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Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise
$L_p$-theory for a Cahn-Hilliard-Gurtin system
1. | Institut für Mathematik, Martin-Luther Universität Halle-Wittenberg, 06099 Halle, Germany |
References:
[1] |
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function Spaces, Differential Operators and Nonlinear Analysis" (eds. H.-J. Schmeisser and H. Triebel), Teubner-Texte Math., Teubner, Stuttgart, 133 (1993), 9-126. |
[2] |
H. Amann, "Linear and Quasilinear Parabolic Problems," Vol. I, Monographs in Mathematics, vol. 89, Birkhäuser Boston Inc., Boston, MA, 1995. |
[3] |
A. Bonfoh and A. Miranville, On Cahn-Hilliard-Gurtin equations, Proceedings of the Third World Congress of Nonlinear Analysts, Part 5 (Catania 2000), 47 (2001), 3455-3466. |
[4] |
R. Chill, E. Fašangová and J. Prüss, Convergence to steady state of solutions of the Cahn-Hilliard and Caginalp equations with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462.
doi: 10.1002/mana.200410431. |
[5] |
R. Denk, M. Hieber and J. Prüss, $\mathcal R$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), viii+114. |
[6] |
R. Denk, M. Hieber and J. Prüss, Optimal $L_p$-$L_q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224.
doi: 10.1007/s00209-007-0120-9. |
[7] |
G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201.
doi: 10.1007/BF01163654. |
[8] |
M. Girardi and L. Weis, Criteria for R-boundedness of operator families, Evolution equations, Lecture Notes in Pure and Appl. Math., Dekker, New York, 234 (2003), 203-221. |
[9] |
M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192.
doi: 10.1016/0167-2789(95)00173-5. |
[10] |
N. J. Kalton and L. Weis, The $H^ \infty$-calculus and sums of closed operators, Math. Ann., 321 (2001), 319-345.
doi: 10.1007/s002080100231. |
[11] |
M. Köhne, J. Prüss and M. Wilke, On quasilinear parabolic evolution equations in weighted $L_p$-spaces, J. Evol. Equ., 10 (2010), 443-463.
doi: 10.1007/s00028-010-0056-0. |
[12] |
A. Miranville, Existence of solutions for a Cahn-Hilliard-Gurtin model, C. R. Acad. Sci. Paris Sér. I Math., 331 (2000), 845-850.
doi: 10.1016/S0764-4442(00)01731-6. |
[13] |
A. Miranville, Generalized Cahn-Hilliard equations based on a microforce balance, J. Appl. Math., (2003), 165-185. |
[14] |
A. Miranville, A. Piétrus and J. M. Rakotoson, Dynamical aspect of a generalized Cahn-Hilliard equation based on a microforce balance, Asymptot. Anal., 16 (1998), 315-345. |
[15] |
A. Miranville and A. Rougirel, Local and asymptotic analysis of the flow generated by the Cahn-Hilliard-Gurtin equations, Z. Angew. Math. Phys., 57 (2006), 244-268.
doi: 10.1007/s00033-005-0017-6. |
[16] |
A. Miranville and S. Zelik, Doubly nonlinear Cahn-Hilliard-Gurtin equations, Hokkaido Math. J., 38 (2009), 315-360. |
[17] |
A. Miranville, Consistent models of Cahn-Hilliard-Gurtin equations with Neumann boundary conditions, Phys. D, 158 (2001), 233-257.
doi: 10.1016/S0167-2789(01)00317-7. |
[18] |
A. Miranville and A. Piétrus, A new formulation of the Cahn-Hilliard equation, Nonlinear Anal. Real World Appl., 7 (2006), 285-307. |
[19] |
J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces, Math. Z., 203 (1990), 429-452.
