August  2017, 37(7): 3625-3699. doi: 10.3934/dcds.2017156

Classical solvability of the multidimensional free boundary problem for the thin film equation with quadratic mobility in the case of partial wetting

Institute of Applied Mathematics and Mechanics NASU, State Institute of Applied Mathematics and Mechanics, R.Luxemburg Str., 74, Donetsk, 83114, Ukraine

Received  November 2015 Revised  February 2017 Published  April 2017

We prove locally in time the existence of the unique smooth solution (including smooth interface) to the multidimensional free boundary problem for the thin film equation with the mobility n = 2 in the case of partial wetting. We also obtain the Schauder estimates and solvability for the Dirichlet and the Neumann problem for a linear degenerate parabolic equation of fourth order.

Citation: Sergey Degtyarev. Classical solvability of the multidimensional free boundary problem for the thin film equation with quadratic mobility in the case of partial wetting. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3625-3699. doi: 10.3934/dcds.2017156
References:
[1]

P. Alvarez-CaudevillaJ. D. Evans and V. A. Galaktionov, Towards optimal regularity for the fourth-order thin film equation in RN: Graveleau-type focusing self-similarity, Journal of Mathematical Analysis and Applications, 431 (2015), 1099-1123.  doi: 10.1016/j.jmaa.2015.06.027.

[2]

D. Andreucci and A. F. Tedeev, Finite speed of propagation for the thin-film equation and other higher-order parabolic equations with general nonlinearity, Interfaces Free Bound., 3 (2001), 233-264.  doi: 10.4171/IFB/40.

[3]

B. V. Bazalii and S. P. Degtyarev, On classical solvability of the multidimensional Stefan problem for convective motion of a viscous incompressible fluid, Math. USSR Sb., 60 (1988), 1-17. 

[4]

F. Bernis, Finite speed of propagation and continuity of the interface for thin viscous flows, Adv. Differential Equations, 1 (1996), 337-368. 

[5]

F. Bernis, Finite speed of propagation for thin viscous flows when 2 ≤ n ≤ 3, C. R. Math. Acad. Sci. Paris, 322 (1996), 1169-1174. 

[6]

F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differential Equations, 83 (1990), 179-206.  doi: 10.1016/0022-0396(90)90074-Y.

[7]

A. Bertozzi and M. Pugh, The lubrication approximation for thin viscous films: Regularity and long-time behavior of eak solutions, Comm. Pure Appl. Math., 49 (1996), 85-123.  doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.3.CO;2-V.

[8]

M. BertschR. Dal PassoH. Garcke and G. Grün, The thin viscous flow equation in higher space dimensions, Adv. Differential Equations, 3 (1998), 417-440. 

[9]

M. BertschL. Giacomelli and G. Karali, Thin-film equations with "partial wetting" energy: Existence of weak solutions, Phys. D., 209 (2005), 17-27.  doi: 10.1016/j.physd.2005.06.012.

[10]

G. I. Bizhanova, Investigation of solvability of the multidimensional two-phase Stefan and the nonstationary filtration Florin problems for second order parabolic equations in weighted H?lder spaces of functions, Journal of Mathematical Sciences, 84 (1997), 823-844.  doi: 10.1007/BF02399935.

[11]

G. I. Bizhanova and V. A. Solonnikov, On problems with free boundaries for second-order parabolic equations, St. Petersburg Math. J., 12 (2001), 949-981. 

[12]

M. BoutatS. HiloutJ. -E. Rakotoson and J. -M. Rakotoson, A generalized thin-film equation in multidimensional space, Nonlinear Anal., 69 (2008), 1268-1286.  doi: 10.1016/j.na.2007.06.028.

[13]

R. Dal PassoH. Garcke and G. Grün, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anall., 29 (1998), 321-342.  doi: 10.1137/S0036141096306170.

[14]

P. Daskalopoulos and R. Hamilton, Regularity of the free boundary for the porous medium equation, J. Amer. Math. Soc., 11 (1998), 899-965.  doi: 10.1090/S0894-0347-98-00277-X.

