On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions

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December 2015, 4(4): 391-429. doi: 10.3934/eect.2015.4.391

On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions

Sergey P. Degtyarev ^1,

1.	Institute for Applied Mathematics and Mechanics NASU, State Institute for Applied Mathematics and Mechanics, R.Luxenburg Str., 74, Donetsk, 83114, Ukraine

Received March 2015 Revised October 2015 Published November 2015

Abstract
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We give relatively simple sufficient conditions on a Fourier multiplier so that it maps functions with the Hölder property with respect to a part of the variables to functions with the Hölder property with respect to all variables. By using these these sufficient conditions we prove solvability in Hölder classes of the initial-boundary value problems for the linearized Cahn-Hilliard equation with dynamic boundary conditions of two types. In addition, Schauders estimates are derived for the solutions corresponding to the problem under study.

Keywords: Cahn-Hilliard equation., Hölder spaces, partial regularity, dynamic boundary conditions, Fourier multipliers.

Mathematics Subject Classification: Primary: 42B15, 42B37; Secondary: 35G15, 35G16, 35R3.

Citation: Sergey P. Degtyarev. On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions. Evolution Equations and Control Theory, 2015, 4 (4) : 391-429. doi: 10.3934/eect.2015.4.391

References:

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References:

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[4]	S. N. Antontsev, C. R. Gonsalves and A. M. Meirmanov, Local existence of classical solutions to the well-posed Helle-Shaw problem, Port. Math. (N.S.), 59 (2002), 435-452.
[5]	J.-H. Bailly, Local existence of classical solutions to first-order parabolic equations describing free boundaries, Nonlinear Anal., 32 (1998), 583-599. doi: 10.1016/S0362-546X(97)00504-X.
[6]	B. V. Basaliy, I. I. Danilyuk and S. P. Degtyarev, Classical solvability of the multidimensional nonstationary filtration problem with free boundary (Russian. English summary), Dokl. Akad. Nauk Ukr. SSR, (1987), 3-7.
[7]	B. V. Bazaliy and S. P. Degtyarev, On classical solvability of the multidimensional Stefan problem for convective motion of a viscous incompresssible fluid, Math. USSR Sb., 60 (1988), 1-17.
[8]	B. V. Bazaliy and S. P. Degtyarev, Solvability of a problem with an unknown boundary between the domains of a parabolic and an elliptic equation, Ukr. Math. J., 41 (1989), 1155-1160. doi: 10.1007/BF01057253.
[9]	B. V. Bazaliy and S. P. Degtyarev, Stefan problem with kinetic and classical conditions at the free boundary, Ukr. Math. J., 44 (1992), 139-148. doi: 10.1007/BF01061735.
[10]	G. I. Bizhanova and V. A. Solonnikov, On problems with free boundaries for second-order parabolic equations, St. Petersburg Mathematical Journal, 12 (2001), 949-981.
[11]	G. I. Bizhanova and V. A. Solonnikov, On some model problems for second order parabolic equations with time derivative in the boundary conditions, St. Petersbg. Math. J., 6 (1995), 1151-1166.
[12]	Y.-K. Cho and D. Kim, A fourier multiplier theorem on the Besov-Lipschits spaces, Korean J. Math., 16 (2008), 85-90.
[13]	P. Constantin, D. Cyrdoba and F. Gancedo, On the global existence for the Muskat problem, J. Fur. Math. Soc. (JEMS), 15 (2013), 201-227. doi: 10.4171/JEMS/360.
[14]	P. Constantin and M. Pugh, Global solutions for small data to the Hele-Shaw problem, Nonlinearity, 6 (1993), 393-415. doi: 10.1088/0951-7715/6/3/004.
[15]	A. Cyrdoba, D. Cyrdoba and F. Gancedo, Porous media: The Muskat problem in three dimensions, Anal. PDE., 6 (1993), 447-497. doi: 10.2140/apde.2013.6.447.
[16]	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 NoDEA, 22 (2015), 185-237. doi: 10.1007/s00030-014-0280-3.
[17]	S. P. Degtyarev, The existence of a smooth interface in the evolutionary elliptic Muskat-Verigin problem with nonlinear source, (Russian) Ukrainian Mathematical Bulletin, 7 (2010), 301-330.
