We prove the global existence of weak solutions for the 2-D compressible Navier-Stokes equations with a density-dependent viscosity coefficient ($λ=λ(ρ)$). Initial data and solutions are small in energy-norm with nonnegative densities having arbitrarily large sup-norm. Then, we show that if there is a vacuum domain at the initial time, then the vacuum domain will retain for all time, and vanishes as time goes to infinity. At last, we show that the condition of $μ=$constant will induce a singularity of the system at vacuum. Thus, the viscosity coefficient $μ$ plays a key role in the Navier-Stokes equations.
We consider the compressible Navier-Stokes system describing the motion of a barotropic fluid with density dependent viscosity confined in a three-dimensional bounded domain $Ω$. We show the convergence of the weak solution to the compressible Navier-Stokes system to the strong solution to the compressible Euler system when the viscosity and the damping coefficients tend to zero.
In this paper, we investigate the compressible Navier-Stokes equations with degenerate, density-dependent, viscosity coefficient driven by multiplicative stochastic noise. We consider three-dimensional periodic domain and prove that the family of weak martingale solutions is sequentially compact.
We consider a compressible Navier-Stokes system for a barotropic fluid with density dependent viscosity in a three-dimensional time-space domain $(0,T)\times Ω_\varepsilon$ where $Ω_\varepsilon = (0,\varepsilon)^2\times (0,1)$. We show that the weak solutions of the 3D system converges to the strong solution of the respective 1D system as $\varepsilon\rightarrow0.$
We identify a class of measure-valued solutions of the barotropic Euler system on a general (un-bounded) spatial domain as a vanishing viscosity limit for the compressible Navier-Stokes system. Then we establish the weak (measure-valued)-strong uniqueness principle, and, as a corollary, we obtain strong convergence to the Euler system on the lifespan of the strong solution.
We study a hydrodynamic limit of a system of coupled kinetic and fluid equations under a strong local alignment force and a strong Brownian motion. More precisely, we consider the Vlasov-Fokker-Planck type equation and compressible Navier-Stokes equations with a density-dependent viscosity. Based on a relative entropy argument, by assuming the existence of weak solutions to that kinetic-fluid system, we rigorously derive a two-phase fluid model consisting of isothermal Euler equations and compressible Navier-Stokes equations with a density-dependent viscosity.
This paper is concerned with the incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients and general initial data in the whole space $\mathbb{R}^d$ $ (d=2$ or 3). It is rigorously showed that, as the Mach number, the shear viscosity coefficient and the magnetic diffusion coefficient simultaneously go to zero, the weak solution of the compressible magnetohydrodynamic equations converges to the strong solution of the ideal incompressible magnetohydrodynamic equations as long as the latter exists.
In this paper, we study the vanishing viscosity limit of one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity, to the isentropic compressible Euler equations. Based on several new uniform estimates to the viscous systems, in addition to the framework recently established by G. Chen and M. Perepelitsa, we justify that the finite energy solution of the isentropic compressible Euler equations for a large class of initial data can be obtained as the inviscid limit of the compressible Navier-Stokes equations even when the viscosity depends on the density.
Global-in-time weak solutions to the Compressible Navier-Stokes-Poisson equations in a three-dimensional torus for large data are considered in this paper. The system takes into account density-dependent viscosity and non-monotone presseur. We prove the existence of global weak solutions to NSP equations with damping term by using the Faedo-Galerkin method and the compactness arguments on the condition that the adiabatic constant satisfies $γ>\frac{4}{3}$.
The vanishing viscosity limit of the two-dimensional (2D) compressible isentropic Navier-Stokes equations is studied in the case that the corresponding 2D inviscid Euler equations admit a planar rarefaction wave solution. It is proved that there exists a family of smooth solutions for the 2D compressible Navier-Stokes equations converging to the planar rarefaction wave solution with arbitrary strength for the 2D Euler equations. A uniform convergence rate is obtained in terms of the viscosity coefficients away from the initial time. In the proof, the hyperbolic wave is crucially introduced to …
In this note, by constructing suitable approximate solutions, we prove the existence of global weak solutions to the compressible Navier-Stokes equations with density-dependent viscosity coefficients in the whole space $\mathbb{R}^N$, $N\geq2$ (or exterior domain), when the initial data are spherically symmetric. In particular, we prove the existence of spherically symmetric solutions to the Saint-Venant model for shallow water in the whole space (or exterior domain).