[15]

S. P. Degtyarev, Liouville property for solutions of the linearized degenerate thin film equation of fourth order in a halfspace, Results in Mathematics, 70 (2016), 137-161.  doi: 10.1007/s00025-015-0467-x.

[16]

S. P, Degtyarev, On some weighted Hölder spaces as a possible functional framework for the thin film equation and other parabolic equations with a degeneration at the boundary of a domain, Pacific journal of mathematics, under consideration.

[17]

S. P. Degtyarev, On some weighted Hölder spaces as a possible functional framework for the thin film equation and other parabolic equations with a degeneration at the boundary of a domain, preprint, https://arxiv.org/pdf/1507.01106.pdf.

[18]

S. P. Degtyarev, Classical solvability of multidimensional two-phase Stefan problem for degenerate parabolic equations and Schauder's estimates for a degenerate parabolic problem with dynamic boundary conditions, Nonlinear Differential Equations and Applications, 22 (2015), 185-237.  doi: 10.1007/s00030-014-0280-3.

[19]

S. P. Degtyarev, Elliptic-parabolic equation and the corresponding problem with free boundary I: Elliptic problem with parameter, Journal of Mathematical Sciences, 200 (2014), 305-329.  doi: 10.1007/s10958-014-1914-z.

[20]

P. -G. de GennesF. Brochard-Wyart and D. Quere, Capillarity and Wetting Phenomena: Bubbles, Pearls, Waves, Springer, New York, (2004). 

[21]

L. GiacomelliH. Knüpfer and F. Otto, Smooth zero-contact-angle solutions to a thin-film equation around the steady state, J. Differential Equations, 245 (2008), 1454-1506.  doi: 10.1016/j.jde.2008.06.005.

[22]

L. Giacomelli and H. Knüpfer, A free boundary problem of fourth order: Classical solutions in weighted H?lder spaces, Commun. Partial Differ. Equations, 35 (2010), 2059-2091.  doi: 10.1080/03605302.2010.494262.

[23]

L. GiacomelliM. V. GnannH. Knüpfer and F. Otto, Well-posedness for the Navier-slip thin-film equation in the case of complete wetting, J. Differ. Equations, 257 (2014), 15-81.  doi: 10.1016/j.jde.2014.03.010.

[24]

L. GiacomelliM. V. Gnann and F. Otto, Regularity of source-type solutions to the thin-film equation with zero contact angle and mobility exponent between 3/2 and 3, Eur. J. Appl. Math., 24 (2013), 735-760.  doi: 10.1017/S0956792513000156.

[25]

K. K. Golovkin, On equivalent normalizations of fractional spaces, Amer. Math. Soc. Transl., 81 (1969), 257-280. 

[26]

C. Goulaouic and N. Shimakura, Regularite holderienne de certains problemes aux limites elliptiques degeneres, (French) [Hölder regularity in a degenerate elliptic problem], Annali della Scuola Normale Superiore de Pisa, 10 (1983), 79-108. 

[27]

H. P. Greenspan, Motion of a small viscous droplet that wets a surface, J.Fluid Mech., 84 (1978), 125-143. 

[28]

G. Grün, Droplet spreading under weak slippage -existence for the Cauchy problem, Comm. Partial Differential Equations, 29 (2005), 1697-1744.  doi: 10.1081/PDE-200040193.

[29]

E. -I. Hanzawa, Classical solutions of the Stefan problem, Tohoku Math.Journ., 33 (1981), 297-335.  doi: 10.2748/tmj/1178229399.

[30]

D. John, On uniqueness of weak solutions for the thin-film equation, Journal of Differential Equations, 259 (2015), 4122-4171.  doi: 10.1016/j.jde.2015.05.013.

[31]

S. Kim and A. K. -Lee, Smooth solution for the porous medium equation in a bounded domain, J.Differ.Equations., 247 (2009), 1064-1095.  doi: 10.1016/j.jde.2009.05.001.

