L 1 Convergence to the Barenblatt Solution for Compressible Euler Equations with Damping

Huang, Feimin; Pan, Ronghua; Wang, Zhen

doi:10.1007/s00205-010-0355-1

Cited by 86 publications

(71 citation statements)

References 35 publications

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“…[8,9,10,17]): so that the Barenblatt self-similar solution defined in I b (t) has the same total mass as that for the solution of (1.1):…”

Section: γ(T) = U(γ(t) T)mentioning

confidence: 99%

“…0 .x/ > 0 for x .0/ < x < x C .0/; 0 .x˙.0// D 0; and 0 <ˇ 1 0 x .x˙.0//ˇ< 1: Let M 2 .0; 1/ be the initial total mass; then the conservation law of mass, (1.1) 1 ,givesZ x C .t / x .t / .x; t /dx D Z x C .0/ x .0/ 0 .x/dx DW M for t > 0:The compressible Euler equations of isentropic flow with damping is closely related to the porous media equation (cf [7][8][9]16]…”

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confidence: 99%

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

Luo

Zeng

2015

Comm Pure Appl Math

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For the physical vacuum free boundary problem with the sound speed being C 1/2 -Hölder continuous near vacuum boundaries of the one-dimensional compressible Euler equations with damping, the global existence of the smooth solution is proved, which is shown to converge to the Barenblatt self-similar solution for the the porous media equation with the same total mass when the initial data is a small perturbation of the Barenblatt solution. The pointwise convergence with a rate of density, the convergence rate of velocity in supreme norm and the precise expanding rate of the physical vacuum boundaries are also given. The proof is based on a construction of higher-order weighted functionals with both space and time weights capturing the behavior of solutions both near vacuum states and in large time, an introduction of a new ansatz, higher-order nonlinear energy estimates and elliptic estimates.

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“…[8,9,10,17]): so that the Barenblatt self-similar solution defined in I b (t) has the same total mass as that for the solution of (1.1):…”

Section: γ(T) = U(γ(t) T)mentioning

confidence: 99%

mentioning

confidence: 99%

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

Luo

Zeng

2015

Comm Pure Appl Math

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“…For the initial-boundary value problems, we can refer to [28,29] for existence of L ∞ entropy weak solutions and to [11,12,21,26] for existence of small smooth solutions. For the asymptotics of solutions, we refer to [14][15][16][17] for L ∞ entropy weak solutions and to [7,24,25,38] for small smooth solutions. Besides, there are some results on the non-isentropic compressible Euler equations with damping, see [9,13,22,27].…”

Section: Introductionmentioning

confidence: 99%

Global solution and large-time behavior of the 3D compressible Euler equations with damping

Tan

Wang

2013

Journal of Differential Equations

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We construct the global unique solution to the compressible Euler equations with damping in R 3 . We assume the H 3 norm of the initial data is small, but the higher order derivatives can be arbitrarily large. When theḢ −s norm (0 s < 3/2) orḂ −s 2,∞ norm (0 < s 3/2) of the initial data is finite, by a regularity interpolation trick, we prove the optimal decay rates of the solution. As an immediate byproduct, the L p -L 2 (1 p 2) type of the decay rates follow without requiring that the L p norm of initial data is small.

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“…After that, such problem was widely studied by many authors, see e.g., among many other works [6,7,11,15,20,22] and references therein for further results. A closely related-though not exactly equivalent-problem is to study the behavior of the solutions to the strong relaxation model as the relaxation time ε in (1) tends to 0.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The strong relaxation limit of the multidimensional Euler equations

Lin

Coulombel

2012

Nonlinear Differ. Equ. Appl.

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Abstract. This paper is devoted to the analysis of global smooth solutions to the multidimensional isentropic Euler equations with stiff relaxation. We show that the asymptotic behavior of the global smooth solution is governed by the porous media equation as the relaxation time tends to zero. The results are proved by combining some classical energy estimates with the so-called Shizuta-Kawashima condition. Mathematics Subject Classification (2010). 35L45, 76N10, 76S05.

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L 1 Convergence to the Barenblatt Solution for Compressible Euler Equations with Damping

Cited by 86 publications

References 35 publications

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

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

Global solution and large-time behavior of the 3D compressible Euler equations with damping

The strong relaxation limit of the multidimensional Euler equations

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