We are concerned about the barotropic compressible Navier-Stokes equations with density-dependent viscosities which may degenerate in vacuum. We show that any classical solution to barotropic compressible Navier-Stokes equations in bounded domains will blow up, when the initial density admits an isolated mass group and the viscousity coefficients satisfy some conditions. A new condition on viscosities is first put forward in this paper.
We consider the Cauchy problem for one-dimensional (1D) barotropic compressible Navier-Stokes equations with density-dependent viscosity and large external force. Under a general assumption on the density-dependent 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.
We consider the free boundary problem of compressible isentropic neo-Hookean viscoelastic fluid equations with surface tension. Under the physical kinetic and dynamic conditions proposed on the free boundary, we investigate regularities of classical solutions to viscoelastic fluid equations in Sobolev spaces which are uniform in viscosity and justify the corresponding vanishing viscosity limits. The key ingredient of our proof is that the deformation gradient tensor in Lagrangian coordinates can be represented as a parameter in terms of flow map so that the inherent structure of the elastic te…
This paper is concerned with a non-isentropic compressible Navier-Stokes/Allen-Cahn system with phase variable dependent viscosity $η(χ)=χ^α$ and temperature dependent heat-conductivity $κ(θ)=θ^β$. We show the global existence of strong solutions under some assumptions on growth exponent $α$ and initial data. It is worth noting that the initial data could be large if $α\ge0$ is small, and the growth exponent $β>0$ can be arbitrary large.
Global Existence and Incompressible Limit for the Three-Dimensional Axisymmetric Compressible Navier-Stokes Equations with Large Bulk Viscosity and Large Initial Data
In this paper we prove the existence of global strong solution for the Navier-Stokes equations with general degenerate viscosity coefficients. The cornerstone of the proof is the introduction of a new effective pressure which allows to obtain an Oleinik-type estimate for the so called effective velocity. In our proof we make use of additional regularizing effects on the velocity which requires to extend the technics developed by Hoff for the constant viscosity case.
We consider Navier-Stokes equations for compressible viscous fluids in one dimension. We prove the existence of global strong solution with large initial data for the shallow water system. The key ingredient of the proof relies to a new formulation of the compressible equations involving a new effective velocity $v$ (see \cite{cras,para,CPAM,CPAM1}) such that the density verifies a parabolic equation. We estimate $v$ in $L^\infty$ norm which enables us to control the vacuum on the density.
The mixed boundary value problem for a compressible Stokes system of partial differential equations in a bounded domain is reduced to two different systems of segregated direct Boundary Integral Equations (BDIEs) expressed in terms of surface and volume parametrix-based potential type operators. Equivalence of the BDIE systems to the mixed BVP and invertibility of the matrix operators associated with the BDIE systems are proved in appropriate Sobolev spaces.
We prove existence and asymptotic stability of the stationary solution for the compressible Navier-Stokes equations for isentropic gas dynamics with a density dependent diffusion in a bounded interval. We present the necessary conditions to be imposed on the boundary data which ensure existence and uniqueness of the steady state, and we subsequent investigate its stability properties by means of the construction of a suitable Lyapunov functional for the system. The Saint-Venant system, modeling the dynamics of a shallow compressible fluid, fits into this general framework.