[32]

H. Knüpfer, Well-posedness for the Navier slip thin-film equation in the case of partial wetting, Comm. Pure Appl. Math., 64 (2011), 1263-1296.  doi: 10.1002/cpa.20376.

[33] O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Equations of Elliptic Type, Academic Press,, New York and London, 1968. 
[34]

O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, R. I. , 1968.

[35]

S. Lange, Real and Functional Analysis, Graduate Texts in Mathematics, 142, SpringerVerlag, New York, 1993. doi: 10.1007/978-1-4612-0897-6.

[36]

B. Liang, Mathematical analysis to a nonlinear fourth-order partial differential equation, Nonlinear Anal., 74 (2011), 3815-3828.  doi: 10.1016/j.na.2011.03.035.

[37]

C. Liu, Qualitative properties for a sixth-order thin film equation, Math. Model. Anal., 15 (2010), 457-471.  doi: 10.3846/1392-6292.2010.15.457-471.

[38]

C. Liu and Y. Tian, Weak solutions for a sixth-order thin film equation, Rocky Mt. J. Math., 41 (2011), 1547-1565.  doi: 10.1216/RMJ-2011-41-5-1547.

[39]

F. Otto, Lubrication approximation with prescribed nonzero contact angle, Comm. Partial Differential Equations, 23 (1998), 2077-2164.  doi: 10.1080/03605309808821411.

[40]

A. E. Shishkov and R. M. Taranets, On the thin-film equation with nonlinear convection in multidimensional domains, Ukr. Math. Bull., 1 (2004), 407-450. 

[41]

L. Simon, Schauder estimates by scaling, Calc. Var. Partial Differ. Equ., 5 (1997), 391-407.  doi: 10.1007/s005260050072.

[42]

V. A. Solonnikov, Estimates for solutions of a non-stationary linearized system of Navier-Stokes equations, Proceedings of the Steklov Institute of Mathematics, 70 (1964), 213-317. 

[43]

V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general form, Proceedings of the Steklov Institute of Mathematics, 83 (1965), 3-163. 

[44]

V. A. Solonnikov, General boundary value problems for Douglis-Nirenberg elliptic systems. Ⅱ, (Russian), Trudy Mat. Inst. Steklov., 92 (1968), 233-297. 

[45]

H. Triebel, Theory of Function Spaces Ⅱ, Reprint of the 1992 edition, Modern Birkhauser Classics, Basel: Birkhauser, 2010.

show all references

References:
[1]

P. Alvarez-CaudevillaJ. D. Evans and V. A. Galaktionov, Towards optimal regularity for the fourth-order thin film equation in RN: Graveleau-type focusing self-similarity, Journal of Mathematical Analysis and Applications, 431 (2015), 1099-1123.  doi: 10.1016/j.jmaa.2015.06.027.

[2]

D. Andreucci and A. F. Tedeev, Finite speed of propagation for the thin-film equation and other higher-order parabolic equations with general nonlinearity, Interfaces Free Bound., 3 (2001), 233-264.  doi: 10.4171/IFB/40.

[3]

B. V. Bazalii and S. P. Degtyarev, On classical solvability of the multidimensional Stefan problem for convective motion of a viscous incompressible fluid, Math. USSR Sb., 60 (1988), 1-17. 

[4]

F. Bernis, Finite speed of propagation and continuity of the interface for thin viscous flows, Adv. Differential Equations, 1 (1996), 337-368. 

[5]

F. Bernis, Finite speed of propagation for thin viscous flows when 2 ≤ n ≤ 3, C. R. Math. Acad. Sci. Paris, 322 (1996), 1169-1174. 

[6]

F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differential Equations, 83 (1990), 179-206.  doi: 10.1016/0022-0396(90)90074-Y.

[7]

A. Bertozzi and M. Pugh, The lubrication approximation for thin viscous films: Regularity and long-time behavior of eak solutions, Comm. Pure Appl. Math., 49 (1996), 85-123.  doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.3.CO;2-V.

