Global solutions of the Navier-Stokes equation with strong viscosity

Following Ebin and Marsden the Navier-Stokes equation is viewed as a perturbation of a geodesic flow on the group of volume preserving diffeomorphisms on a compact Riemannian manifold. Existence and uniqueness of bounded solutions for all position time is shown by taking a higher order diffusion term.Partial y supported by Alexander von Humboldt Foundation     

Inverse viscosity problem for the Navier-Stokes equation

We consider an inverse problem of determining a viscosity coefficient in the Navier-Stokes equation by observation data in a neighborhood of the boundary. We prove the Lipschitz stability by the Carleman estimates in Sobolev spaces of negative order.     

Global behavior of spherically symmetric Navier-Stokes equations with density-dependent viscosity

In this paper, we study a free boundary problem for compressible spherically symmetric Navier-Stokes equations without a solid core. Under certain assumptions imposed on the initial data, we obtain the global existence and uniqueness of the weak solution, give some uniform bounds (with respect to time) of the solution and show that it converges to a stationary one as time tends to infinity. Moreover, we obtain the stabilization rate estimates of exponential type in L^∞-norm and weighted H¹-norm of the solution by constructing some Lyapunov functionals. The results show that such system is stable under the small perturbations, and could be applied to the astrophysics.     

On non-stationary Navier-Stokes flows with vanishing viscosity associated with a variational inequality

Let g is a positive increasing function with 1?g(0). The existence of a unique solution of the Navier-Stokes flow associated with K_ε,γ and the convergence of the solution to that of the Euler equations as the viscosity goes to zero are established.     

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.     

Remarks on one-dimensional Compressible Navier-Stokes equations with density-dependent viscosity and vacuum

The Navier-Stokes system for one-dimensional compressible fluids with density-dependent viscosities when the initial density connects to vacuum continuously is considered in the present paper. When the viscosity coefficient u is proportional to pθ with 0 〈 θ 〈 1, the global existence and the uniqueness of weak solutions are proved which improves the previous results in [Vong, S. W., Yang, T., Zhu, C. J.： Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum II. J. Differential Equations, 192（2）, 475-501 （2003）]. Here p is the density. Moreover, a stabilization rate estimate for the density as t → ＋∞ for any θ 〉 0 is also given.     

Global smooth solutions of the compressible Navier-Stokes equations with density-dependent viscosity

In this paper we study a free boundary problem for the viscous, compressible, heat conducting, one-dimensional real fluids. More precisely, the viscosity is assumed to be a power function of density, i.e., μ(ρ)=ρα, where ρ denotes the density of fluids and α is a positive constant. In addition, the equations of state include and are more general than perfect flows which only depend linearly on temperature. The global existence (uniqueness) of smooth solutions is established with for general, large initial data, which improves the previous results. Moreover, it is also shown that the solutions will not develop vacuum, mass concentration or heat concentration in a finite time provided the initial data are bounded and smooth, and do not contain vacuum.     

Global well-posedness of 3-D density-dependent Navier-Stokes system with variable viscosity

Given initial data(ρ0, u0) satisfying 0 m ρ0≤ M, ρ0- 1 ∈ L2∩˙W1,r(R3) and u0 ∈˙H-2δ∩ H1(R3) for δ∈ ]1/4, 1/2[ and r ∈ ]6, 3/1- 2δ[, we prove that: there exists a small positive constant ε1,which depends on the norm of the initial data, so that the 3-D incompressible inhomogeneous Navier-Stokes system with variable viscosity has a unique global strong solution(ρ, u) whenever‖ u0‖ L2 ‖▽u0 ‖L2 ≤ε1 and ‖μ(ρ0)- 1‖ L∞≤ε0 for some uniform small constant ε0. Furthermore, with smoother initial data and viscosity coefficient, we can prove the propagation of the regularities for such strong solution.     

Dynamical behaviors for 1D compressible Navier-Stokes equations with density-dependent viscosity

The dynamical behaviors of vacuum states for one-dimensional compressible Navier-Stokes equations with density-dependent viscosity coefficient are considered. It is first shown that a unique strong solution to the free boundary value problem exists globally in time, the free boundary expands outwards at an algebraic rate in time, and the density is strictly positive in any finite time but decays pointwise to zero time-asymptotically. Then, it is proved that there exists a unique global weak solution to the initial boundary value problem when the initial data contains discontinuously a piece of continuous vacuum and is regular away from the vacuum. The solution is piecewise regular and contains a piece of continuous vacuum before the time T_∗>0, which is compressed at an algebraic rate and vanishes at the time T_∗, meanwhile the weak solution becomes either a strong solution or a piecewise strong one and tends to the equilibrium state exponentially.     

Determination of viscosity in the stationary Navier-Stokes equations

In this paper we consider the stationary Navier-Stokes equations in a bounded domain with a variable viscosity. We prove that one can uniquely determine the viscosity function from the knowledge of boundary data.     

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 .     

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→+∞.     

Equivalence of the Navier-Stokes equation and the infinite-dimensional Burgers equation

In this paper, we prove the equivalence of the Navier-Stokes equation and the infinite-dimensional Burgers-type equation with the Laplace-Lévy operator. An explicit formula for the solution of a certain system of linear equations arising in studying the circulation of the solution of the Navier-Stokes equation is presented. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 109–120, 2006.     

Decay rate of quasilinear wave equation with viscosity

This paper is concerned with the decay rate of solutions for a quasilinear wave equation with viscosity. We use a so-called energy perturbation method to establish decay rate of solutions in terms of energy norm for a class of nonlinear functions. With the help of a comparison lemma of differential inequalities, we establish a relationship between decay rate of solutions and f .     

Vanishing viscosity of isentropic Navier-Stokes equations for interacting shocks

We study the vanishing viscosity of the Navier-Stokes equations for interacting shocks. Given an entropy solution to p-system which consists of two different families of shocks interacting at some positive time,we show that such entropy solution is the vanishing viscosity limit of a family of global smooth solutions to the isentropic Navier-Stokes equations. The key point of the proofs is to derive the estimates separately before and after the interaction time and connect the incoming and outgoing viscous shock profiles.