Global spherically symmetric classical solution to compressible Navier-Stokes equations with large initial data and vacuum

In this paper, we obtain a result on the existence and uniqueness of global spherically symmetric classical solutions to the compressible isentropic Navier-Stokes equations with vacuum in a bounded domain or exterior domain {\Omega} of Rn(n >= 2). Here, the initial data could be large. Besides, the regularities of the solutions are better than those obtained in [H.J. Choe and H. Kim, Math. Methods Appl. Sci., 28 (2005), pp. 1-28; Y. Cho and H. Kim, Manuscripta Math., 120 (2006), pp. 91-129; S.J. Ding, H.Y.Wen, and C.J. Zhu, J. Differential Equations, 251 (2011), pp. 1696-1725]. The analysis is based on some new mathematical techniques and some new useful energy estimates. This is an extension of the work of Choe and Kim, Cho and Kim, and Ding, Wen, and Zhu, where the global radially symmetric strong solutions, the local classical solutions in three dimensions, and the global classical solutions in one dimension were obtained, respectively. This paper can be viewed as the first result on the existence of global classical solutions with large initial data and vacuum in higher dimensionComment: 22 pages. arXiv admin note: substantial text overlap with arXiv:1103.142

Global Classical Large Solutions to Navier-Stokes Equations for Viscous Compressible and Heat Conducting Fluids with Vacuum

In this paper, we consider the 1D Navier-Stokes equations for viscous compressible and heat conducting fluids (i.e., the full Navier-Stokes equations). We get a unique global classical solution to the equations with large initial data and vacuum. Because of the strong nonlinearity and degeneration of the equations brought by the temperature equation and by vanishing of density (i.e., appearance of vacuum) respectively, to our best knowledge, there are only two results until now about global existence of solutions to the full Navier-Stokes equations with special pressure, viscosity and heat conductivity when vacuum appears (see \cite{Feireisl-book} where the viscosity $\mu=$const and the so-called {\em variational} solutions were obtained, and see \cite{Bresch-Desjardins} where the viscosity $\mu=\mu(\rho)$ degenerated when the density vanishes and the global weak solutions were got). It is open whether the global strong or classical solutions exist. By applying our ideas which were used in our former paper \cite{Ding-Wen-Zhu} to get $H^3-$estimates of $u$ and $\theta$ (see Lemma \ref{non-le:3.14}, Lemma \ref{non-le:3.15}, Lemma \ref{non-rle:3.12} and the corresponding corollaries), we get the existence and uniqueness of the global classical solutions (see Theorem \ref{non-rth:1.1})

Global symmetric classical and strong solutions of the full compressible Navier-Stokes equations with vacuum and large initial data

First of all, we get the global existence of classical and strong solutions of the full compressible Navier-Stokes equations in three space dimensions with initial data which is large and spherically or cylindrically symmetric. The appearance of vacuum is allowed. In particular, if the initial data is spherically symmetric, the space dimension can be taken not less than two. The analysis is based on some delicate {\it a priori} estimates globally in time which depend on the assumption $\kappa=O(1+\theta^q)$ where $q>r$ ($r$ can be zero), which relaxes the condition $q\ge2+2r$ in [14,29,42]. This could be viewed as an extensive work of [18] where the equations hold in the sense of distributions in the set where the density is positive with initial data which is large, discontinuous, and spherically or cylindrically symmetric in three space dimension. Finally, with the assumptions that vacuum may appear and that the solutions are not necessarily symmetric, we establish a blow-up criterion in terms of $\|\rho\|_{L^\infty_tL_x^\infty}$ and $\|\rho\theta\|_{L^4_tL^(12/5)_x}$ for strong solutions.Comment: arXiv admin note: text overlap with arXiv:1103.142

On the global existence and stability of 3-D viscous cylindrical circulatory flows

In this paper, we are concerned with the global existence and stability of a 3-D perturbed viscous circulatory flow around an infinite long cylinder. This flow is described by 3-D compressible Navier-Stokes equations. By introducing some suitably weighted energy spaces and establishing a priori estimates, we show that the 3-D cylindrical symmetric circulatory flow is globally stable in time when the corresponding initial states are perturbed suitably small.Comment: 21 page

Local existence and uniqueness of strong solutions to the free boundary problem of the full compressible Navier-Stokes equations in 3D

In this paper we establish the local-in-time existence and uniqueness of strong solutions to the free boundary problem of the full compressible Navier-Stokes equations in three-dimensional space. The vanishing density and temperature condition is imposed on the free boundary, which captures the motions of the non-isentropic viscous gas surrounded by vacuum with bounded entropy. We also assume some proper decay rates of the density towards the boundary and singularities of derivatives of the temperature across the boundary on the initial data, which coincides with the physical vacuum condition for the isentropic flows. This extends the previous result of Liu [ArXiv:1612.07936] by removing the spherically symmetric assumption and considering more general initial density and temperature profiles

Local well-posedness of the vacuum free boundary of 3-D compressible Navier-Stokes equations

In this paper, we consider the 3-D motion of viscous gas with the vacuum free boundary. We use the conormal derivative to establish local well-posedness of this system. One of important advantages in the paper is that we do not need any strong compatibility conditions on the initial data in terms of the acceleration.Comment: 31 page

Symmetric flows for compressible heat-conducting fluids with temperature dependent viscosity coefficients

We consider the Navier--Stokes equations for compressible heat-conducting ideal polytropic gases in a bounded annular domain when the viscosity and thermal conductivity coefficients are general smooth functions of temperature. A global-in-time, spherically or cylindrically symmetric, classical solution to the initial boundary value problem is shown to exist uniquely and converge exponentially to the constant state as the time tends to infinity under certain assumptions on the initial data and the adiabatic exponent $\gamma$. The initial data can be large if $\gamma$ is sufficiently close to 1. These results are of Nishida--Smoller type and extend the work [Liu et al., SIAM J. Math. Anal. 46 (2014), 2185--2228] restricted to the one-dimensional flows

Global well-posedness for the full compressible Navier-Stokes equations

In this paper, we mainly study the Cauchy problem for the full compressible Navier-Stokes equations in Sobolev spaces. We establish the global well-posedness of the equations with small initial data by using Friedrich's method and compactness arguments.Comment: arXiv admin note: text overlap with arXiv:1407.4661, arXiv:1109.5328 by other author

Global well-posedness of 2D compressible Navier-Stokes equations with large data and vacuum

In this paper, we study the global well-posedness of the 2D compressible Navier-Stokes equations with large initial data and vacuum. It is proved that if the shear viscosity $\mu$ is a positive constant and the bulk viscosity \l is the power function of the density, that is, \l(\r)=\r^\b with \b>3, then the 2D compressible Navier-Stokes equations with the periodic boundary conditions on the torus $\mathbb{T}^2$ admit a unique global classical solution $(\r,u)$ which may contain vacuums in an open set of $\mathbb{T}^2$. Note that the initial data can be arbitrarily large to contain vacuum states.Comment: 42 page

Global classical solution to the Cauchy problem of 2D baratropic compressible Navier-Stokes system with large initial data

For periodic initial data with initial density, we establish the global existence and uniqueness of strong and classical solutions for the two-dimensional compressible Navier-Stokes equations with no restrictions on the size of initial data provided the shear viscosity is a positive constant and the bulk one is \lam=\rho^{\b} with \b>1.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1205.5342, arXiv:1207.3746 by other author