[18]	S. P. Degtyarev, The existence of a smooth interface in the evolutionary elliptic Muskat-Verigin problem with nonlinear source,, , ().
[19]	R. Denk, J. Prüss and R. Zacher, Maximal $L_p$ - regularity of parabolic problems with boundary dynamics of relaxation type, J. Funct. Anal., 255 (2008), 3149-3187. doi: 10.1016/j.jfa.2008.07.012.
[20]	R. Denk and R. Volevich, A new class of parabolic problems connected with Newton's polygon, Discrete Cont. Dyn. Syst., Dynamical Systems and Differential Equations, Proceedings of the 6th AIMS International Conference, suppl., (2007), 294-303.
[21]	R. Denk and L. R. Volevich, Parabolic boundary value problems connected with Newton's polygon and some problems of crystallization, J. Evol. Equ., 8 (2008), 523-556. doi: 10.1007/s00028-008-0392-5.
[22]	R. Denk and M. Kaip, General Parabolic Mixed Order Systems in $L_p$ and Applications, Operator Theory: Advances and Applications, 239, Birkhäuser/Springer, 2013. doi: 10.1007/978-3-319-02000-6.
[23]	P. Dintelmann, Classes of Fourier multipliers and Besov-Nikolskij spaces, Math. Nachr., 173 (1995), 115-130. doi: 10.1002/mana.19951730108.
[24]	H. Dong, Gradient estimates for parabolic and elliptic systems from linear laminates, Arch. Ration. Mech. Anal., 205 (2012), 119-149. doi: 10.1007/s00205-012-0501-z.
[25]	H. Dong and S. Kim, Partial Scauder estimates for second-order elliptic and parabolic equations, Calc. Var. Partial Differential Equations, 40 (2011), 481-500. doi: 10.1007/s00526-010-0348-9.
[26]	J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), 1309-1364. doi: 10.1080/03605309308820976.
[27]	J. Escher and B.-V. Matioc, On the parabolicity of the Muskat problem: Well-posedness, fingering, and stability results, Z. Anal. Anwend., 30 (2011), 193-218. doi: 10.4171/ZAA/1431.
[28]	J. Escher and G. Simonett, Classical solutions of multidimensional Hele-Shaw models, SIAM J. Math. Anal., 28 (1997), 1028-1047. doi: 10.1137/S0036141095291919.
[29]	J. Escher and G. Simonett, Classical solutions for Hele-Shaw models with surface tension, Adv. Differential Equations, 2 (1997), 619-642.
[30]	P. Fife, Schauder estimates under incomplete Hölder continuity assumptions, Pacific J. Math., 13 (1963), 511-550. doi: 10.2140/pjm.1963.13.511.
[31]	A. Friedman, B. Hu and J. J. L. Velazquez, A Stefan problem for a protocell model with symmetry-breaking bifurcations of analitic solutions, Interfaces Free Bound., 3 (2001), 143-199. doi: 10.4171/IFB/37.
[32]	A. Friedman and J. J. L. Velazquez, A free boundary problem associated with crystallization of polymers in a temperature field, Indiana Univ. Math. J., 50 (2001), 1609-1649. doi: 10.1512/iumj.2001.50.2118.
[33]	E. Frolova, Solvability in Sobolev spaces of a problem for a second order parabolic equation with time derivative in the boundary condition, Portugal Math., 56 (1999), 419-441.
[34]	C. G. Gal and H. Wu, Asymptotic behavior of Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation, Discrete Contin. Dyn. Syst., 22 (2008), 1041-1063. doi: 10.3934/dcds.2008.22.1041.
[35]	S. Gindikin and L. R. Volevich, The Method of Newton's Polyhedron in the Theory of Partial Differential Equations, Mathematics and its Applications, 86, Kluwer Academic Publishers Group, Dordrecht, 1992. doi: 10.1007/978-94-011-1802-6.
[36]	M. Girardi and L. Weis, Operator-valued Fourier multiplier theorems on Besov spaces, Math. Nachr., 251 (2003), 34-51. doi: 10.1002/mana.200310029.
[37]	G. R. Goldstein and A. Miranville, A Cahn-Hilliard-Gurtin model with dynamic boundary conditions, Discrete Contin. Dyn. Syst. Ser. S., 6 (2013), 387-400. doi: 10.3934/dcdss.2013.6.387.
[38]	K. K. Golovkin, On equivalent normalizations of fractional spaces, in Automatic Programming, Numerical Methods and Functional Analysis, Trudy Mat. Inst. Steklov., 66, Acad. Sci. USSR, Moscow-Leningrad, 1962, 364-383.
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