Singularities and Similarities in Interface Flows

Bertozzi, Andrea L.; Brenner, Michael P.; Dupont, Todd; Kadanoff, Leo P.

doi:10.1007/978-1-4612-0859-4_6

Cited by 131 publications

(250 citation statements)

References 19 publications

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“…For arbitrary β > 0, l > 4, s > 0, δ > 0 such that s + 2δ < a, we find for σ = β + α + 1 that 9) where v = v(x, t) = (uζ l s+δ,δ ) (n+α+1)/4 . By the Gagliardo-Nirenberg interpolation inequality,…”

mentioning

confidence: 87%

“…t > 0) nonnegative solutions was demonstrated by Beretta, Bertsch & Dal Passo [2] and by Bertozzi & Pugh [12], and the solutions were seen to possess the regularity of the source type solutions at the contact line. Positivity properties were seen to depend strongly on the value of n [7,2], with "touchdown" being possible for small values of n [9,2]. Solutions were proven to become positive in finite time, and to converge to their mean as t → ∞ [2,12].…”

Section: Introductionmentioning

confidence: 95%

See 1 more Smart Citation

The thin film equation with backwards second order diffusion

Novick-Cohen¹,

Shishkov²

2011

Interfaces Free Bound.

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We focus on the thin film equation with lower order "backwards" diffusion which can describe, for example, the evolution of thin viscous films in the presence of gravity and thermo-capillary effects, or the thin film equation with a "porous media cutoff" of van der Waals forces. We treat in detail the equationwhere ν = ±1, n > 0, M > m, and A 0. Global existence of weak nonnegative solutions is proven when m − n > −2 and A > 0 or ν = −1, and when −2 < m − n < 2, A = 0, ν = 1. From the weak solutions, we get strong entropy solutions under the additional constraint that m−n > −3/2 if ν = 1. A local energy estimate is obtained when 2 n < 3 under some additional restrictions. Finite speed of propagation is proven when m > n/2, for the case of "strong slippage", 0 < n < 2, when ν = 1 based on local entropy estimates, and for the case of "weak slippage", 2 n < 3, when ν = ±1 based on local entropy and energy estimates.

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mentioning

confidence: 87%

Section: Introductionmentioning

confidence: 95%

The thin film equation with backwards second order diffusion

Novick-Cohen¹,

Shishkov²

2011

Interfaces Free Bound.

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show abstract

“…However, as our algorithm is approaching steady state, a large number of iterations in the propagation phase pose a burden on the computational time. To successfully speed up the convergence of our algorithm, we used adaptive timestepping, a modified form of the scheme proposed in [1].…”

Section: Adaptive Timesteppingmentioning

confidence: 99%

“…We also estimate the dimensionless error in every iteration of the algorithm, and if we find an error to be smaller than Tol 2 the initial sub-timestep in the propagation phase of the next iteration will be increased be 10 %. However, we never allow the initial sub-timestep to be larger than 1 8 of the timestep. Notice that we are not adapting the timestep, the total propagation time in each iteration is the same.…”

Section: Adaptive Timesteppingmentioning

confidence: 99%

Statistical Density Estimation Using Threshold Dynamics for Geometric Motion

Kostic¹,

Bertozzi²

2012

J Sci Comput

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Our goal is to estimate a probability density based on discrete point data via segmentation techniques. Since point data may represent certain activities, such as crime, our method can be successfully used for detecting regions of high activity. In this work we design a binary segmentation version of the well-known Maximum Penalized Likelihood Estimation (MPLE) model, as well as a minimization algorithm based on thresholding dynamics originally proposed by Merriman et al. (The Computational Crystal Growers, pp. 73-83, 1992). We also present some computational examples, including one with actual residential burglary data from the San Fernando Valley.

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“…This class of model equation is related to many physical systems involving fluid interfaces. When n = 1, m = 1, it describes a thin jet in a Hele-Shaw cell [1], [5], [8], [9]; when n = m = 3 it describes fluid droplets hanging from a ceiling [10]; when n = 0 and m = 1, it describes solidification of a hyper-cooled melt [3], [4]; and when n = 3, m = −1, it models van der Waals force driven thin film [7], [12], [18], [19], [20], when the space dimension is one R. Laugesen and M. Pugh [16] studied rigorously, in a general setting, positive periodic steady states and touchdown steady states solution. F. Bernis and A. Friedman in [2] established the existence of weak solutions and showed that the support of the thin film will expand with time.…”

Section: Introductionmentioning

confidence: 99%

Point Rupture Solutions of Singular Elliptic Equations in N-D

Miloua¹

2016

JAAM

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We consider the elliptic equationwhere f is a positive continuous function satisfyingMotivated by the thin film equations, a solution u is said to be a point rupture solution if for some p ∈ Ω, u (p) = 0 and u (p) > 0 in Ω\ {p}. Solving the associated ordinary differential equations confirm our main results of sufficient conditions on f for the existence uniquness of radial point rupture solution and its asymptotic behavior. Furthermore, we can prove that our results can be applied to the point rupture solutions for a class of quasi-linear elliptic equations of the form div (a (u) ∆u) = a (u) 2 |∇u| 2 + f (u)

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Singularities and Similarities in Interface Flows

Cited by 131 publications

References 19 publications

The thin film equation with backwards second order diffusion

The thin film equation with backwards second order diffusion

Statistical Density Estimation Using Threshold Dynamics for Geometric Motion

Point Rupture Solutions of Singular Elliptic Equations in N-D

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