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Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions
Semilinear stochastic equations with bilinear fractional noise
1. | Dpto. Ecuaciones Diferenciales y Análisis Numérico , Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla |
2. | Charles University in Prague, Faculty of Mathematics and Physics, Sokolovska 83, Prague 8, Czech Republic, Czech Republic |
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-662-12878-7. |
[2] |
H. Bessaih, M. J. Garrido-Atienza and B. Schmalfuß, Stochastic Shell Models driven by a multiplicative fractional Brownian motion, Physica D: Nonlinear Phenomena, 320 (2016), 38-56.
doi: 10.1016/j.physd.2016.01.008. |
[3] |
C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin, 1977. |
[4] |
Y. Chen, H. Gao, M. J. Garrido-Atienza and B. Schmalfuß, Pathwise solutions of SPDEs and random dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 79-98.
doi: 10.3934/dcds.2014.34.79. |
[5] |
J. W. Cholewa and T. Dlotko, Cauchy problems in weighted Lebesgue spaces, Czechoslovak Math. J., 54 (2004), 991-1013.
doi: 10.1007/s10587-004-6447-z. |
[6] |
G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996.
doi: 10.1017/CBO9780511662829. |
[7] |
T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise, Stochastic Process. Appl., 115 (2005), 1357-1383.
doi: 10.1016/j.spa.2005.03.011. |
[8] |
F. Flandoli and B. Schmalfuß, Random attractors for the $3$D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics Stochastics Rep., 59 (1996), 21-45.
doi: 10.1080/17442509608834083. |
[9] |
H. Gao, M. J. Garrido-Atienza, B. Schmalfuß, Random attractors for stochastic evolution equations driven by fractional Brownian motion, SIAM J. Math. Anal., 46 (2014), 2281-2309.
doi: 10.1137/130930662. |
[10] |
M. J. Garrido-Atienza, P. Kloeden and A. Neuenkirch, Discretization of stationary solutions of stochastic systems driven by fractional Brownian motion, Applied Mathematics and Optimization, 60 (2009), 151-172.
doi: 10.1007/s00245-008-9062-9. |
[11] |
M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion, Discrete and Continuous Dynamical System, Series B, 14 (2010), 473-493.
doi: 10.3934/dcdsb.2010.14.473. |
[12] |
M. J. Garrido-Atienza, B. Maslowski and B. Schmalfuß, Random attractors for stochastic equations driven by a fractional Brownian motion, International Journal of Bifurcation and Chaos, 20 (2010), 2761-2782.
doi: 10.1142/S0218127410027349. |
[13] |
M. J. Garrido-Atienza and B. Schmalfuß, Ergodicity of the infinite dimensional fractional Brownian motion, Journal of Dynamics and Differential Equations, 23 (2011), 671-681.
doi: 10.1007/s10884-011-9222-5. |
[14] |
B. Gess, Random Attractors for Stochastic Porous Media Equations perturbed by space-time linear multiplicative noise, C.R. Acad. Sci. Paris, Ser. I, 350 (2012), 299-302.
doi: 10.1016/j.crma.2012.02.004. |
[15] |
B. Gess, W. Liu and M. Röckner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, Journal of Differential Equations, 251 (2011), 1225-1253.
doi: 10.1016/j.jde.2011.02.013. |
[16] |
H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics, 24, Cambridge University Press, Cambridge, 1990. |
[17] |
M. B. Marcus and J. Rosen, Markov Processes, Gaussian Processes, and Local Times, Cambridge Studies in Advanced Mathematics, 100, Cambridge University Press, Cambridge, 2006.
doi: 10.1017/CBO9780511617997. |
[18] |
B. Maslowski and D. Nualart, Evolution equations driven by a fractional Brownian motion, J. Funct. Anal., 202 (2003), 277-305.
doi: 10.1016/S0022-1236(02)00065-4. |
[19] |
B. Maslowski and B. Schmalfuß, Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion, Stochastic Anal. Appl., 22 (2004), 1577-1607.
doi: 10.1081/SAP-200029498. |
[20] |
B. Maslowski and J. Šnupárková, Stochastic equations with multiplicative fractional noise in Hilbert space, preprint, arXiv:1609.00582. |
[21] |
D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81. |
[22] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[23] |
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, 1993. |
[24] |
B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations, in Int. Seminar on Applied Mathematics Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V. Reitmann, T. Riedrich and N. Koksch), 1992, 185-192. |
[25] |
B. Schmalfuß, Attractors for the nonautonomous dynamical systems, in Int. Conf. Differential Equations, Vol. 1, 2 (Berlin, 1999), World Sci. Publ., River Edge, NJ, 2000, 684-689. |
[26] |
J. Šnupárková, Stochastic bilinear equations with fractional Gaussian noise in Hilbert space, Acta Univ. Carolin. Math. Phys., 51 (2010), 49-67. |
[27] |
M. Zähle, Integration with respect to fractal functions and stochastic calculus I, Probab. Theory Relat. Fields, 111 (1998), 333-374.
