zbMATH — the first resource for mathematics

On uniqueness of weak solutions for the thin-film equation. (English) Zbl 1322.35084
The existence of non-negative weak solutions to the thin film equation (TFE) \[ \partial_t h + \mathrm{div} \left( h \nabla\Delta h \right) = 0 \quad\text{ in }\quad (0,\infty)\times\mathbb{R}^n \] is now well-known but uniqueness is a more delicate issue, partly due to the degenerate diffusion when \(h\) vanishes and the lack of a comparison principle. It is shown herein that, if the initial condition \(h_0\) is sufficiently close to the stationary solution \(h_{\mathrm{st}}: y=(y_1,\ldots,y_n) \mapsto (y_n)_+^2\), then the thin film equation has a unique solution in a suitable class which stays close to \(h_{\mathrm{st}}\) in the sense that \(\sup\{ |\nabla\sqrt{h(s,x)} - e_n|\;:\;x\in\mathcal{P}(s)\}\) is small for all times \(s\geq 0\). Here, \(e_n\) is the unit vector corresponding to the \(y_n\)-axis and \(\mathcal{P}(s)\) denotes the positivity set of \(h(s)\) at time \(s\geq 0\). A by-product of the analysis is the analyticity of the level sets of the solution.
The proof relies on the expected monotonicity with respect to \(y_n\) of solutions close to \(h_{\mathrm{st}}\) which allows the author to use the so-called von Mises transformation \[ \sqrt{h}(s,y',v(s,y',z)) = z\;, \quad y'=(y_1,\ldots,y_{n-1})\in \mathbb{R}^{n-1}\;, \quad z>0\;. \] Introducing \(u(t,x) = v(t,x) - x_n\) for \(t>0\) and \(x=(x_1,\ldots,x_n) \in H = \mathbb{R}^{n-1}\times (0,\infty)\), the function \(v\) solves a semilinear equation of the form \[ \partial_t u + Lu = f(u) \quad\text{ in }\quad (0,\infty)\times H\;, \] where \(L\) is the non-uniform parabolic operator \[ Lu(x) = \frac{1}{x_n} \Delta \left( x_n^3 \Delta u(x) \right) - 4 \Delta_{x'} u(x)\;. \] A suitable functional framework is identified in which the linear evolution equation associated to \(L\) is well-posed. Various regularity estimates and a representation formula featuring the Green’s function are then derived. A fixed point argument is then used to construct a unique solution \(u\) to the nonlinear equation for small initial data.

35K65 Degenerate parabolic equations
35K46 Initial value problems for higher-order parabolic systems
35B20 Perturbations in context of PDEs
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI arXiv
[1] Angenent, S., Nonlinear analytic semiflows, Proc. Roy. Soc. Edinburgh Sect. A, 115, 1-2, 91-107, (1990) · Zbl 0723.34047
[2] Bernis, F., Finite speed of propagation and continuity of the interface for thin viscous flows, Adv. Differential Equations, 1, 3, 337-368, (1996) · Zbl 0846.35058
[3] Bernis, F., Finite speed of propagation for thin viscous flows when \(2 \leq n \leq 3\), C. R. Math. Acad. Sci. Paris, 322, 12, 1169-1174, (1996) · Zbl 0853.76018
[4] Bernis, F.; Friedman, A., Higher order nonlinear degenerate parabolic equations, J. Differential Equations, 83, 1, 179-206, (1990) · Zbl 0702.35143
[5] Bertozzi, A.; Pugh, M., The lubrication approximation for thin viscous films: regularity and long-time behavior of weak solutions, Comm. Pure Appl. Math., 49, 2, 85-123, (1996) · Zbl 0863.76017
[6] Bertsch, M.; Dal Passo, R.; Garcke, H.; Grün, G., The thin viscous flow equation in higher space dimensions, Adv. Differential Equations, 3, 3, 417-440, (1998) · Zbl 0954.35035
