Global Existence of Smooth Solutions and Convergence to Barenblatt Solutions for the Physical Vacuum Free Boundary Problem of Compressible Euler Equations with Damping

Luo, Tao; Zeng, Huihui

doi:10.1002/cpa.21562

Cited by 77 publications

(68 citation statements)

References 19 publications

(66 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…See [22,10] for the viscous case. The nonlinear stabilities of some special solutions in variant settings with the physical vacuum condition can be found in [17,18,36,37,40,41,42,46,59,60]. However, all the above works are concerned on the isentropic case and seldom results in the non-isentropic case have been obtained.…”

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

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: 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

“…We transform the system (19) into Lagrangian variables. We let ξpx, tq denote the "position" of the gas particle x at time t. Thus, B t ξ " u˝ξ for t ą 0 and ξpx, 0q " x where˝denotes composition so that ru˝ξspx, tq :" upξpx, tq, tq .…”

Section: The Euler Equationsmentioning

confidence: 99%

“…Having defined e γ ptq and E γ ptq, the identical proof as for the case γ " 2 provides us with the following Theorem 4. For 1 ă γ ď 3 and ǫ ą 0 taken sufficiently small, if the initial data satisfies e γ p0q ă ǫ, then there exists a global unique solution of the Euler equations (19), such that E γ ptq ď ǫ for all t ě 0. Furthermore, ξpx, tq " αptqηpx, tq is a global solution to the 1-d Euler equations (19).…”

Section: Global Existence For All 1 ă γ ďmentioning

confidence: 99%

Global Existence of Near-Affine Solutions to the Compressible Euler Equations

Shkoller

Sideris

2019

Arch Rational Mech Anal

View full text Add to dashboard Cite

We establish global existence of solutions to the compressible Euler equations, in the case that a finite volume of ideal gas expands into vacuum. Vacuum states can occur with either smooth or singular sound speed, the latter corresponding to the so-called physical vacuum singularity when the enthalpy vanishes on the vacuum wave front like the distance function. In this instance, the Euler equations lose hyperbolicity and form a degenerate system of conservation laws, for which a local existence theory has only recently been developed. Sideris [25] found a class of expanding finite degree-of-freedom global-in-time affine solutions, obtained by solving nonlinear ODEs. In three space dimensions, the stability of these affine solutions, and hence global existence of solutions, was established by Hadžić & Jang [7] with the pressure-density relation p " ρ γ with the constraint that 1 ă γ ď 5 3 . They asked if a different approach could go beyond the γ ą 5 3 threshold. We provide an affirmative answer to their question, and prove stability of affine flows and global existence for all γ ą 1, thus also establishing global existence for the shallow water equations when γ " 2.

show abstract

Global Existence of Smooth Solutions and Convergence to Barenblatt Solutions for the Physical Vacuum Free Boundary Problem of Compressible Euler Equations with Damping

Cited by 77 publications

References 19 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

Global Existence of Near-Affine Solutions to the Compressible Euler Equations

Contact Info

Product

Resources

About