Nonlinear Asymptotic Stability of the Lane-Emden Solutions for the Viscous Gaseous Star Problem with Degenerate Density Dependent Viscosities

Luo, Tao; Xin, Zhouping; Zeng, Huihui

doi:10.1007/s00220-016-2753-1

Cited by 93 publications

(55 citation statements)

References 44 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, if considering the non-isentropic Euler equations (µ = λ = κ = 0) of ρ, u and S, then the sound speed is given by ∂p(ρ,S) ∂ρ = γ p ρ = √ γRθ. Therefore, the singular condition on the derivatives of the temperature profiles (2.9) coincides with the physical vacuum condition for the isentropic fluids ( [24,8,39]) in certain sense.…”

Section: The Reference Domainmentioning

confidence: 72%

“…As it is pointed out by Liu in [33], such a singularity of the density makes the standard hyperbolic method fail in establishing the local wellposedness theory for the inviscid flow. Only recently, the local well-posedness theory for inviscid flows is developed by Coutand, Lindblad and Shkoller [7,8,9], Jang and Masmoudi [24,25], Gu and Lei [13,14], Luo, Xin and Zeng [39] in variant settings. See [22,10] for the viscous case.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Local Existence and Uniqueness of Strong Solutions to the Free Boundary Problem of the Full Compressible Navier--Stokes Equations in three dimensions

Liu¹,

Yuan²

2019

SIAM J. Math. Anal.

View full text Add to dashboard Cite

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.|∂ 2 t a| J −1 |Dv t ||a| 2 + J −2 |Dv| 2 |a| 3 , |∂a| J

show abstract

Section: The Reference Domainmentioning

confidence: 72%

Section: Introductionmentioning

confidence: 99%

Local Existence and Uniqueness of Strong Solutions to the Free Boundary Problem of the Full Compressible Navier--Stokes Equations in three dimensions

Liu¹,

Yuan²

2019

SIAM J. Math. Anal.

View full text Add to dashboard Cite

show abstract

“…We emphasize that the physical vacuum profiles mentioned here commonly exist in a lot of physical models, such as, the gaseous star problem, the Euler damping equations, etc. We refer these results to [29,30,31,50].However, the studies of free boundary problems mentioned above mainly concern the isentropic flows. We start the study of free boundary problem for non-isentropic flows by studying the equilibria of the radiation gaseous stars in [22], in which we establish the corresponding degeneracy of density and temperature near the vacuum boundary.…”

mentioning

confidence: 99%

The self-similar solutions to full compressible Navier–Stokes equations without heat conductivity

Liu

Yuan

2019

Math. Models Methods Appl. Sci.