We are concerned with spherically symmetric solutions of the Euler equations for multidimensional compressible fluids, which are motivated by many important physical situations. Various evidences indicate that spherically symmetric solutions of the compressible Euler equations may blow up near the origin at certain time under some circumstance. The central feature is the strengthening of waves as they move radially inward. A longstanding open, fundamental question is whether concentration could form at the origin. In this paper, we develop a method of vanishing viscosity and related estimate t…
We study the zero viscosity and heat conductivity limit of an initial boundary problem for the linearized Navier-Stokes-Fourier equations of a compressible viscous and heat conducting fluid in the half plane. We consider the case that the viscosity and thermal diffusivity converge to zero at the same order. The approximate solution of the linearized Navier-Stokes-Fourier equations with inner and boundary expansion terms is analyzed formally first by multiscale analysis. Then the pointwise estimates of the error terms of the approximate solution are obtained by energy methods, thus establish th…
We consider the compressible Navier-Stokes system where the viscosity depends on density and the heat conductivity is proportional to a positive power of the temperature under stress-free and thermally insulated boundary conditions. Under the same conditions on the initial data as those of the constant viscosity and heat conductivity case ([Kazhikhov-Shelukhin. J. Appl. Math. Mech. 41 (1977)], we obtain the existence and uniqueness of global strong solutions. Our result can be regarded as a natural generalization of the Kazhikhov's theory for the constant heat conductivity case to the degenera…
We are concerned with the large time behavior of solutions to the Cauchy problem of the one-dimensional compressible micropolar fluid model without viscosity, where the far-field states of the initial data are prescribed to be different. If the corresponding Riemann problem of the compressible Euler system admits a contact discontinuity and two rarefaction waves solutions, we show that for such a non-viscous model, the combination of the viscous contact wave with two rarefaction waves is time-asymptotically stable provided that the strength of the composite wave and the initial perturbation ar…
In this paper, we consider the small viscosity limit problem for the isentropic compressible Navier-Stokes equations in a 2D exterior domain with impermeable boundary conditions , and the corresponding Euler equations have vortex sheet solutions.We obtain that away from the boundary and the contact discontinuous the isentropic compressible viscous flow can be approximated by the corresponding inviscid flow, near the boundary (the contact discontinuous) there is a boundary layer (vortex layer)for the angular velocity in the leading order expansion of solution, while the radial velocity and the …
Motivated by a recent paper by Barrett and Süli [J.W. Barrett & E. Süli: Existence of global weak solutions to compressible isentropic finitely extensible bead-spring chain models for dilute polymers, Math. Models Methods Appl. Sci., 26 (2016)], we consider the compressible Navier--Stokes system coupled with a Fokker--Planck type equation describing the motion of polymer molecules in a viscous compressible fluid occupying a bounded spatial domain, with polymer-number-density-dependent viscosity coefficients. The model arises in the kinetic theory of dilute solutions of nonhomogeneous polymeric…
We prove weak-strong uniqueness results for the compressible Navier-Stokes system with degenerate viscosity coefficient and with vacuum in one dimension. In other words, we give conditions on the weak solution constructed in \cite{Jiu} so that it is unique. The novelty consists in dealing with initial density $ρ_0$ which contains vacuum. To do this we use the notion of relative entropy developed recently by Germain, Feireisl et al and Mellet and Vasseur (see \cite{PG,Fei,15}) combined with a new formulation of the compressible system (\cite{cras,CPAM,CPAM1,para}) (more precisely we introduce a…
Global uniform regularity and vanishing vertical viscosity limit for the compressible Navier--Stokes equations in the half-space
School of Mathematics, Sun Yat-sen University（中山大学数学学院）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）；Department of Mathematics, University of Notre Dame（圣母大学数学系）；Sun Yat-sen University Institute of Advanced Studies Hong Kong（中山大学高等研究院（香港））
In this paper, we study the uniform regularity and vanishing viscosity limit for the compressible nematic liquid crystal flows in three dimensional bounded domain. It is shown that there exists a unique strong solution for the compressible nematic liquid crystal flows with boundary condition in a finite time interval which is independent of the viscosity coefficient. The solutions are uniform bounded in a conormal Sobolev space. Furthermore, we prove that the density and velocity are uniform bounded in $W^{1, \infty}$, and the director field is uniform bounded in $W^{3,\infty}$ respectively. B…
We prove the convergence of the vanishing viscosity limit of the one-dimensional, isentropic, compressible Navier-Stokes equations to the isentropic Euler equations in the case of a general pressure law. Our strategy relies on the construction of fundamental solutions to the entropy equation that remain controlled for unbounded densities, and employs an improved reduction framework to show that measure-valued solutions constrained by the Tartar commutation relation (but with possibly unbounded support) reduce to a Dirac mass. As the Navier-Stokes equations do not admit an invariant region, we …
We derive two systems of boundary-domain integral equations (BDIEs) equivalent to the Dirichlet problem for the compressible Stokes system using the potential method with an explicit parametrix (Levi function). The BDIEs are given in terms of surface and volume hydrodynamic potentials. We study the mapping properties of these integral potential operators and used them to prove existence and uniqueness of solution of the two systems of BDIEs obtained taking into account the non-trivial kernels of the single layer and hypersingular hydrodynamic surface potentials.