doi: 10.1007/s004400050171. |
[28] |
M. Zähle, Integration with respect to fractal functions and stochastic calculus II, Math. Nachr., 225 (2001), 145-183.
doi: 10.1002/1522-2616(200105)225:1<145::AID-MANA145>3.0.CO;2-0. |
show all references
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-662-12878-7. |
[2] |
H. Bessaih, M. J. Garrido-Atienza and B. Schmalfuß, Stochastic Shell Models driven by a multiplicative fractional Brownian motion, Physica D: Nonlinear Phenomena, 320 (2016), 38-56.
doi: 10.1016/j.physd.2016.01.008. |
[3] |
C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin, 1977. |
[4] |
Y. Chen, H. Gao, M. J. Garrido-Atienza and B. Schmalfuß, Pathwise solutions of SPDEs and random dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 79-98.
doi: 10.3934/dcds.2014.34.79. |
[5] |
J. W. Cholewa and T. Dlotko, Cauchy problems in weighted Lebesgue spaces, Czechoslovak Math. J., 54 (2004), 991-1013.
doi: 10.1007/s10587-004-6447-z. |
[6] |
G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996.
doi: 10.1017/CBO9780511662829. |
[7] |
T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise, Stochastic Process. Appl., 115 (2005), 1357-1383.
doi: 10.1016/j.spa.2005.03.011. |
[8] |
F. Flandoli and B. Schmalfuß, Random attractors for the $3$D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics Stochastics Rep., 59 (1996), 21-45.
doi: 10.1080/17442509608834083. |
[9] |
H. Gao, M. J. Garrido-Atienza, B. Schmalfuß, Random attractors for stochastic evolution equations driven by fractional Brownian motion, SIAM J. Math. Anal., 46 (2014), 2281-2309.
doi: 10.1137/130930662. |
[10] |
M. J. Garrido-Atienza, P. Kloeden and A. Neuenkirch, Discretization of stationary solutions of stochastic systems driven by fractional Brownian motion, Applied Mathematics and Optimization, 60 (2009), 151-172.
doi: 10.1007/s00245-008-9062-9. |
[11] |
M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion, Discrete and Continuous Dynamical System, Series B, 14 (2010), 473-493.
doi: 10.3934/dcdsb.2010.14.473. |
[12] |
M. J. Garrido-Atienza, B. Maslowski and B. Schmalfuß, Random attractors for stochastic equations driven by a fractional Brownian motion, International Journal of Bifurcation and Chaos, 20 (2010), 2761-2782.
doi: 10.1142/S0218127410027349. |
[13] |
M. J. Garrido-Atienza and B. Schmalfuß, Ergodicity of the infinite dimensional fractional Brownian motion, Journal of Dynamics and Differential Equations, 23 (2011), 671-681.
doi: 10.1007/s10884-011-9222-5. |
[14] |
B. Gess, Random Attractors for Stochastic Porous Media Equations perturbed by space-time linear multiplicative noise, C.R. Acad. Sci. Paris, Ser. I, 350 (2012), 299-302.
doi: 10.1016/j.crma.2012.02.004. |
[15] |
B. Gess, W. Liu and M. Röckner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, Journal of Differential Equations, 251 (2011), 1225-1253.
doi: 10.1016/j.jde.2011.02.013. |
[16] |
H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics, 24, Cambridge University Press, Cambridge, 1990. |
[17] |
M. B. Marcus and J. Rosen, Markov Processes, Gaussian Processes, and Local Times, Cambridge Studies in Advanced Mathematics, 100, Cambridge University Press, Cambridge, 2006.
doi: 10.1017/CBO9780511617997. |
[18] |
B. Maslowski and D. Nualart, Evolution equations driven by a fractional Brownian motion, J. Funct. Anal., 202 (2003), 277-305.
doi: 10.1016/S0022-1236(02)00065-4. |
[19] |
B. Maslowski and B. Schmalfuß, Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion, Stochastic Anal. Appl., 22 (2004), 1577-1607.
doi: 10.1081/SAP-200029498. |
[20] |
B. Maslowski and J. Šnupárková, Stochastic equations with multiplicative fractional noise in Hilbert space, preprint, arXiv:1609.00582. |
[21] |
D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81. |
[22] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[23] |
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, 1993. |
[24] |
B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations, in Int. Seminar on Applied Mathematics Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V. Reitmann, T. Riedrich and N. Koksch), 1992, 185-192. |
[25] |
B. Schmalfuß, Attractors for the nonautonomous dynamical systems, in Int. Conf. Differential Equations, Vol. 1, 2 (Berlin, 1999), World Sci. Publ., River Edge, NJ, 2000, 684-689. |
[26] |
J. Šnupárková, Stochastic bilinear equations with fractional Gaussian noise in Hilbert space, Acta Univ. Carolin. Math. Phys., 51 (2010), 49-67. |
[27] |
M. Zähle, Integration with respect to fractal functions and stochastic calculus I, Probab. Theory Relat. Fields, 111 (1998), 333-374.
doi: 10.1007/s004400050171. |
[28] |
M. Zähle, Integration with respect to fractal functions and stochastic calculus II, Math. Nachr., 225 (2001), 145-183.
doi: 10.1002/1522-2616(200105)225:1<145::AID-MANA145>3.0.CO;2-0. |
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