July  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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[4]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.   Google Scholar

[27]

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

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[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.   Google Scholar

[41]

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

[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.   Google Scholar

[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.   Google Scholar

[44]

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

[45]

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

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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[4]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.   Google Scholar

[27]

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

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[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.   Google Scholar

[41]

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

[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.   Google Scholar

[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.   Google Scholar

[44]

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

[45]

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

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