[8]

M. BertschR. Dal PassoH. Garcke and G. Grün, The thin viscous flow equation in higher space dimensions, Adv. Differential Equations, 3 (1998), 417-440. 

[9]

M. BertschL. Giacomelli and G. Karali, Thin-film equations with "partial wetting" energy: Existence of weak solutions, Phys. D., 209 (2005), 17-27.  doi: 10.1016/j.physd.2005.06.012.

[10]

G. I. Bizhanova, Investigation of solvability of the multidimensional two-phase Stefan and the nonstationary filtration Florin problems for second order parabolic equations in weighted H?lder spaces of functions, Journal of Mathematical Sciences, 84 (1997), 823-844.  doi: 10.1007/BF02399935.

[11]

G. I. Bizhanova and V. A. Solonnikov, On problems with free boundaries for second-order parabolic equations, St. Petersburg Math. J., 12 (2001), 949-981. 

[12]

M. BoutatS. HiloutJ. -E. Rakotoson and J. -M. Rakotoson, A generalized thin-film equation in multidimensional space, Nonlinear Anal., 69 (2008), 1268-1286.  doi: 10.1016/j.na.2007.06.028.

[13]

R. Dal PassoH. Garcke and G. Grün, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anall., 29 (1998), 321-342.  doi: 10.1137/S0036141096306170.

[14]

P. Daskalopoulos and R. Hamilton, Regularity of the free boundary for the porous medium equation, J. Amer. Math. Soc., 11 (1998), 899-965.  doi: 10.1090/S0894-0347-98-00277-X.

[15]

S. P. Degtyarev, Liouville property for solutions of the linearized degenerate thin film equation of fourth order in a halfspace, Results in Mathematics, 70 (2016), 137-161.  doi: 10.1007/s00025-015-0467-x.

[16]

S. P, Degtyarev, On some weighted Hölder spaces as a possible functional framework for the thin film equation and other parabolic equations with a degeneration at the boundary of a domain, Pacific journal of mathematics, under consideration.

[17]

S. P. Degtyarev, On some weighted Hölder spaces as a possible functional framework for the thin film equation and other parabolic equations with a degeneration at the boundary of a domain, preprint, https://arxiv.org/pdf/1507.01106.pdf.

[18]

S. P. Degtyarev, Classical solvability of multidimensional two-phase Stefan problem for degenerate parabolic equations and Schauder's estimates for a degenerate parabolic problem with dynamic boundary conditions, Nonlinear Differential Equations and Applications, 22 (2015), 185-237.  doi: 10.1007/s00030-014-0280-3.

[19]

S. P. Degtyarev, Elliptic-parabolic equation and the corresponding problem with free boundary I: Elliptic problem with parameter, Journal of Mathematical Sciences, 200 (2014), 305-329.  doi: 10.1007/s10958-014-1914-z.

[20]

P. -G. de GennesF. Brochard-Wyart and D. Quere, Capillarity and Wetting Phenomena: Bubbles, Pearls, Waves, Springer, New York, (2004). 

[21]

L. GiacomelliH. Knüpfer and F. Otto, Smooth zero-contact-angle solutions to a thin-film equation around the steady state, J. Differential Equations, 245 (2008), 1454-1506.  doi: 10.1016/j.jde.2008.06.005.

[22]

L. Giacomelli and H. Knüpfer, A free boundary problem of fourth order: Classical solutions in weighted H?lder spaces, Commun. Partial Differ. Equations, 35 (2010), 2059-2091.  doi: 10.1080/03605302.2010.494262.

[23]

L. GiacomelliM. V. GnannH. Knüpfer and F. Otto, Well-posedness for the Navier-slip thin-film equation in the case of complete wetting, J. Differ. Equations, 257 (2014), 15-81.  doi: 10.1016/j.jde.2014.03.010.

