## The Thin Viscous Flow Equation In Higher Space Dimensions (1997)

Venue: | Adv. Differential Equations |

Citations: | 12 - 4 self |

### BibTeX

@ARTICLE{Bertsch97thethin,

author = {Michiel Bertsch and Roberta Dal Passo and Harald Garcke and Günther Grün},

title = {The Thin Viscous Flow Equation In Higher Space Dimensions},

journal = {Adv. Differential Equations},

year = {1997},

volume = {3},

pages = {417--440}

}

### Years of Citing Articles

### OpenURL

### Abstract

We prove local integral (entropy) estimates for nonnegative solutions of the fourth order degenerate parabolic equation u t + div(u n r\Deltau) = 0 in space dimensions two and three. These estimates enable us to show that solutions have finite speed of propagation if n 2 ( 1 8 ; 2) and that the support cannot shrink if the growth exponent n is larger than 3=2. In addition, we prove decay estimates for solutions of the Cauchy problem and a growth estimate for their support. 1.

### Citations

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A.: Higher order nonlinear degenerate parabolic equations
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(Show Context)
Citation Context ... overview on degenerate parabolic equations of higher order and their applications we refer to Bernis [2]. The mathematical investigation of problem (P T ) started with a paper by Bernis and Friedman =-=[5]-=-. In one space dimension they were able to show the existence of a nonnegative Holder continuous solution for all values ns1. The Holder continuity of the solution is important for their analysis beca... |

53 | The lubrication approximation for thin viscous films --- regularity and long-time behavior of weak solutions
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Citation Context ...mbers ff satisfying 1 2sff + ns2. In the paper by Bernis and Friedman [5] identity (E) was applied only for ff = 1 \Gamma n. Using the new estimates Beretta, Bertsch, Dal Passo [1] and Bertozzi, Pugh =-=[7]-=- were able to prove regularity results that are optimal in the sense that they are sharp for the source type similarity solutions. Integral estimates derived from a local version of identity (E) are u... |

49 |
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(Show Context)
Citation Context ... ; (P T ) in space dimensions two and three. This problem appears in the lubrication theory for thin viscous films that are driven by surface tension and the function u is the height of the film (cf. =-=[15]-=-). The above partial differential equation is a fourth order parabolic equation that degenerates for u = 0. In recent years also other examples of similar degenerate parabolic equations of higher orde... |

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34 |
The Cahn–Hilliard equation with a concentration dependent mobility: motion by minus the laplacian of the mean curvature
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Citation Context ...abolic equations of higher order appeared in the physics and materials science literature. We only mention models for phase separation in alloys (Cahn--Hilliard equation with degenerate mobility, cf. =-=[8]-=-, [10], [11]) and models for the evolution of dislocation densities in the theory of plasticity (Norton--Hoff type models, cf. [14]). For an overview on degenerate parabolic equations of higher order ... |

32 |
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Citation Context ... + 2N ands= (2 \Gamma q)N 2q : If B r (0) is replaced by IR N , then (24) holds with the constant K 1 = 0. The Gagliardo--Nirenberg inequality was independently proved by Gagliardo [13] and Nirenberg =-=[18, 19]-=-. Bernis proved the lemma for the nonstandard case q 2 (0; 1) in one space dimension. The generalization to higher space dimensions is straightforward. The explicit dependence of the constants on r fo... |

29 |
Source type solutions of a fourth order nonlinear degenerate parabolic equation. Nonlinear Analysis 18: 217±234
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Citation Context ... Holder continuity of the solution is important for their analysis because it ensures the smoothness of the solution where it is positive and it implies its boundedness. Bernis, Peletier and Williams =-=[6]-=- studied the question whether self-similar source type solutions of the Cauchy problem corresponding to (P T ) exist. They showed that only for n 2 (0; 3) self-similar source type solutions with finit... |

28 |
Finite speed of propagation and continuity of the interface for thin viscous flows
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(Show Context)
Citation Context ...prove regularity results that are optimal in the sense that they are sharp for the source type similarity solutions. Integral estimates derived from a local version of identity (E) are used by Bernis =-=[3]-=- and Kersner, Shiskov [16] to show that in one space dimension solutions to (P T ) have the property of finite speed of propagation of their support if n 2 (0; 2). In addition, Bernis [3] obtained reg... |

26 |
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Citation Context ...i estimates for real numbers ff satisfying 1 2sff + ns2. In the paper by Bernis and Friedman [5] identity (E) was applied only for ff = 1 \Gamma n. Using the new estimates Beretta, Bertsch, Dal Passo =-=[1]-=- and Bertozzi, Pugh [7] were able to prove regularity results that are optimal in the sense that they are sharp for the source type similarity solutions. Integral estimates derived from a local versio... |

