Global solutions of the Navier-Stokes equations for compressible flow with density-dependent viscosity and discontinuous initial data

In this paper, we study the evolutions of the interfaces between the gas and the vacuum for viscous one-dimensional isentropic gas motions. We prove the global existence and uniqueness for discontinuous solutions of the Navier-Stokes equations for compressible flow with density-dependent viscosity coefficient. Precisely, the viscosity coefficient μ is proportional to ρ^θ with 0<θ<1. Specifically, we require that the initial density be piecewise smooth with arbitrarily large jump discontinuities, bounded above and below away from zero, in the interior of gas. We show that the discontinuities in the density persist for all time, and give a decay result for the density as t→+∞.     

Regularity and asymptotic behavior of 1D compressible Navier-Stokes-Poisson equations with free boundary

In this paper, firstly, we consider the regularity of solutions in to the 1D Navier-Stokes-Poisson equations with density-dependent viscosity and the initial density that is connected to vacuum with discontinuities, and the viscosity coefficient is proportional to ρθ with 0<θ<1. Furthermore, we get the asymptotic behavior of the solutions when the viscosity coefficient is a constant. This is a continuation of [S.J. Ding, H.Y. Wen, L. Yao, C.J. Zhu, Global solutions to one-dimensional compressible Navier-Stokes-Poisson equations with density-dependent viscosity, J. Math. Phys. 50 (2009) 023101], where the existence and uniqueness of global weak solutions in H¹([0,1]) for both cases: μ(ρ)=ρθ, 0<θ<1 and μ=constant have been established.     

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.     

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 weak solutions and asymptotic behavior to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum

This paper is concerned with existence of global weak solutions to a class of compressible Navier-Stokes equations with density-dependent viscosity and vacuum. When the viscosity coefficient μ is proportional to ρθ with , a global existence result is obtained which improves the previous results in Fang and Zhang (2004) [4], Vong et al. (2003) [27], Yang and Zhu (2002) [30]. Here ρ is the density. Moreover, we prove that the domain, where fluid is located on, expands outwards into vacuum at an algebraic rate as the time grows up due to the dispersion effect of total pressure. It is worth pointing out that our result covers the interesting case of the Saint-Venant model for shallow water (i.e., θ=1, γ=2).     

A remark on 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 the free boundary. The viscosity coefficient μ is proportional to ρθ with θ>0, where ρ is the density. The existence, uniqueness, regularity of global weak solutions in H¹([0,1]) have been established by Xin and Yao in [Z. Xin, Z. Yao, The existence, uniqueness and regularity for one-dimensional compressible Navier-Stokes equations, preprint]. Furthermore, under certain assumptions imposed on the initial data, we improve the regularity result obtained in [Z. Xin, Z. Yao, The existence, uniqueness and regularity for one-dimensional compressible Navier-Stokes equations, preprint] by driving some new a priori estimates.     

Global solutions to isentropic planar magnetohydrodynamic equations with density‐dependent viscosity

In this paper, we study the isentropic compressible planar magnetohydrodynamic equations with viscosity depending on density and with free boundaries. Precisely, when the viscosity coefficient λ(ρ) is proportional to ρ^θ with θ > 0, where ρ is the density, we establish the existence of global solutions under certain assumptions on the initial data by deriving some new a priori estimates.     

Global solutions of real compressible reactive gas with density-dependent viscosity and self-gravitation for higher-order kinetics

A mathematical model for viscous, real, compressible, reactive fluid flows is considered. The existence of global solutions for the free boundary problem with species diffusion in dynamic combustion is established when the viscosity λ depends on the density i.e., λ(ρ)=Aρα (), where A is a generic positive constant. Furthermore, the equations of state depend nonlinearly on density and temperature unlike the case of perfect gases or radiative flows. In addition, the shock wave, turbulence, vacuum, mass concentration or extremely hot spot will not be developed in any finite time if the initial data do not contain vacuum.     

Global strong solutions of the Cauchy problem for 1D compressible Navier-Stokes equations with density-dependent viscosity

We consider the Cauchy problem for one-dimensional compressible isentropic Navier-Stokes equations with density-dependent viscosity μ(ρ) = Aρ ^α , where α > 0 and A > 0. The global existence of strong solutions is obtained, which improves the previous results by enlarging the interval of α. Moreover, our result shows that no vacuum is developed in a finite time provided the initial data does not contain vacuum.     

