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

Sergey P.  Degtyarev

doi:10.3934/eect.2015.4.391

Article Contents

2015, Volume 4, Issue 4: 391-429. Doi: 10.3934/eect.2015.4.391

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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

Received: February 28, 2015

Revised: September 30, 2015

Published: November 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 42B15, 42B37; Secondary: 35G15, 35G16, 35R35.

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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, arXiv:1307.0109v2.
[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]	K. K. Golovkin and V. A. Solonnikov, Bounds for integral operators in translation-invariant norms, in Boundary Value Problems of Mathematical Physics, Part 1, Collection of Articles, Trudy Mat. Inst. Steklov. Nauka, 70, Moscow-Leningrad, 1964, 47-58.
[40]	K. K. Golovkin and V. A. Solonnikov, Estimates of integral operators in translation-invariant norms. II, in Boundary Value Problems of Mathematical Physics. Part 4, Trudy Mat. Inst. Steklov. 92, 1966, 5-30.
[41]	K. K. Golovkin and V. A. Solonnikov, On some estimates of convolutions, in Boundary-Value Problems of Mathematical Physics and Related Problems of Function Theory. Part 2, Zap. Nauchn. Sem. LOMI, "Nauka", 7, Leningrad. Otdel., Leningrad, 1968, 6-86.
[42]	B. Grec and E. V. Radkevich, Newton's polygon method and the local solvability of free boundary problems, Journal of Mathematical Sciences, 143 (2007), 3253-3292.doi: 10.1007/s10958-007-0208-0.
[43]	D. Guidetti, The parabolic mixed Cauchy-Dirichlet problem in spaces of functions which are Hölder continuous with respect to space variables, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 7 (1996), 161-168.
[44]	D. Guidetti, Optimal regularity for mixed parabolic problems in spaces of functions which are Hölder continuous with respect to space variables, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 763-790.
[45]	V. N. Gusakov and S. P. Degtyarev, Existence of a smooth solution in a filtration problem, Ukr. Math. J., 41 (1989), 1027-1032.doi: 10.1007/BF01056273.
[46]	L. Hörmander, The Analysis of Linear Partial Differential Operators. I, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, 1983.doi: 10.1007/978-3-642-96750-4.
[47]	S. D. Ivasishen, Green's matrices of boundary value problems for Petrovskii parabolic systems of general form. I, Mathematics of the USSR-Sbornik, 42 (1982), 93-144.
[48]	S. D. Ivasishen, Green's matrices of boundary value problems for Petrovskii parabolic systems of general form. II, Mathematics of the USSR-Sbornik, 42 (1982), 461-498.
[49]	A. I. Komech, Linear partial differential equations with constant coefficients, in Partial Differential Equations II: Elements of the Modern Theory. Equations with Constant Coefficients, Encyclopaedia of Mathematical Sciences, 31, Springer, 2013, 121-255.doi: 10.1007/978-3-642-57876-2_2.
[50]	J. Kovats, Real analytic solutions of parabolic equations with time-measurable coefficients, Proc. Amer. Math. Soc., 130 (2002), 1055-1064.doi: 10.1090/S0002-9939-01-06163-9.
[51]	S. N. Kruzkov, A. Castro and M. M. Lopez, Schauder-type estimates and existence theorems for the solution of basic problems for linear and nonlinear parabolic equations, Sov. Math. Dokl., 16 (1975), 60-64.
[52]	N. Krylov, The Calderon-Zygmund theorem and parabolic equations in $L_p(\mathbbR,C^{2+\alpha})$- spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 1 (2002), 799-820.
[53]	N. V. Krylov, Parabolic equations in $L_p$-spaces with mixed norms, St. Petersburg Mathematical Journal, 14 (2003), 603-614.
[54]	N. Krylov and E. Priola, Elliptic and parabolic second-order PDEs with growing coefficients, Comm. Partial Differential Equations, 35 (2010), 1-22.doi: 10.1080/03605300903424700.
[55]	Y. Kusaka and A. Tani, On the classical solvability of the two-phase Stefan problem in a viscous incompressible fluid flow, Math. Models Methods Appl. Sci., 12 (2002), 365-391.doi: 10.1142/S0218202502001696.
[56]	Y. Kusaka, On a free boundary problem describing the phase transition in an incompressible viscous fluid, Interfaces Free Bound., 12 (2010), 157-185.doi: 10.4171/IFB/231.
[57]	O. A. Ladyzhenskaya, A theorem on multiplicators in nonhomogeneous holder spaces and some of its applications, Journal of Mathematical Sciences (New York), 115 (2003), 2792-2802.doi: 10.1023/A:1023373920221.
[58]	O. A. Ladyzhenskaya, On multiplicators in Hölder spaces with nonhomogeneous metric, Methods. Appl. Anal., 7 (2000), 465-472.
[59]	G. M. Lieberman, Intermediate Schauder theory for second order parabolic equations. IV. Time irregularity and regularity, Differential Integral Equations, 5 (1992), 1219-1236.
