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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
1. | Institute for Applied Mathematics and Mechanics NASU, State Institute for Applied Mathematics and Mechanics, R.Luxenburg Str., 74, Donetsk, 83114, Ukraine |
References:
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H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications,, Math. Nachr., 186 (1997), 5.
doi: 10.1002/mana.3211860102. |
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H. Amann, Anisotropic Function Spaces and Maximal Regularity for Parabolic Problems. Part 1: Function Spaces,, Jindrich Necas Center for Mathematical Modeling Lecture Notes, (2009).
|
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H. Amann, Linear and Quasilinear Parabolic Problems, Volume I, Abstract Linear Theory,, Monographs in Mathematics, (1995).
doi: 10.1007/978-3-0348-9221-6. |
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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.
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J.-H. Bailly, Local existence of classical solutions to first-order parabolic equations describing free boundaries,, Nonlinear Anal., 32 (1998), 583.
doi: 10.1016/S0362-546X(97)00504-X. |
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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.
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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.
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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.
doi: 10.1007/BF01057253. |
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B. V. Bazaliy and S. P. Degtyarev, Stefan problem with kinetic and classical conditions at the free boundary,, Ukr. Math. J., 44 (1992), 139.
doi: 10.1007/BF01061735. |
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G. I. Bizhanova and V. A. Solonnikov, On problems with free boundaries for second-order parabolic equations,, St. Petersburg Mathematical Journal, 12 (2001), 949.
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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.
|
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Y.-K. Cho and D. Kim, A fourier multiplier theorem on the Besov-Lipschits spaces,, Korean J. Math., 16 (2008), 85. Google Scholar |
[13] |
P. Constantin, D. Cyrdoba and F. Gancedo, On the global existence for the Muskat problem,, J. Fur. Math. Soc. (JEMS), 15 (2013), 201.
doi: 10.4171/JEMS/360. |
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P. Constantin and M. Pugh, Global solutions for small data to the Hele-Shaw problem,, Nonlinearity, 6 (1993), 393.
doi: 10.1088/0951-7715/6/3/004. |
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A. Cyrdoba, D. Cyrdoba and F. Gancedo, Porous media: The Muskat problem in three dimensions,, Anal. PDE., 6 (1993), 447.
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.
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.
|
[18] |
S. P. Degtyarev, The existence of a smooth interface in the evolutionary elliptic Muskat-Verigin problem with nonlinear source,, , ().
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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.
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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.
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H. Dong, Gradient estimates for parabolic and elliptic systems from linear laminates,, Arch. Ration. Mech. Anal., 205 (2012), 119.
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H. Dong and S. Kim, Partial Scauder estimates for second-order elliptic and parabolic equations,, Calc. Var. Partial Differential Equations, 40 (2011), 481.
doi: 10.1007/s00526-010-0348-9. |
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J. Escher, Quasilinear parabolic systems with dynamical boundary conditions,, Comm. Partial Differential Equations, 18 (1993), 1309.
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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.
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J. Escher and G. Simonett, Classical solutions of multidimensional Hele-Shaw models,, SIAM J. Math. Anal., 28 (1997), 1028.
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J. Escher and G. Simonett, Classical solutions for Hele-Shaw models with surface tension,, Adv. Differential Equations, 2 (1997), 619.
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P. Fife, Schauder estimates under incomplete Hölder continuity assumptions,, Pacific J. Math., 13 (1963), 511.
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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.
doi: 10.4171/IFB/37. |
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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.
doi: 10.1512/iumj.2001.50.2118. |
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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.
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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.
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G. R. Goldstein and A. Miranville, A Cahn-Hilliard-Gurtin model with dynamic boundary conditions,, Discrete Contin. Dyn. Syst. Ser. S., 6 (2013), 387.
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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.
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, (2009).
|
[3] |
H. Amann, Linear and Quasilinear Parabolic Problems, Volume I, Abstract Linear Theory,, Monographs in Mathematics, (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.
|
[5] |
J.-H. Bailly, Local existence of classical solutions to first-order parabolic equations describing free boundaries,, Nonlinear Anal., 32 (1998), 583.
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] |
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.
|
[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.
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.
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.
|
[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.
|
[12] |
Y.-K. Cho and D. Kim, A fourier multiplier theorem on the Besov-Lipschits spaces,, Korean J. Math., 16 (2008), 85. Google Scholar |
[13] |
P. Constantin, D. Cyrdoba and F. Gancedo, On the global existence for the Muskat problem,, J. Fur. Math. Soc. (JEMS), 15 (2013), 201.
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.
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.
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.
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.
|
[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.
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., (2007), 294.
|
[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.
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, (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.
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.
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.
doi: 10.1007/s00526-010-0348-9. |
[26] |
J. Escher, Quasilinear parabolic systems with dynamical boundary conditions,, Comm. Partial Differential Equations, 18 (1993), 1309.
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.
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.
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.
|
[30] |
P. Fife, Schauder estimates under incomplete Hölder continuity assumptions,, Pacific J. Math., 13 (1963), 511.
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.
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.
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.
|
[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.
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, (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.
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.
doi: 10.3934/dcdss.2013.6.387. |
[38] |
K. K. Golovkin, On equivalent normalizations of fractional spaces,, in Automatic Programming, (1962), 364.
|
[39] |
K. K. Golovkin and V. A. Solonnikov, Bounds for integral operators in translation-invariant norms,, in Boundary Value Problems of Mathematical Physics, (1964), 47.
|
[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, (1966), 5.
|
[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, (1968), 6.
|
[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.
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.
|
[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.
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