The one-dimensional Darcy's law as the limit of a compressible Euler flow

Marcati, Pierangelo; Milani, Albert

doi:10.1016/0022-0396(90)90130-h

Cited by 156 publications

(109 citation statements)

References 16 publications

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“…We refer the reader to [8] for the existence of global smooth solutions in the isentropic case. We also refer the reader to [6] for the derivation of the porous media equation as the limit of the isentropic Euler equations in one space dimension. In this paper, we show that the result of [10] can be made independent of the relaxation time τ .…”

Section: Introductionmentioning

confidence: 99%

The strong relaxation limit of the multidimensional isothermal Euler equations

Coulombel

Goudon

2006

Trans. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 99%

The strong relaxation limit of the multidimensional isothermal Euler equations

Coulombel

Goudon

2006

Trans. Amer. Math. Soc.

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“…Because of Proposition 3, this implies that ρ 2 ε ρ 2 in the sense of distributions, which implies in turn that the family ρ ε is relatively compact in L 2 ([0, T ] × R) strong thanks to the argument of [5], Section 4.…”

Section: Thanks To Proposition 1 By Banach-alaoglu Theorem There Exmentioning

confidence: 92%

The diffusive limit of Carleman-type models in the range of very fast diffusion equations

Salvarani

Toscani

2009

J. Evol. Equ.

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We improve the existing results on the limiting behavior of the Cauchy problem for a class of Carleman-like models with power-type interaction rate in the diffusive scaling with data in the spaces L p , 1 ≤ p ≤ ∞. The convergence result, which has been carefully established before for exponents of the interaction rate α ≤ 1, is extended here to the range of exponents 1 < α < 4/3. In addition, we discuss the problem of establishing a good theory in the still remaining range α ∈ [4/3, 2), by introducing a modified kinetic system which admits an explicit self-similar solution. The analysis of this solution clarifies the role of the exponentᾱ = 4/3.

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“…The above formal derivation of heat equation has been justified by many authors, see [1][2][3] and the references therein. In [2], Junca and Rascle studied the convergence of the solutions to (1.1) towards those of (1.6) for arbitrary large initial data in BV (R) space.…”

Section: Introductionmentioning

confidence: 99%

“…In [2], Junca and Rascle studied the convergence of the solutions to (1.1) towards those of (1.6) for arbitrary large initial data in BV (R) space. Marcati and Milani [3] showed the derivation of the porous media equation as the limit of the isentropic Euler equations in one space dimension. Recently, Coulombel and Goudon [1] constructed the uniform smooth solutions to (1.1) in the multidimensional case and proved this relaxation-time limit in some Sobolev space…”

Section: Introductionmentioning

confidence: 99%