This paper is concerned with an initial and boundary value problem of the one-dimensional planar MHD equations for viscous, heat-conducting, compressible, ideal polytropic fluids with constant transport coefficients and large data. The vanishing shear viscosity limit is justified and the convergence rates are obtained. More important, to capture the behavior of the solutions at vanishing shear viscosity, both the boundary-layer thickness and the boundary-layer solution are discussed. As by-products, the global well-posedness of strong solutions with large data is established. The proofs are ba…
This paper is devoted to the global existence of weak solutions to the three-dimensional compressible Navier-Stokes equations with heat-conducting effects in a bounded domain. The viscosity and the heat conductivity coefficients are assumed to be functions of the temperature, and the shear viscosity coefficient may vanish as the temperature goes to zero. The proof is to apply Galerkin method to a suitable approximate system with several parameters and obtain uniform estimates for the approximate solutions. The key ingredient in obtaining the required estimates is to apply De Giorgi's iteration…
Global Existence and Incompressible Limit of the Cauchy Problem for 2D Compressible Navier-Stokes Equations with Large Bulk Viscosity and Large Initial Data
University of Chinese Academy of Sciences（中国科学院大学）；Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院应用数学研究所）
We are concerned with the uniform regularity estimates and vanishing viscosity limit of solution to two dimensional viscous compressible magnetohydrodynamics (MHD) equations with transverse background magnetic field. When the magnetic field is assumed to be transverse to the boundary and the tangential component of magnetic field satisfies zero Neumann boundary condition, even though the velocity is imposed the no-slip boundary condition, the uniform regularity estimates of solution and its derivatives still can be achieved in suitable conormal Sobolev spaces in the half plane $\mathbb{R}^2_+$…
The long-time regularity and asymptotic of weak solutions are studied for compressible Navier-Stokes equations with degenerate viscosity in a bounded periodic domain in two and three dimensions. It is shown that the density keeps strictly positive from below and above after a finite period of time. Moreover, higher velocity regularity is obtained via a parabolic type iteration technique. Since then the weak solution conserves its energy equality, and decays exponentially to the equilibrium in $L^{2}$-norm as time goes to infinity. In addition, assume that the initial momentum is zero, the expo…
In this paper, we study the asymptotic stability of rarefaction waves for the compressible isentropic Navier-Stokes equations with density-dependent viscosity. First, a weak solution around a rarefaction wave to the Cauchy problem is constructed by approximating the system and regularizing the initial values which may contain vacuum state. Then some global in time estimates on the weak solution are obtained. Based on these uniform estimates, the vacuum states are shown to vanish in finite time and the weak solution we constructed becomes a unique strong one. Consequently, the stability of the …
In this paper, we are concerned with the Cauchy problem for one-dimensional compressible isentropic Navier-Stokes equations with density-dependent viscosity $μ(ρ)=ρ^α(α>0)$ and pressure $P(ρ)=ρ^γ\ (γ>1)$. We will establish the global existence and asymptotic behavior of weak solutions for any $α>0$ and $γ>1$ under the assumption that the density function keeps a constant state at far fields. This enlarges the ranges of $α$ and $γ$ and improves the previous results presented by Jiu and Xin. As a result, in the case that $0<α<\frac12$, we obtain the large time behavior of the strong solution obt…
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 $γ$. The initial data can be large if $γ$ is sufficiently close to 1. These res…
The aim of this article is to justify mathematically, in the two-dimensional periodic setting, a generalization of a two-phase model with pressure dependent viscosity first proposed by A. Lefebvre-Lepot and B. Maury to describe a system in one dimension of aligned spheres interacting through lubrication forces. This model involves an adhesion potential, apparent only on the congested domain, which keeps track of history of the flow. The solutions are constructed (through a singular limit) from a compressible Navier-Stokes system with viscosity and pressure both singular close to a maximal volu…
