On the expanding configurations of viscous radiation gaseous stars: the isentropic model

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...

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“…Then after integrating (1.17) 1 in (x, 1) for x ∈ (0, 1), it holds 24) supposed that η L ∞ , xη x L ∞ are small enough. Notice, on the one hand,…”

Section: Comments and Methodologymentioning

confidence: 99%

“…In addition, inspired by our previous works [24] (also by [10,9]), we impose the boundedness of η, xη x and the decay of η τ , xη xτ as a priori assumptions. However, instead of imposing boundedness of pq directly, we only impose the decay of the quantity B.…”

Section: Comments and Methodologymentioning

confidence: 99%

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

Liu

Yuan

2019

Math. Models Methods Appl. Sci.

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Variational rotating solutions to non-isentropic Euler-Poisson equations with prescribed total mass

Yuan

2021

Sci. China Math.

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On the expanding configurations of viscous radiation gaseous stars: the isentropic model

Cited by 2 publications

References 38 publications

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

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

Variational rotating solutions to non-isentropic Euler-Poisson equations with prescribed total mass

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