The thin-film equation close to self-similarity

Seis, Christian

doi:10.2140/apde.2018.11.1303

Cited by 23 publications

(24 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A rather natural question in the context of parabolic problems is how the regularity of the free boundary propagates. This question has been studied by Kienzler [52], Koch [59], and Seis [72] for the porous-medium equation (1.3) and by John [50] and Seis [73] in the case of the thin-film equation with linear mobility (i.e., for the case of (1.2) with n " 1), but it has not been addressed in physical dimensions for the more realistic case n " 2 with complete-wetting boundary conditions treated here. We do not prove a regularizing effect in the tangential variables and we only prove partial regularity in the normal variables.…”

Section: Main Resultmentioning

confidence: 99%

“…Well-posedness and classical solutions. Well-posedness and regularity for zero contact angles in the Hele-Shaw case (equation (1.2) with n " 1) have been treated by Bringmann, Giacomelli, Knüpfer, and Otto [13,35], Giacomelli and Knüpfer [34], the first author [38], and the first author, Ibrahim, and Masmoudi [40] in 1`1 dimensions and by John [50] and Seis [73] in any number of spatial dimensions while nonzero contact angles have been the subject of the works of Knüpfer and Masmoudi [57,58] for 1`1 dimensions only. The remarkable result is that solutions are smooth functions in the distance to the free boundary only for the linear-mobility case, i.e., for the case n " 1.…”

Section: 4mentioning

confidence: 99%

“…We are aware of only three works (which have been mentioned afore) establishing well-posedness and regularity of the free boundary in more than 1`1 dimensions. The first two, John [50] and Seis [73], treat the linear mobility case in arbitrary dimensions, showing that level sets (and therefore also the free boundary) are analytic manifolds of time and space. The third paper due to Degtyarev [25] treats the quadratic mobility case in arbitrary dimensions under partial wetting conditions, i.e., condition (1.1b) is replaced by the less degenerate condition pν, ∇hq " g for a function satisfying point-wise bounds g ě ε ą 0.…”

mentioning

confidence: 99%

See 2 more Smart Citations

The Navier-slip thin-film equation for 3D fluid films: Existence and uniqueness

Gnann

Petrache

2018

Journal of Differential Equations

View full text Add to dashboard Cite

We consider the thin-film equation Bth`∇¨`h 2 ∇∆h˘" 0 in physical space dimensions (i.e., one dimension in time t and two lateral dimensions with h denoting the height of the film in the third spatial dimension), which corresponds to the lubrication approximation of the Navier-Stokes equations of a three-dimensional viscous thin fluid film with Navier-slip at the substrate. This equation can have a free boundary (the contact line), moving with finite speed, at which we assume a zero contact angle condition (complete-wetting regime). Previous results have focused on the 1`1-dimensional version, where it has been found that solutions are not smooth as a function of the distance to the free boundary. In particular, a well-posedness and regularity theory is more intricate than for the second-order counterpart, the porousmedium equation, or the thin-film equation with linear mobility (corresponding to Darcy dynamics in the Hele-Shaw cell). Here, we prove existence and uniqueness of classical solutions that are perturbations of an asymptotically stable traveling-wave profile. This leads to control on the free boundary and in particular its velocity.1.3. Weak solutions to the thin-film equation. We emphasize that a well-established global existence theory of weak solutions to (1.2) has been developed, starting with Bernis and Friedman [7] and later on upgraded to the stronger entropy-weak solutions by Beretta, Bertsch, and Dal Passo [4], and independently by Bertozzi and Pugh [9], which also exist in higher dimensions (cf. [21,45]). An alternative gradient-flow approach leading to generalized minimizing-movement solutions (that are weak solutions as well) is due to Loibl, Matthes, and Zinsl [61], Matthes, McCann, and Savaré [63], and Otto [69]. Qualitative properties of weak solutions have been the subject of for instance the works of Bernis [5, 6], Grün [43, 44], and Hulshof and Shishkov [48], where finite speed of propagation has been proved, Bertsch, Dal Passo, Garcke, and Grün [10], Dal Passo, Giacomelli, and Grün [22], Fischer [28, 29, 30], Giacomelli and Grün [33], and Grün [46], where waiting-time phenomena have been considered, or Carlen and Ulusoy [19], Carrillo and Toscani [20], and Matthes, McCann, and Savaré [63], where the intermediate asymptotics of (1.2) have been investigated. Partial-wetting boundary conditions have been considered by Bertsch, Giacomelli, and Karali [11], Esselborn [27], Mellet [64], and Otto [69]. We refer to Ansini and Giacomelli [2], Bertozzi [8], and Giacomelli and Shishkov [37] for detailed reviews. Nevertheless, unlike in the porous-medium case (1.3), this theory does neither give uniqueness of solutions nor enough control at the free boundary to give an expression like (1.1d) a classical meaning. Furthermore, the regularity of the free boundary as a sub-manifold of p0, 8qˆR 2 appears to be inaccessible within this theory. This explains the interest in a well-posedness and regularity theory of classical solutions to (1.1). 1.4.Well-posedness and classical solutions. Well-posednes...

