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

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

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.
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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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.

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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.

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show all references

References:
[1]

H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr., 186 (1997), 5-56. doi: 10.1002/mana.3211860102.

[2]

H. Amann, Anisotropic Function Spaces and Maximal Regularity for Parabolic Problems. Part 1: Function Spaces, Jindrich Necas Center for Mathematical Modeling Lecture Notes, 6, Matfyzpress, Prague, 2009.

[3]

H. Amann, Linear and Quasilinear Parabolic Problems, Volume I, Abstract Linear Theory, Monographs in Mathematics, 89, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9221-6.

[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.

[39]

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