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

Liu, Xin; Yuan, Yuan

doi:10.1137/18m1180426

Cited by 16 publications

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“…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. Also, we establish the local well-posedness with such degenerate profiles in [22,26].This work is part of our project on studying the large time dynamics of flows with bounded entropy in the setting of free boundary problems. In particular, as a starting point, we are investigating the flows without heat conductivity in spherically symmetric motions.…”

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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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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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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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“…Later in [33], we first proved the existence and uniqueness of the local-in-time strong solutions to the free boundary problem of the full CNS (constants µ, 2µ `3λ, κ ą 0). In [33], we impose more general decay rates of the initial density and temperature near the vacuum boundary, i.e., the first line of (1.10) and ´8 ă ∇ n pθ 0 q ă 0 on Γ 0 .…”

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Immediate blowup of classical solutions to the vacuum free boundary problem of non-isentropic compressible Navier--Stokes equations

Liu¹,

Yuan²

2022

Preprint

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This paper considers the immediate blowup of classical solutions to the vacuum free boundary problem of non-isentropic compressible Navier-Stokes equations, where the viscosities and the heat conductivity could be constants, or more physically, the degenerate, temperature-dependent functions which vanish on the vacuum boundary (i.e., µ " μθ α , λ " λθ α , κ " κθ α , for constants 0 ď α ď 1{pγ ´1q, μ ą 0, 2μ `nλ ě 0, κ ě 0, and adiabatic exponent γ ą 1). For such vacuum free boundary problems, imposing proper decaying or singular conditions on the density or temperature profiles across the vacuum boundary is important to the existence of the classical solutions.In our previous study (Liu and Yuan, Math. Models Methods Appl. Sci. ( 9) 12, 2019 ), with three-dimensional spherical symmetry and constant shear viscosity, vanishing bulk viscosity and heat conductivity, we established a class of globalin-time large solutions, with bounded entropy and entropy derivatives, under the condition the decaying rate of the initial density to the vacuum boundary is of any positive power of the distance function to the boundary. In this paper we prove that such classical solutions do not exist for any small time for non-vanishing bulk viscosity, provided the initial velocity is expanding near the boundary.When the heat conductivity does not vanish, it is automatically satisfied that the normal derivative of the temperature of the classical solution across the free boundary does not degenerate, which supports that such singular condition imposed on the initial data in our previous local well-posedness study (Liu and Yuan, SIAM J. Math. Anal. (2) 51, 2019 ) is physical; meanwhile, the entropy of the classical solution immediately blowups if the decaying rate of the initial density is not of 1{pγ ´1q power of the distance function to the boundary.We also have a similar non-existence result for the one-dimensional case, but with more restrictions on the parameter decaying rate of the initial density. These results provide a first-step study of the well-posedness theory of the vacuum free boundary problem of the viscous gas with degenerate, temperature-dependent transport coefficients. The proofs are obtained by the analyzing the boundary behaviors of the velocity, entropy and temperature, and investigating the maximum principles for parabolic equations with degenerate coefficients.

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“…We note that a global existence of weak solutions was established in [12] for the spherically symmetric motions, where the density is positive on the free boundary. For more related results of the compressible Navier-Stokes equations with vacuum boundary, one can refer to [11,15,33,34,42,45] and references therein. Next, we state the original free boundary problem in three space dimensions in cylindrically coordinates as follows.…”

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Global wellposedness of vacuum free boundary problem of isentropic compressible magnetohydrodynamic equations with axisymmetry

Li¹,

Ou²

2022

DCDS-B

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In this paper, we prove the global existence of the strong solutions to the vacuum free boundary problem of isentropic compressible magnetohydrodynamic equations with small initial data and axial symmetry, where the solutions are independent of the axial variable and the angular variable. The solutions capture the precise physical behavior that the sound speed is C 1/2 -Hölder continuous across the vacuum boundary provided that the adiabatic exponent γ ∈ (1, 2). The main difficulties of this problem lie in the singularity at the symmetry axis, the degeneracy of the system near the free boundary and the strong coupling of the magnetic field and the velocity. We overcome the obstacles by constructing some new weighted nonlinear functionals (involving both lower-order and higher-order derivatives) and establishing the uniformin-time weighted energy estimates of solutions by delicate analysis, in which the balance of pressure and self-gravitation, and the dissipation of velocity are crucial.

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

Cited by 16 publications

References 60 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

Immediate blowup of classical solutions to the vacuum free boundary problem of non-isentropic compressible Navier--Stokes equations

Global wellposedness of vacuum free boundary problem of isentropic compressible magnetohydrodynamic equations with axisymmetry

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