2021
DOI: 10.48550/arxiv.2108.09032
|View full text |Cite
Preprint
|
Sign up to set email alerts
|

The porous medium equation as a singular limit of the thin film Muskat problem

Abstract: The singular limit of the thin film Muskat problem is performed when the density (and possibly the viscosity) of the lighter fluid vanishes and the porous medium equation is identified as the limit problem. In particular, the height of the denser fluid is shown to converge towards the solution to the porous medium equation and an explicit rate for this convergence is provided in space dimension d ≤ 4. Moreover, the limit of the height of the lighter fluid is determined in a certain regime and is given by the c… Show more

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
3

Citation Types

0
4
0

Year Published

2021
2021
2023
2023

Publication Types

Select...
2
1

Relationship

2
1

Authors

Journals

citations
Cited by 3 publications
(4 citation statements)
references
References 11 publications
0
4
0
Order By: Relevance
“…On the other hand, non-negative global weak solutions to (1.2), which are also bounded, are constructed in [20,22], but the uniqueness of such solutions is an open problem, even in dimension N = 1. This is in sharp contrast with the porous medium equation for which several uniqueness results for weak solutions are available in the literature, see [1,3,7,21,24,25,27] and the references therein. It is actually the strong coupling in (1.2a) which makes it difficult to generalize the methods from the above references to this two-phase version of the porous medium equation.…”
Section: Introductionmentioning
confidence: 96%
“…On the other hand, non-negative global weak solutions to (1.2), which are also bounded, are constructed in [20,22], but the uniqueness of such solutions is an open problem, even in dimension N = 1. This is in sharp contrast with the porous medium equation for which several uniqueness results for weak solutions are available in the literature, see [1,3,7,21,24,25,27] and the references therein. It is actually the strong coupling in (1.2a) which makes it difficult to generalize the methods from the above references to this two-phase version of the porous medium equation.…”
Section: Introductionmentioning
confidence: 96%
“…On the other hand, non-negative global weak solutions to (1.2), which are also bounded, are constructed in [20,22], but the uniqueness of such solutions is an open problem, even in dimension N = 1. This is in sharp contrast with the porous medium equation for which several uniqueness results for weak solutions are available in the literature, see [1,3,7,21,24,25,27] and the references therein. It is actually the strong coupling in (1.2a) which makes it difficult to generalize the methods from the above references to this two-phase version of the porous medium equation.…”
Section: Introductionmentioning
confidence: 96%
“…(B.13)We now infer from (B.5) thatb 11 (u)∂ x ϕ 1 ∂ x ϕ 1,+ = b 11 (u)1 (−∞,0) (u 1 )|∂ x u 1 | 2 ≥ 0 , b 12 (u)∂ x ϕ 2 ∂ x ϕ 1,+ = b 12 (u)1 (−∞,0) (u 1 )∂ x u 1 ∂ x u 2 = 0 , b 21 (u)∂ x ϕ 1 ∂ x ϕ 2,+ = b 21 (u)1 (−∞,0) (u 2 )∂ x u 1 ∂ x u 2 = 0 , b 22 (u)∂ x ϕ 2 ∂ x ϕ 2,+ = b 22 (u)1 (−∞,0) (u 2 )|∂ x u 2 | 2 ≥ 0 ,so that the second term on the left-hand side of (B.13) is non-negative. Consequently, (B 13). givesΩ |ϕ 1,+ | 2 + |ϕ 2,+ | 2 dx ≤ 0 ,which implies that ϕ 1,+ = ϕ 2,+ = 0 a.e.…”
mentioning
confidence: 94%
“…As a consequence of Theorem 1.2 and of the estimate (A.13), we obtain uniform L ∞ -bounds for the height f of the denser fluid in the regime where R → 0 and µ is fixed. Such an estimate has been used recently in [13,Corollary 1.4] when performing the singular limit R → 0 (while µ is kept fixed or µ → ∞) in the thin film Muskat problem (1.1) in order to recover the porous medium equation ∂ t f = div f ∇f in the limit.…”
Section: Introductionmentioning
confidence: 99%