View full text Add to dashboard Cite

In this work, we establish a class of globally defined, large solutions to the free boundary problem of compressible Navier-Stokes equations with constant shear viscosity and vanishing bulk viscosity. We establish such solutions with initial data perturbed around any self-similar solution when γ > 7/6. In the case when 7/6 < γ < 7/3, as long as the self-similar solution has bounded entropy, a solution with bounded entropy can be constructed. It should be pointed out that the solutions we obtain in this fashion do not in general keep being a small perturbation of the selfsimilar solution due to the second law of thermodynamics, i.e., the growth of entropy. If in addition, in the case when 11/9 < γ < 5/3, we can construct a solution as a global-in-time small perturbation of the self-similar solution and the entropy is uniformly bounded in time. Contents 1 Introduction 1 2 Energy estimates 13 3 Interior estimates 20 4 Point-wise estimates 27 5 Regularity 33representing the non-negativity of the shear and bulk viscosities, respectively. Also, γ > 1 is the thermodynamic coefficient of the fluid. In particular, for the ideal gas, the pressure potential p and the specific inner energy e can be expressed as, in terms of the temperature θ and the density ρ, p = Kρθ, e = c ν θ, for some positive constants K and c ν , referred to as the thermodynamic and specific inner energy coefficients, respectively. Then γ is given by, in this case,Huang, Li, Xin [12,11] established the global well-posedness for the isentropic and heat conductive flows. However, These solutions fail to track the entropy in the vacuum area. In fact, as pointed out by Xin and Yan [39,40], the classical solutions to non-heat-conductive CNS with bounded entropy will blow up in finite time due to the appearance of vacuum states. Also, as pointed out in [41] by Liu, Xin, Yang, the vacuum states for CNS may not produce physically desirable solutions. More recently, Li, Wang, Xin [18] showed that with vacuum states, the classical solutions to CNS do not exist with finite entropy. Motivated by the studies mentioned above, in order to establish a solution with bounded entropy and vacuum states, we are working on the free boundary problem of CNS. Before moving on to our works, it is worth mentioning some previous works, if not all of them, in the following. When the density connects to the vacuum area on the moving boundary with a jump, the local well-posedness theory and the global stability of equilibria can be tracked back to Solonnikov, Tani, Zadrzyńska, and Zajaczkowski,[36,45,44,46,47,48,43,42]. On the other hand, when the density profile connects continuously to vacuum across the moving boundary, the degeneracy of the density causes singularities when establishing derivative estimates. This has been pointed out by Liu [20] for inviscid flows with physical vacuum. With weighted energy estimates, Jang, Masmoudi, Coutand, Lindblad, Shkoller established the local well-posedness in [17,16,6,5,4] for such flows. See also [7,8,28,10,9]. In the case of viscous flo...

show abstract

“…Due to the balance between flow pressure and gravity force, it was shown that there is a critical value γ c = 4 3 in spatial three-dimension, the stationary Lane-Emden solution is expected to be stable for γ > 4 3 and instable for γ < 4 3 . Indeed, Jang and Tice [9] prove the instability theory of the NSP equations for 6 5 < γ < 4 3 , and the nonlinear asymptotic stability of the Lane-Emden solutions for the viscous gaseous star is established by Luo-Xin-Zeng [14,15] for 4 3 < γ < 2. For γ = 6 5 , the nonlinear instability is proved for gravitational Euler-Poisson system in [7].…”

Section: Introduction and Main Resultsmentioning

confidence: 94%

“…On the other hand, for 1 < γ < 6 5 , there are no stationary solutions with finite total mass. Recently, many important study on the asymptotic stability/instability of stationary solutions has been made, for instance, in [2,12,9,7,14,15,16]. Due to the balance between flow pressure and gravity force, it was shown that there is a critical value γ c = 4 3 in spatial three-dimension, the stationary Lane-Emden solution is expected to be stable for γ > 4 3 and instable for γ < 4 3 .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Free boundary value problem to 3D spherically symmetric compressible Navier–Stokes–Poisson equations

Kong

2017

Z. Angew. Math. Phys.

View full text Add to dashboard Cite

In the paper, we consider the free boundary value problem to 3D spherically symmetric compressible isentropic Navier-Stokes-Poisson equations for self-gravitating gaseous stars with γlaw pressure density function for 6 5 < γ ≤ 4 3 . For stress free boundary condition and zero flow density continuously across the free boundary, the global existence of spherically symmetric weak solutions is shown, and the regularity and long time behavior of global solution are investigated for spherically symmetric initial data with the total mass smaller than a critical mass.Mathematics Subject Classification (2010). 35Q35;76N15.

show abstract

Nonlinear Asymptotic Stability of the Lane-Emden Solutions for the Viscous Gaseous Star Problem with Degenerate Density Dependent Viscosities

Cited by 93 publications

References 44 publications

Local Existence and Uniqueness of Strong Solutions to the Free Boundary Problem of the Full Compressible Navier--Stokes Equations in three dimensions

Local Existence and Uniqueness of Strong Solutions to the Free Boundary Problem of the Full Compressible Navier--Stokes Equations in three dimensions

The self-similar solutions to full compressible Navier–Stokes equations without heat conductivity

Free boundary value problem to 3D spherically symmetric compressible Navier–Stokes–Poisson equations

Contact Info

Product

Resources

About