[24]

L. GiacomelliM. V. Gnann and F. Otto, Regularity of source-type solutions to the thin-film equation with zero contact angle and mobility exponent between 3/2 and 3, Eur. J. Appl. Math., 24 (2013), 735-760.  doi: 10.1017/S0956792513000156.

[25]

K. K. Golovkin, On equivalent normalizations of fractional spaces, Amer. Math. Soc. Transl., 81 (1969), 257-280. 

[26]

C. Goulaouic and N. Shimakura, Regularite holderienne de certains problemes aux limites elliptiques degeneres, (French) [Hölder regularity in a degenerate elliptic problem], Annali della Scuola Normale Superiore de Pisa, 10 (1983), 79-108. 

[27]

H. P. Greenspan, Motion of a small viscous droplet that wets a surface, J.Fluid Mech., 84 (1978), 125-143. 

[28]

G. Grün, Droplet spreading under weak slippage -existence for the Cauchy problem, Comm. Partial Differential Equations, 29 (2005), 1697-1744.  doi: 10.1081/PDE-200040193.

[29]

E. -I. Hanzawa, Classical solutions of the Stefan problem, Tohoku Math.Journ., 33 (1981), 297-335.  doi: 10.2748/tmj/1178229399.

[30]

D. John, On uniqueness of weak solutions for the thin-film equation, Journal of Differential Equations, 259 (2015), 4122-4171.  doi: 10.1016/j.jde.2015.05.013.

[31]

S. Kim and A. K. -Lee, Smooth solution for the porous medium equation in a bounded domain, J.Differ.Equations., 247 (2009), 1064-1095.  doi: 10.1016/j.jde.2009.05.001.

[32]

H. Knüpfer, Well-posedness for the Navier slip thin-film equation in the case of partial wetting, Comm. Pure Appl. Math., 64 (2011), 1263-1296.  doi: 10.1002/cpa.20376.

[33] O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Equations of Elliptic Type, Academic Press,, New York and London, 1968. 
[34]

O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, R. I. , 1968.

[35]

S. Lange, Real and Functional Analysis, Graduate Texts in Mathematics, 142, SpringerVerlag, New York, 1993. doi: 10.1007/978-1-4612-0897-6.

[36]

B. Liang, Mathematical analysis to a nonlinear fourth-order partial differential equation, Nonlinear Anal., 74 (2011), 3815-3828.  doi: 10.1016/j.na.2011.03.035.

[37]

C. Liu, Qualitative properties for a sixth-order thin film equation, Math. Model. Anal., 15 (2010), 457-471.  doi: 10.3846/1392-6292.2010.15.457-471.

[38]

C. Liu and Y. Tian, Weak solutions for a sixth-order thin film equation, Rocky Mt. J. Math., 41 (2011), 1547-1565.  doi: 10.1216/RMJ-2011-41-5-1547.

[39]

F. Otto, Lubrication approximation with prescribed nonzero contact angle, Comm. Partial Differential Equations, 23 (1998), 2077-2164.  doi: 10.1080/03605309808821411.

[40]

A. E. Shishkov and R. M. Taranets, On the thin-film equation with nonlinear convection in multidimensional domains, Ukr. Math. Bull., 1 (2004), 407-450. 

[41]

L. Simon, Schauder estimates by scaling, Calc. Var. Partial Differ. Equ., 5 (1997), 391-407.  doi: 10.1007/s005260050072.

[42]

V. A. Solonnikov, Estimates for solutions of a non-stationary linearized system of Navier-Stokes equations, Proceedings of the Steklov Institute of Mathematics, 70 (1964), 213-317. 

[43]

V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general form, Proceedings of the Steklov Institute of Mathematics, 83 (1965), 3-163. 

[44]

V. A. Solonnikov, General boundary value problems for Douglis-Nirenberg elliptic systems. Ⅱ, (Russian), Trudy Mat. Inst. Steklov., 92 (1968), 233-297. 

[45]

H. Triebel, Theory of Function Spaces Ⅱ, Reprint of the 1992 edition, Modern Birkhauser Classics, Basel: Birkhauser, 2010.

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