Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping

Hsiao, Ling; Liu, Tai-Ping

doi:10.1007/bf02099268

Cited by 357 publications

(310 citation statements)

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“…We will see that an easy lemma plays an important role. It should be pointed out that the key approach in [13][14][15][16][17][18][19][20][21] is to compare the solution of (1.2)-(1.3) with the similarity solution of (1.4) via energy estimates. Unfortunately, the exponential decay rate cannot be achieved by this approach, due to the boundary effects.…”

Section: Preliminaries and Main Resultsmentioning

confidence: 99%

“…In this direction, the readers are referred to [31], [13], [14] and [12] for existence of small smooth solutions; to [27], [4], and [6] for solutions in BV ; to [7] and [20] for L ∞ solutions. For large time behavior of solutions, we refer [13], [14], [33], [34] and [45] fore small smooth solutions; and we refer [19], [21], [39] and [47] for weak solutions. For initial boundary value problems, see [16], [28] and [35] for small smooth solutions.…”

Section: Introductionmentioning

confidence: 99%

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Initial boundary value problem for compressible Euler equations with damping

Pan¹,

Zhao²

2008

Indiana Univ. Math. J.

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Abstract. We construct global L ∞ entropy weak solutions to the initial boundary value problem for the damped compressible Euler equations on bounded domain with physical boundaries. Time asymptotically, the density is conjectured to satisfy the porous medium equation and the momentum obeys to the classical Darcy's law. Based on entropy principle, we showed that the physical weak solutions converges to steady states exponentially fast in time. We also proved that the same is true for the related initial boundary value problems of porous medium equation and thus justified the validity of Darcy's law in large time. *

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Section: Preliminaries and Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Initial boundary value problem for compressible Euler equations with damping

Pan¹,

Zhao²

2008

Indiana Univ. Math. J.

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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%

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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“…When there are no phase transitions in the flow, then the third equation is missing and p only depends on v. The corresponding 2 × 2 system has been studied by many authors: see [36,13,28,29,37,14,30], for the homogeneous case or with continuous source terms and [32,33] for the case of discontinuous source terms.…”

Section: Introductionmentioning

confidence: 99%

Phase transitions for laminar-turbulent flows in a pipeline or through porous media

Corli

Fan

2013

Communications in Information and Systems

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Liquid/vapor phase changes for a fluid flow through a porous medium or a pipeline are considered. In particular, the model covers both laminar and turbulent flows. The presence of both laminar and turbulent flows causes jump discontinuities in the friction coefficient. Classical trajectories of traveling waves terminate when they intersect the discontinuity. We construct traveling wave solutions by monotonically smoothing the discontinuity and then taking a limiting process. The limit is independent of the monotonepreserving smoothing. This uniqueness justifies the construction of the traveling wave via this smoothing and limiting approach. Existence of traveling waves is established in a wide range of situations; in particular, the end states may be formed either by pure phases or mixtures.

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Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping

Cited by 357 publications

References 1 publication

Initial boundary value problem for compressible Euler equations with damping

Initial boundary value problem for 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

Phase transitions for laminar-turbulent flows in a pipeline or through porous media

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