by establishing some relations among some physical quantities such as mass, momentum, momentum of inertia, internal energy, potential energy, total energy, and some combined functionals of these quantities, see the extended results in [19]. So, the second aim of this note is to generalize the blowup results of [6,19] to the compressible Navier-Stokes equations with Maxwell's law.…”