[7] Bertsch, M.; Giacomelli, L.; Karali, G., Thin-film equations with “partial wetting” energy: existence of weak solutions, Phys. D, 209, 1, 17-27, (2005) · Zbl 1079.76011
[8] Boatto, S.; Kadanoff, L. P.; Olla, P., Traveling-wave solutions to thin-film equations, Phys. Rev. E, 48, 6, 4423-4431, (1993)
[9] Dal Passo, R.; Garcke, H.; Grün, G., On a fourth-order degenerate parabolic equation: global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anal., 29, 2, 321-342, (1998) · Zbl 0929.35061
[10] Daskalopoulos, P.; Hamilton, R., Regularity of the free boundary for the porous medium equation, J. Amer. Math. Soc., 11, 4, 899-966, (1998) · Zbl 0910.35145
[11] Fabes, E.; Stroock, D., A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash, Arch. Ration. Mech. Anal., 96, 4, 327-338, (1986) · Zbl 0652.35052
[12] Folland, G., Real analysis: modern techniques and their applications, (1984), John Wiley & Sons NY · Zbl 0549.28001
[13] Giacomelli, L.; Knüpfer, H.; Otto, F., Smooth zero-contact-angle solutions to a thin-film equation around the steady state, J. Differential Equations, 245, 6, 1454-1506, (2008) · Zbl 1159.35039
[14] Giacomelli, L.; Gnann, M.; Otto, F., Regularity of source-type solutions to the thin-film equation with zero contact angle and mobility exponent between 3/2 and 3, European J. Appl. Math., 24, 5, 1-26, (2013)
[15] Giacomelli, L.; Knüpfer, H., A free boundary problem of fourth order: classical solutions in weighted Hölder spaces, Comm. Partial Differential Equations, 35, 11, 2059-2091, (2010) · Zbl 1223.35208
[16] Grün, G., Droplet spreading under weak slippage - existence for the Cauchy problem, Comm. Partial Differential Equations, 29, 11-12, 1697-1744, (2005) · Zbl 1156.35388
[17] Huh, C.; Scriven, L. E., Hydrodynamic model of steady movement of a solid/liquid/fluid contact line, J. Colloid Interface Sci., 35, 1, 85-101, (1971)
[18] Huxley, M., Exponential sums and lattice points III, Proc. Lond. Math. Soc., 87, 3, 591-609, (2003) · Zbl 1065.11079
[19] Knüpfer, H., Well-posedness for the Navier slip thin-film equation in the case of partial wetting, Comm. Pure Appl. Math., 64, 9, 1263-1296, (2011) · Zbl 1227.35146
[20] H. Knüpfer, N. Masmoudi, Darcy flow on a plate with prescribed contact angle - well-posedness and lubrication approximation, preprint, 2010.
[21] Koch, H., Non-Euclidean singular integrals and the porous medium equation, (1999), Universität Heidelberg, Habilitation thesis
[22] Koch, H., Partial differential equations and singular integrals, (Dispersive Nonlinear Problems in Mathematical Physics, Quaderni di Matematica, vol. 15, (2004)), 59-122 · Zbl 1141.42012
[23] Koch, H.; Lamm, T., Geometric flows with rough initial data, Asian J. Math., 16, 2, 209-235, (2012) · Zbl 1252.35159
[24] Kufner, A., Weighted Sobolev spaces, (1985), John Wiley & Sons NY · Zbl 0567.46009
[25] Oron, A.; Davis, S. H.; Bankoff, S. G., Long-scale evolution of thin liquid films, Rev. Modern Phys., 69, 3, 931-980, (1997)
[26] Otto, F., Lubrication approximation with prescribed nonzero contact angle, Comm. Partial Differential Equations, 23, 11-12, 63-103, (1998) · Zbl 0923.35211
[27] Otto, F., The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26, 1-2, 101-174, (2001) · Zbl 0984.35089
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.