Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density (original) (raw)
Nonlinear Analysis: Theory, Methods & Applications, 2013
We consider blowup of classical solutions to compressible Navier-Stokes equations with revised Maxwell's law which can be regarded as a relaxation to the classical Newtonian flow. For this new model, we show that for some special large initial data, the life span of any C 1 solution must be finite. This shows much difference with Newtonian flow where the global existence of C 1 solutions is still an open and famous problem in fluid dynamics. We exploit the property of finite propagation speed and the methods from Sideris (1985) to prove our results.
Journal of Differential Equations, 2008
We prove that the smooth solutions to the Cauchy problem for the Navier-Stokes equations with conserved total mass, finite total energy and finite momentum of inertia lose the initial smoothness within a finite time in the case of space of dimension 3 or greater even if the initial data are not compactly supported. The cases of isentropic and incompressible fluids are also considered.
High Regularity of Solutions of Compressible Navier-Stokes Equations
Advances in Differential Equations, 2007
We study the Navier-Stokes equations for compressible barotropic fluids in a bounded or unbounded domain Ω of R 3. The initial density may vanish in an open subset of Ω or to be positive but vanish at space infinity. We first prove the local existence of solutions (ρ (j) , u (j)) in C([0, T * ]; H 2(k−j)+3 × D 1 0 ∩ D 2(k−j)+3 (Ω)), 0 ≤ j ≤ k, k ≥ 1 under the assumptions that the data satisfy compatibility conditions and that the initial density is sufficiently small. To control the nonnegativity or decay at infinity of density, we need to establish a boundary value problem of (k+1)-coupled elliptic system which may not be in general solvable. The smallness condition of initial density is necessary for the solvability, which is not necessary in case that the initial density has positive lower bound. Secondly, we prove the global existence of smooth radial solutions of isentropic compressible Navier-Stokes equations on a bounded annulus or a domain which is the exterior of a ball under a smallness condition of initial density.
Blow-up of the viscous heat-conducting compressible flow
Preprint Series of Department of Mathematics, Hokkaido University, 2005
We show the blow-up of smooth solution of viscous heat-conducting flow when the initial density is compactly supported. This is an extension of Z. Xin's result[4] to the case of positive heat conduction coefficient but we do not need any information for the lower bound of the entropy. We control the lower bound of second moment by total energy.
Blow-up of viscous heat-conducting compressible flows
Journal of Mathematical Analysis and Applications, 2006
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.
A Survey of the Compressible Navier-Stokes Equations
Taiwanese Journal of Mathematics
This paper presents mathematical properties of solutions to the Navier-Stokes equations for compressible fluids. We first review existence results for the Cauchy problem, and describe some regularity properties of solutions in the presence of possibly vanishing densities. Finally, we address the problem of the low Mach number limit leading to incompressible models.
A Beale–Kato–Majda blow-up criterion for the 3-D compressible Navier–Stokes equations
Journal de Mathématiques Pures et Appliquées, 2011
We prove a blow-up criterion in terms of the upper bound of the density for the strong solution to the 3-D compressible Navier-Stokes equations. The initial vacuum is allowed. The main ingredient of the proof is a priori estimate for an important quantity under the assumption that the density is upper bounded, whose divergence can be viewed as the effective viscous flux.
Remarks on the rate of decay of solutions to linearized compressible Navier-Stokes equations
2015
We consider the Lp−Lq estimates of solutions to the Cauchy problem of linearized compressible Navier–Stokes equation. Especially, we investigate the diffusion wave property of the compressible Navier–Stokes flows, which was studied by D. Hoff and K. Zumbrum and Tai-P. Liu and W. Wang. 1. Introduction. In this paper, we consider the Cauchy problem of the following linearized compressible Navier-Stokes equations: ρt + γdiv v = 0 in (0,∞) × Rn,(1.1) vt − α∆v − β∇div v + γ∇ρ = 0 in (0,∞) × Rn,