[60]	L. Lorenzi, Optimal Schauder estimates for parabolic problems with data measurable with respect to time, SIAM J. Math. Anal., 32 (2000), 588-615.doi: 10.1137/S0036141098342842.
[61]	L. Lorenzi, Optimal Hölder regularity for nonautonomous Kolmogorov equations, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 169-191.doi: 10.3934/dcdss.2011.4.169.
[62]	A. Lunardi, Maximal space regularity in nonhomogeneous initial-boundary value parabolic problem, Numer. Funct. Anal. Optim., 10 (1989), 323-349.doi: 10.1080/01630568908816306.
[63]	A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, 16, Birkhäuser Verlag, Basel, 1995.doi: 10.1007/978-3-0348-9234-6.
[64]	A. M. Meirmanov, On the classical solution of the multidimensional Stefan problem for quasilinear parabolic equations, Math. USSR Sb., 40 (1981), 157-178.
[65]	A. Meirmanov, The Muskat problem for a viscoelastic filtration, Interfaces Free Bound., 13 (2011), 463-484.doi: 10.4171/IFB/268.
[66]	I. Sh. Mogilevskiĭ and V. A. Solonnikov, Solvability of a noncoercive initial-boundary value problem for the Stokes system in Hölder classes of functions (the half-space case), Z. Anal. Anwendungen, 8 (1989), 329-347.
[67]	J. Prüss, R. Racke and S. Zheng, Maximal regularity and a symptotic behavior of solutions for the Cahn-Hilliard equation with dynamic boundary conditions, Anali di Matematica, 185 (2006), 627-648.doi: 10.1007/s10231-005-0175-3.
[68]	J. Prüss, G. Simonett and R. Zacher, Qualitative behavior of solutions for thermodinamically consistent Strfan problems, Arch. Ration. Mech. Anal., 207 (2013), 611-667.doi: 10.1007/s00205-012-0571-y.
[69]	R. Racke and S. Zheng, The Cahn-Hilliard equation with dynamic boundary conditions, Adv. Differential Equations, 8 (2003), 83-110.
[70]	E. V. Radkevich, On conditions for the existence of a classical solution of the modified Stefan problem (Gibbs-Thomson low), Russ. Ac. Sc. Sb. Math., 75 (1993), 221-246.doi: 10.1070/SM1993v075n01ABEH003381.
[71]	E. V. Radkevich, On the spectrum of the pencil in the Verigin-Muskat problem, Russ. Ac. Sc. Sb. Math., 80 (1995), 33-73.doi: 10.1070/SM1995v080n01ABEH003513.
[72]	J. F. Rodrigues and V. A. Solonnikov, On a parabolic system with time derivative in the boundary conditions and related free boundary problems, Math. Ann., 315 (1999), 61-95.doi: 10.1007/s002080050318.
[73]	E. Sinestrari, On the solutions of the first boundary value problem for the linear parabolic equations, Proc. Roy. Soc. Edinburgh, Sect.A., 108 (1988), 339-355.doi: 10.1017/S0308210500014712.
[74]	V. A. Solonnikov, Estimates for solutions of a non-stationary linearized system of Navier-Stokes equations, in Boundary Value Problems of Mathematical Physics. Part 1, Collection of Articles, Trudy Mat. Inst. Steklov. Nauka, 70, Moscow-Leningrad, 1964, 213-317.
[75]	V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general form, in Proceedings of the Steklov Institute of Mathematics, 83, 1965, 3-163.
[76]	V. A. Solonnikov, Estimates of solutions to some noncoercive initial-boundary value problems with the help of a theorem about multipliers in Laplace-Fourier integrals, (Russian) in Functional and Numerical Methods of Mathematical Physics, Naukova Dumka, Kyjiv, 1988, 220-228.
[77]	V. A. Solonnikov, Estimates of solutions of the second initial-boundary problem for the Stokes system in the spaces of functions with Holder continuous derivatives with respect to spatial variables, Journal of Mathematical Sciences (New York), 109 (2002), 1997-2017.doi: 10.1023/A:1014456711451.
[78]	G. Tian and X.-J. Wang, Partial regularity for elliptic equatios, Discrete Contin. Dyn. Syst., 28 (2010), 899-913.doi: 10.3934/dcds.2010.28.899.
[79]	H. Triebel, Theory of Function Spaces II, Modern Birkhauser Classics, Birkhäuser, Basel, 2010.
[80]	J. L. Vázquez and E. Vitillaro, On the Laplace equation with dynamical boundary conditions of reactive-diffusive type, J. Math. Anal. Appl., 354 (2009), 674-688.doi: 10.1016/j.jmaa.2009.01.023.
[81]	J. L. Vázquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive type, Comm. Partial Differential Equations, 33 (2008), 561-612.doi: 10.1080/03605300801970960.
[82]	H. Wu, Convergence to equilibrium for a Cahn-Hilliard model with Wentzell boundary condition, Asymptot. Anal., 54 (2007), 71-92.
[83]	D. Yang, W. Yuan and C. Zhuo, Musielak-Orlicz Besov-type and Triebel-Lizorkin-type spaces, Rev. Mat. Complut., 27 (2014), 93-157.doi: 10.1007/s13163-013-0120-8.
[84]	F. Yi, Local classical solution of Muskat free boundary problem, J. Partial Differential Equations., 9 (1996), 84-96.
[85]	F. Yi, Global classical solution of Muskat free boundary problem, J. Math. Anal. Appl., 288 (2003), 442-461.doi: 10.1016/j.jmaa.2003.09.003.