doi: 10.1007/BF02570748. |
[20] |
J. Prüss and M. Wilke, On conserved Penrose-Fife type systems, Parabolic Problems, The Herbert Amann Festschrift, Progress in nonlinear differential equations and their applications, Birkhäuser, Basel, 80 (2011), 541-576. |
[21] |
R. Seeley, Interpolation in $L^p$ with boundary conditions, Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I. Studia Math., 44 (1972), 47-60. |
show all references
References:
[1] |
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function Spaces, Differential Operators and Nonlinear Analysis" (eds. H.-J. Schmeisser and H. Triebel), Teubner-Texte Math., Teubner, Stuttgart, 133 (1993), 9-126. |
[2] |
H. Amann, "Linear and Quasilinear Parabolic Problems," Vol. I, Monographs in Mathematics, vol. 89, Birkhäuser Boston Inc., Boston, MA, 1995. |
[3] |
A. Bonfoh and A. Miranville, On Cahn-Hilliard-Gurtin equations, Proceedings of the Third World Congress of Nonlinear Analysts, Part 5 (Catania 2000), 47 (2001), 3455-3466. |
[4] |
R. Chill, E. Fašangová and J. Prüss, Convergence to steady state of solutions of the Cahn-Hilliard and Caginalp equations with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462.
doi: 10.1002/mana.200410431. |
[5] |
R. Denk, M. Hieber and J. Prüss, $\mathcal R$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), viii+114. |
[6] |
R. Denk, M. Hieber and J. Prüss, Optimal $L_p$-$L_q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224.
doi: 10.1007/s00209-007-0120-9. |
[7] |
G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201.
doi: 10.1007/BF01163654. |
[8] |
M. Girardi and L. Weis, Criteria for R-boundedness of operator families, Evolution equations, Lecture Notes in Pure and Appl. Math., Dekker, New York, 234 (2003), 203-221. |
[9] |
M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192.
doi: 10.1016/0167-2789(95)00173-5. |
[10] |
N. J. Kalton and L. Weis, The $H^ \infty$-calculus and sums of closed operators, Math. Ann., 321 (2001), 319-345.
doi: 10.1007/s002080100231. |
[11] |
M. Köhne, J. Prüss and M. Wilke, On quasilinear parabolic evolution equations in weighted $L_p$-spaces, J. Evol. Equ., 10 (2010), 443-463.
doi: 10.1007/s00028-010-0056-0. |
[12] |
A. Miranville, Existence of solutions for a Cahn-Hilliard-Gurtin model, C. R. Acad. Sci. Paris Sér. I Math., 331 (2000), 845-850.
doi: 10.1016/S0764-4442(00)01731-6. |
[13] |
A. Miranville, Generalized Cahn-Hilliard equations based on a microforce balance, J. Appl. Math., (2003), 165-185. |
[14] |
A. Miranville, A. Piétrus and J. M. Rakotoson, Dynamical aspect of a generalized Cahn-Hilliard equation based on a microforce balance, Asymptot. Anal., 16 (1998), 315-345. |
[15] |
A. Miranville and A. Rougirel, Local and asymptotic analysis of the flow generated by the Cahn-Hilliard-Gurtin equations, Z. Angew. Math. Phys., 57 (2006), 244-268.
doi: 10.1007/s00033-005-0017-6. |
[16] |
A. Miranville and S. Zelik, Doubly nonlinear Cahn-Hilliard-Gurtin equations, Hokkaido Math. J., 38 (2009), 315-360. |
[17] |
A. Miranville, Consistent models of Cahn-Hilliard-Gurtin equations with Neumann boundary conditions, Phys. D, 158 (2001), 233-257.
doi: 10.1016/S0167-2789(01)00317-7. |
[18] |
A. Miranville and A. Piétrus, A new formulation of the Cahn-Hilliard equation, Nonlinear Anal. Real World Appl., 7 (2006), 285-307. |
[19] |
J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces, Math. Z., 203 (1990), 429-452.
doi: 10.1007/BF02570748. |
[20] |
J. Prüss and M. Wilke, On conserved Penrose-Fife type systems, Parabolic Problems, The Herbert Amann Festschrift, Progress in nonlinear differential equations and their applications, Birkhäuser, Basel, 80 (2011), 541-576. |
[21] |
R. Seeley, Interpolation in $L^p$ with boundary conditions, Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I. Studia Math., 44 (1972), 47-60. |
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