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26 | Lubrication approximation with prescribed nonzero contact angle
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Citation Context ...tegral estimates resulting from identity (E) imply that the solutions have a zero contact angle for almost all t. The only result for a non--zero contact angle that is known to the authors is by Otto =-=[20]-=- who shows in space dimension N = 1 for the special case of growth exponent n = 1 existence of weak solutions with a prescribed contact angle. In higher space dimensions there are existence results fo... |

15 | Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix, Phys
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(Show Context)
Citation Context ...tions of higher order appeared in the physics and materials science literature. We only mention models for phase separation in alloys (Cahn--Hilliard equation with degenerate mobility, cf. [8], [10], =-=[11]-=-) and models for the evolution of dislocation densities in the theory of plasticity (Norton--Hoff type models, cf. [14]). For an overview on degenerate parabolic equations of higher order and their ap... |

14 |
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Citation Context ...nd Grun [14]. But in these papers the growth exponent n was restricted to the interval [1; 2) if one wants to prescribe initial data with compact support. In a recent paper Dal Passo, Garcke and Grun =-=[9]-=- were able to show a global version of the integral estimates in higher space dimensions (cf. also Section 2). These estimates make it possible to show existence of weak solutions to problem (P T ) if... |

13 | Finite speed of propagation for thin viscous flows when 2 - Bernis - 1996 |

11 |
Degenerate parabolic differential equations of fourth order and a plasticity model with nonlocal hardening
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Citation Context ...tion in alloys (Cahn--Hilliard equation with degenerate mobility, cf. [8], [10], [11]) and models for the evolution of dislocation densities in the theory of plasticity (Norton--Hoff type models, cf. =-=[14]-=-). For an overview on degenerate parabolic equations of higher order and their applications we refer to Bernis [2]. The mathematical investigation of problem (P T ) started with a paper by Bernis and ... |

9 |
Source-type solutions to thin-film equations in higher dimensions
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Citation Context ...lutions to problem (P T ) in higher space dimensions are bounded or continuous. Recently Bernis and Ferreira have constructed self-similar source-type solutions in the case of higher space dimensions =-=[12]-=-. Let us shortly describe the outline of this paper. In Section 2, we present -- basically following the spirit of [9] -- the main ingredients and results concerning the construction of solutions to p... |

4 |
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Citation Context ...a q)N (4 \Gamma N)q + 2N ands= (2 \Gamma q)N 2q : If B r (0) is replaced by IR N , then (24) holds with the constant K 1 = 0. The Gagliardo--Nirenberg inequality was independently proved by Gagliardo =-=[13]-=- and Nirenberg [18, 19]. Bernis proved the lemma for the nonstandard case q 2 (0; 1) in one space dimension. The generalization to higher space dimensions is straightforward. The explicit dependence o... |

2 |
Existence of Free Boundaries in Thin Film Theory, Preprint of
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(Show Context)
Citation Context ...that are optimal in the sense that they are sharp for the source type similarity solutions. Integral estimates derived from a local version of identity (E) are used by Bernis [3] and Kersner, Shiskov =-=[16]-=- to show that in one space dimension solutions to (P T ) have the property of finite speed of propagation of their support if n 2 (0; 2). In addition, Bernis [3] obtained regularity results for the re... |

1 |
An extended interpolation inequality, Ann
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(Show Context)
Citation Context ...3 `\Gamma1 C 1 4 (r 0 ) 4 T j 4 E 0 (r 0 ; T ) `\Gamma1 4 : But this property only holds as long as r 1sr 0 2 . With the help of the integral estimate (1) and the Gagliardo--Nirenberg inequality (cf. =-=[19]-=-) we estimate E 0 (r 0 ; T )sZ \Omega T jD 2 u ff+n+1 2 j 2 C 1 ff(ff+1) Z \Omega u ff+1 0 + C 1 C 2 Z T 0 Z \Omega u ff+n+1 C 1 ff(ff+1) Z \Omega u ff+1 0 + C Z T 0 i kru(t)k a(ff+n+1) 2 ku 0 k (1\Ga... |

1 |
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Citation Context ...ma : Let us collect some properties of the solution u. The regularity of u implies that u 2 C([0; T ]; L p (\Omega\Gamma5 for p 2 [1; 1) if N = 2 and for p 2 [1; 6) if N = 3 (cf. Corollary 4 in Simon =-=[21]). He-=-nce, u attains its initial values in the sense of iv). Since u 2 C i [0; T ]; \Gamma H 1;q \Delta 0 j " L 1 \Gamma 0; T ; H 1(\Omega\Gamma \Delta for q ? 4N 2N+(2\GammaN)n 4 we can conclude that ... |