show abstract

Section: Main Resultmentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

The Navier-slip thin-film equation for 3D fluid films: Existence and uniqueness

Gnann

Petrache

2018

Journal of Differential Equations

View full text Add to dashboard Cite

show abstract

“…We give brief arguments as to see that the controllability study in Section 1.2 may also be applied to the one-dimensional thin-film equation linearized around its self-similar profile, derived in [29,35]. The thin-film equation…”

Section: Null-controllability Of the Linearized Thin-film Equationmentioning

confidence: 99%

Null-controllability of perturbed porous medium gas flow

Geshkovski

2020

ESAIM: COCV

View full text Add to dashboard Cite

In this work, we investigate the null-controllability of a nonlinear degenerate parabolic equation, which is the equation satisfied by a perturbation around the self-similar solution of the porous medium equation in Lagrangian-like coordinates. We prove a local null-controllability result for a regularized version of the nonlinear problem, in which singular terms have been removed from the nonlinearity. We use spectral techniques and the source-term method to deal with the linearized problem and the conclusion follows by virtue of a Banach fixed-point argument. The spectral techniques are also used to prove a null-controllability result for the linearized thin-film equation, a degenerate fourth order analog of the problem under consideration.

show abstract

“…The main difficulty in the analysis is again caused by the fact that the equation's parabolicity degenerates, as the film height u tends to zero. A rich amount of research has been dedicated to this problem in the last decade, mostly prescribing a constant slope at the free boundary; see [13,[16][17][18][19][22][23][24][27][28][29]36] for strong solutions in (weighted) Sobolev or Hölder spaces.…”

Section: Introductionmentioning

confidence: 99%

Local strong solutions to a quasilinear degenerate fourth-order thin-film equation

Lienstromberg

Müller

2020

Nonlinear Differ. Equ. Appl.

View full text Add to dashboard Cite

We study the problem of existence and uniqueness of strong solutions to a degenerate quasilinear parabolic non-Newtonian thin-film equation. Originating from a non-Newtonian Navier-Stokes system, the equation is derived by lubrication theory and under the assumption that capillarity is the only driving force. The fluid's shear-thinning rheology is described by the so-called Ellis constitutive law. For flow behaviour exponents α ≥ 2 the corresponding initial boundary value problem fits into the abstract setting of Amann (Function Spaces, Differential Operators and Nonlinear Analysis, Vieweg Teubner Verlag, Stuttgart, 1993). Due to a lack of regularity this is not true for flow behaviour exponents α ∈ (1, 2). For this reason we prove an existence theorem for abstract quasilinear parabolic evolution problems with Hölder continuous dependence. This result provides existence of strong solutions to the non-Newtonian thinfilm problem in the setting of fractional Sobolev spaces and (little) Hölder spaces. Uniqueness of strong solutions is derived by energy methods and by using the particular structure of the equation.

show abstract

The thin-film equation close to self-similarity

Cited by 23 publications

References 42 publications

The Navier-slip thin-film equation for 3D fluid films: Existence and uniqueness

The Navier-slip thin-film equation for 3D fluid films: Existence and uniqueness

Null-controllability of perturbed porous medium gas flow

Local strong solutions to a quasilinear degenerate fourth-order thin-film equation

Contact Info

Product

Resources

About