Compressible Navier–Stokes equations with vacuum state in the case of general pressure law

In this paper, we consider the one‐dimensional compressible isentropic Navier–Stokes equations with a general ‘pressure law’ and the density‐dependent viscosity coefficient when the density connects to vacuum continuously. Precisely, the viscosity coefficient µ is proportional to ρ^θ and 0<θ<1, where ρ is the density. And the pressure P = P(ρ) is a general ‘pressure law’. The global existence and the uniqueness of weak solutions is proved, and a decay result for the pressure as t→ + ∞ is given. It is also proved that no vacuum states and no concentration states develop, and the free boundary do not expand to infinite. Copyright © 2006 John Wiley & Sons, Ltd.     

Vacuum states on compressible Navier-Stokes equations with general density-dependent viscosity and general pressure law

In this paper,we study the one-dimensional motion of viscous gas with a general pres- sure law and a general density-dependent viscosity coefficient when the initial density connects to the vacuum state with a jump.We prove the global existence and the uniqueness of weak solutions to the compressible Navier-Stokes equations by using the line method.For this,some new a priori estimates are obtained to take care of the general viscosity coefficientμ(ρ)instead ofρ~θ.     

Global classical solution to the 3D isentropic compressible Navier–Stokes equations with general initial data and a density‐dependent viscosity coefficient

In this paper, we study the global existence of classical solutions to the three‐dimensional compressible Navier–Stokes equations with a density‐dependent viscosity coefficient (λ = λ(ρ)). For the general initial data, which could be either vacuum or non‐vacuum, we prove the global existence of classical solutions, under the assumption that the viscosity coefficient μ is large enough. Copyright © 2014 John Wiley & Sons, Ltd.     

Global Smooth Solutions to the 2-D Inhomogeneous Navier-Stokes Equations with Variable Viscosity

Under the assumptions that the initial density ρ0 is close enough to 1 and ρ0 -- 1∈ H^s＋1（R^2）, u0 ∈ H^s（R^2） ∩ H^-ε（R^2） for s 〉 2 and 0 〈 ε 〈 1, the authors prove the global existence and uniqueness of smooth solutions to the 2-D inhomogeneous Navier-Stokes equations with the viscous coefficient depending on the density of the fluid. Furthermore, the L^2 decay rate of the velocity field is obtained.     

Discontinuous solutions of the compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum

In this paper, we study the evolutions of the interfaces between gas and the vacuum for one-dimensional viscous gas motions when the initial density connects to vacuum continuously. The degeneracy appears in the initial data and has effect on the viscosity coefficient because the coefficient is assumed to be a power function of the density. Using some new a priori estimates, we establish the new local (in time) existence and uniqueness results under minimal hypotheses on the initial density, so that the interval for the power of the density in the viscosity coefficient is enlarged to (0,γ). In particular, we include the important case that the initial density could be piecewise smooth with arbitrarily large jump discontinuities, and could degenerate to zero.     

Regularity of 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity

In this paper, we consider one-dimensional compressible isentropic Navier-Stokes equations with the viscosity depending on density and with free boundary. The viscosity coefficient μ is proportional to ρθ with 0<θ<1, where ρ is the density. The existence and uniqueness of global weak solutions in H¹([0,1]) have been established in [S. Jiang, Z. Xin, P. Zhang, Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity, Methods Appl. Anal. 12 (2005) 239-252]. We will establish the regularity of global solution under certain assumptions imposed on the initial data by deriving some new a priori estimates.     

Global existence of strong solutions of the Navier-Stokes equations for isentropic compressible fluids with density-dependent viscosity

This paper is concerned with global strong solutions of the isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient in one-dimensional bounded intervals. Precisely, the viscosity coefficient μ=μ(ρ) and the pressure P is proportional to ργ with γ>1. The important point in this paper is that the initial density may vanish in an open subset. We also show that the strong solution obtained above is unique provided that the initial data satisfies additional regularity and a compatible condition. Compared with former results obtained by Hyunseok Kim in [H. Kim, Global existence of strong solutions of the Navier-Stokes equations for one-dimensional isentropic compressible fluids, available at: http://com2mac.postech.ac.kr/papers/2001/01-38.pdf], we deal with density-dependent viscosity coefficient.     

Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum (II)

This is a continuation of the paper (Comm. Math. Phys. 230 (2002) 329) on the study of the compressible Navier-Stokes equations for isentropic flow when the initial density connects to vacuum continuously. The degeneracy appears in the initial data and has effect on the viscosity coefficient because the coefficient is assumed to be a power function of the density. This assumption comes from physical consideration and it also gives the well-posedness of the Cauchy problem. A new global existence result is established by some new a priori estimates so that the interval for the power of the density in the viscosity coefficient is enlarged to .     

粘性依赖于温度的MHD方程组整体经典解的正则性

本文主要研究可压缩非等熵平面磁流体动力学方程组的Cauchy问题整体经典解的正则性,其中方程组的粘性系数λ,μ,磁扩散系数η和热传导系数κ都是比容v和温度θ的函数,正比于h(v)θα,h是满足一定条件的非退化光滑函数.在正则性准则■的条件下,当α适当小时,我们证明了大初值整体经典解的存在性.     

Navier–Stokes equations with degenerate viscosity,vacuum and gravitational force

The global existence of weak solutions to the compressible Navier–Stokes equations with vacuum attracts many research interests nowadays. For the isentropic gas, the viscosity coefficient depends on density function from physical point of view. When the density function connects to vacuum continuously, the vacuum degeneracy gives some analytic difficulties in proving global existence. In this paper, we consider this case with gravitational force and fixed boundary condition. By giving a series of a priori estimates on the solution coping with the degeneracy of vacuum, gravitational force and boundary effect, we give global existence and uniqueness results similar to the case without force and boundary. Copyright © 2006 John Wiley & Sons, Ltd.     

Navier-Stokes方程的一种等阶稳定化亏量校正有限元法

本文对粘性不可压缩Navier-Stokes方程提出了一种等阶稳定化亏量校正有限元法.将通常的压力投影稳定化方法与亏量校正思想相结合,建立了一种稳定的有限元格式,绕开了inf-sup条件的限制,并且克服了当粘性系数很小时造成的不稳定性.对速度/压力采用等阶多项式空间,证明了解的存在唯一性,给出了误差估计.误差估计的结果表明,每校正一步误差的精度提高一阶.     

有关广义中心三项式系数的3个超同余式

设b,c为整数,定义广义中心三项式系数T_n(b,c)=[xⁿx²+bx+c]ⁿ=[π/2]∑k=0(n 2k)(2k n)b^n-2kck(n∈N={0,1...}),这里[xn]P(x)表示多项式P(x)中xn项的系数.特别地,中心Delannoy多项式Dn(x)=T_n(2x+1,x2+x)(n∈N),中心三项式系数T_n=T_n(1,1)(n∈N).本文研究了孙智伟在[南京大学学报:数学半年刊,2019,36(1):1-99]中提出的猜想,即完全证明了两个关于Dn(x)和的超同余式和一个关于中心三项式系数的超同余式的特殊情形.例如,设p为素数,r,m为正整数满足p■m条件.则对于任何p-adic整数x,有1/m²p^3r-3(p^rm-1∑k=0(2k+1)Dk(x)²-P²p^r-1m-1∑k=0(2k+1)Dk(x)²)=0(mod p³).     

Hölder continuity of interfaces for the porous medium equation with absorption

In this paper, we study the one-dimensional motion of viscous gas connecting to vacuum state with a jump in density when the viscosity depends on the density. Precisely, when the viscosity coefficient μ is proportional to ρ^θ and 0 < θ < 1/2, where ρ is the density, the global existence and the uniqueness of weak solutions are proved. This improves the previous results by enlarging the interval of θ.     