Global existence and numerical approximation in adiabatic shearing of incompressible fluids

We first study the existence of global solutions to an adiabatic shearing model of incompressible fluids between two parallel plates when 1⩽α⩽3, where the viscosity is μ(θ)=θ^−α. Then we propose two finite difference schemes for μ(θ)⩾μ₀>0 (0<θ<∞) and for μ(θ)↓0 as θ→∞, respectively. Numerical experiments are presented which demonstrate the high accuracy of the schemes.     

Meet-irreducible submaximal clones determined by nontrivial equivalence relations

Let θ and ρ be a nontrivial equivalence relation and a binary relation on a finite set A, respectively. It is known from Rosenberg’s classification theorem (1965) that the clone Pol θ, which consists of all operations on A that preserve θ, is among the maximal clones on A. In the present paper, we find all binary relations ρ such that the clone Pol ρ is a meet-irreducible maximal subclone of Pol θ.     

Global classical solution to 1D compressible Navier–Stokes equations with no vacuum at infinity

C. Miao In this paper, we are concerned with the 1D Cauchy problem of the compressible Navier–Stokes equations with the viscosity μ(ρ) = 1+ρ^β(β≥0). The initial density can be arbitrarily large and keep a non‐vacuum state at far fields. We will establish the global existence of the classical solution for 0≤β < γ via a priori estimates when the initial density contains vacuum in interior interval or is away from the vacuum. We will show that the solution will not develop vacuum in any finite time if the initial density is away from the vacuum. To study the well‐posedness of the problem, it is crucial to obtain the upper bound of the density. Some new weighted estimates are applied to obtain our main results. Copyright © 2015 John Wiley & Sons, Ltd.     

Approximation by polynomials with bounded coefficients

Let θ be a real number satisfying 1<θ<2, and let A(θ) be the set of polynomials with coefficients in {0,1}, evaluated at θ. Using a result of Bugeaud, we prove by elementary methods that θ is a Pisot number when the set (A(θ)−A(θ)−A(θ)) is discrete; the problem whether Pisot numbers are the only numbers θ such that 0 is not a limit point of (A(θ)−A(θ)) is still unsolved. We also determine the three greatest limit points of the quantities , where C(θ) is the set of polynomials with coefficients in {−1,1}, evaluated at θ, and we find in particular infinitely many Perron numbers θ such that the sets C(θ) are discrete.     

Global existence, asymptotic behavior and blow-up of solutions for coupled Klein-Gordon equations with damping terms

This paper studies the Cauchy problem for the coupled system of nonlinear Klein-Gordon equations with damping terms. We first state the existence of standing wave with ground state, based on which we prove a sharp criteria for global existence and blow-up of solutions when E(0)<d. We then introduce a family of potential wells and discuss the invariant sets and vacuum isolating behavior of solutions for 0<E(0)<d and E(0)≤0, respectively. Furthermore, we prove the global existence and asymptotic behavior of solutions for the case of potential well family with 0<E(0)<d. Finally, a blow-up result for solutions with arbitrarily positive initial energy is obtained.     

Global well-posedness for the density-dependent incompressible magnetohydrodynamic flows in bounded domains

In this paper, we study the three-dimensional incompressible magnetohydrodynamic equations in a smooth bounded domains, in which the viscosity of the fluid and the magnetic diffusivity are concerned with density. The existence of global strong solutions is established in vacuum cases, provided the assumption that(|| ?μ(ρ0)|| Lp +|| ?ν(ρ0) ||Lq + ||b0|| L3+ ||ρ0|| L∞)(p, q 3) is small enough, there is not any smallness condition on the velocity.     

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 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.     

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.     

Quelques propriétés topologiques des espaces d'interpolation dans le cadre général

We study, in the general case, the reflexivity of the real Lions-Peetre interpolation spaces (A₀, A₁)_{θ, p} (0 < θ < 1, 1 < p < ∞). We prove that these spaces are reflexive if and only if the canonical injection i, from the intersection A₀ ∩ A₁ into the sum A₀ + A₁, is weakly compact.     

Generalized M-Matrices

Let ρ?Rⁿ be a proper cone. From the theory of M-matrices (see e.g. [1]) it is known that if there exist α > 0 and a matrix B: ρ→ρ such that A = B?αI, then the following conditions are equivalent: (i) ? A is ρ-monotone,(ii) A is ρ-seminegative, (iii) Re[Spectrum(A)]<0. In this paper we show that while the condition (e) e^tAρ?ρ ?t≥0 is more general than the structural assumption A = B?αI, conditions (i)-(iii) are nevertheless all equivalent to (iv) {x∈ρ: Ax∈ρ}={0} when (e) holds.