This paper concerns the global existence for classical solutions problem to the 3D Density-Dependent Viscosity barotropic compressible Navier-Stokes in $Ω$ with slip boundary condition, where $Ω$ is a simply connected bounded $C^{\infty}$ domain in $\mathbb{R}^3$ and its boundary only has a finite number of 2-dimensional connected components. By a series of a priori estimates, we show that the classical solution to the system exists globally in time under the assumption that the initial energy is suitably small. The initial density of such a classical solution is allowed to have large oscillat…
We consider the one-dimensional compressible Navier--Stokes system for a viscous and heat-conducting ideal polytropic gas when the viscosity $μ$ and the heat conductivity $κ$ depend on the specific volume $v$ and the temperature $θ$ and are both proportional to $h(v)θ^α$ for certain non-degenerate smooth function $h$. We prove the existence and uniqueness of a global-in-time non-vacuum solution to its Cauchy problem under certain assumptions on the parameter $α$ and initial data, which imply that the initial data can be large if $|α|$ is sufficiently small. Our result appears to be the first g…
In this paper we consider the global well-posedness of compressible magnetohydrodynamic system in $\R^d$ with $d\geq2$, in the framework of the critical Besov spaces. We can show that if the initial data, the shear viscosity and the magnetic diffusion coefficient are small comparing with the volume viscosity, then compressible magnetohydrodynamic system has a unique global solution.
This paper is concerned with an initial and boundary value problem of the one-dimensional planar MHD equations for viscous, heat-conducting, compressible, ideal polytropic fluids with constant transport coefficients and large data. The vanishing shear viscosity limit is justified and the convergence rates are obtained. To capture the behavior of the solutions at small shear viscosity, we also discuss the boundary-layer thickness and the boundary-layer solution. As by-products, the global well-posedness of strong solutions with large data is established. The proofs are based on the global (unif…
This paper is devoted to investigating the global existence of weak solutions for the compressible primitive equations (CPE) with damping term in a three-dimensional torus for large initial data. The system takes into account density-dependent viscosity. In our proof, we represent the vertical velocity as a function of the density and the horizontal velocity which will play a role to use the Faedo-Galerkin method to obtain the global existence of the approximate solutions. Motivated by Vasseur and Yu [yucheng2016], we obtain the key estimates of lower bound of the density, the Bresch-Desjardin…
For the initial boundary value problem of compressible barotropic Navier-Stokes equations in one-dimensional bounded domains with general density-dependent viscosity and large external force, we prove that there exists a unique global classical solution with large initial data containing vacuum. Furthermore, we show that the density is bounded from above independently of time which in particular yields the large time behavior of the solution as time tends to infinity: the density and the velocity converge to the steady states in $L^p$ and in $W^{1,p}$ ($1\le p<+\infty$) respectively. Moreover,…
We study the compressible quantum Navier-Stokes (QNS) equations with degenerate viscosity in the three dimensional periodic domains. On the one hand, we consider QNS with additional damping terms. Motivated by the recent works [Li-Xin, arXiv:1504.06826] and [Antonelli-Spirito, Arch. Ration. Mech. Anal., 203(2012), 499--527], we construct a suitable approximate system which has smooth solutions satisfying the energy inequality and the BD entropy estimate. Using this system, we obtain the global existence of weak solutions to the compressible QNS equations with damping terms for large initial da…
We obtain existence and conormal Sobolev regularity of strong solutions to the 3D compressible isentropic Navier-Stokes system on the half-space with a Navier boundary condition, over a time that is uniform with respect to the viscosity parameters when these are small. These solutions then converge globally and strongly in $L^2$ towards the solution of the compressible isentropic Euler system when the viscosity parameters go to zero.
In this paper, we investigate the uniform regularity for the isentropic compressible Navier-Stokes system with general Navier-slip boundary conditions (1.6) and the inviscid limit to the compressible Euler system. It is shown that there exists a unique strong solution of the compressible Navier-Stokes equations with general Navier-slip boundary conditions in an interval of time which is uniform in the vanishing viscosity limit. The solution is uniformly bounded in a conormal Sobolev space and is uniform bounded in $W^{1,\infty}$. It is also shown that the boundary layer for the density is weak…