R^N中的非线性退化椭圆方程粘性解的存在性

本文研究了Ｒ＾Ｎ中的非线性退化椭圆型方程Ｆ（Ｄｕ，Ｄ＾２ｕ）＋ｕｓ＝ｆ的非负粘性解的存在性，其中ｓ〉０，Ｆ满足某些关于ｐ的条件，本文在下面的条件下证明了存在性；１．ｓ〉ｐ－１，ｆ在无穷远处不需要增长条件；２．０〈ｓ≤ｐ－１，ｆ在无穷远处具有某种增长条件。     

On the surface viscosity at the boundary between phases

The equations of motion of the interphase boundary are considered. It is shown that the conditions at the surface separating the phases obtained in /1, 2/ by different methods, are identical. The study of the dynamics of the fluid-fluid interface was initiated by Bussinesq /3/ who postulated a linear relationship between the surface stress tensor T_β and the strain rate tensor S_β, assigning two viscosity coefficients to the surface, the dilatation coefficient k (the analog of volume viscosity) and the two-dimensional shear viscosity . In the three-dimensional coordinate system two of whose axes u¹ and usu2 coincide with the axes of any coordinate system at the surface and whose third axis u³ is perpendicular to the surface, his results have the form Tβ = [γ + (k - )θ]a_β + S_β , θ = a^βS_β, V_{, β} = r_{. β} — v_sb_β,      

Remark on compressible Navier-Stokes equations with density-dependent viscosity and discontinuous initial data

In this paper, we study the free boundary problem for 1D compressible Navier-Stokes equations with density-dependent viscosity. We focus on the case where the viscosity coefficient vanishes on vacuum. We prove the global existence and uniqueness for discontinuous solutions to the Navier-Stokes equations when the initial density is a bounded variation function, and give a decay result for the density as t→+∞.     

Global well-posedness and large-time behavior of 1D compressible Navier-Stokes equations with density-depending viscosity and vacuum in unbounded domains

We consider the Cauchy problem for one-dimensional(1D) barotropic compressible Navier-Stokes equations with density-depending viscosity and large external forces. Under a general assumption on the densitydepending viscosity, we prove that the Cauchy problem admits a unique global strong(classical) solution for the large initial data with vacuum. Moreover, the density is proved to be bounded from above time-independently.As a consequence, we obtain the large time behavior of the solution without external forces.     

Compressible Navier-Stokes equations with ripped density

We are concerned with the Cauchy problem for the two-dimensional compressible Navier-Stokes equations  supplemented with general H¹ initial velocity and bounded initial density not necessarily strictly positive: it may be the characteristic function of any set, for instance. In the perfect gas case, we establish global-in-time existence and uniqueness, provided the volume (bulk) viscosity coefficient is large enough. For more general pressure laws (like e.g., $P={\rho }^{\gamma }$ with $\gamma >1$), we still get global existence, but uniqueness remains an open question. As a by-product of our results, we give a rigorous justification of the convergence to the inhomogeneous incompressible Navier-Stokes equations when the bulk viscosity tends to infinity. In the three-dimensional case, similar results are proved for short time without restriction on the viscosity, and for large time if the initial velocity field is small enough.     

Unique solvability for the density‐dependent non‐Newtonian compressible fluids with vacuum

The paper is devoted to the existence and uniqueness of local solutions for the density‐dependent non‐Newtonian compressible fluids with vacuum in one‐dimensional bounded intervals. The important points in this paper are that the initial density may vanish in an open subset and the viscosity coefficient is nonlinearly dependent of density and shear rate.     

Uniform regularity and vanishing viscosity limit for the chemotaxis‐Navier‐Stokes system in a 3D bounded domain

We investigate the uniform regularity and vanishing viscosity limit for the incompressible chemotaxis‐Navier‐Stokes system with Navier boundary condition for velocity field and Neumann boundary condition for cell density and chemical concentration in a 3D bounded domain. It is shown that there exists a unique strong solution of the incompressible chemotaxis‐Navier‐Stokes system in a finite time interval, which is independent of the viscosity coefficient. Moreover, this solution is uniformly bounded in a conormal Sobolev space, which allows us to take the vanishing viscosity limit to obtain the incompressible chemotaxis‐Euler system.