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Lp-solutions of the Navier-Stokes equation with fractional Brownian noise

1 Dipartimento di Matematica, Universitàdi Pavia, via Ferrata 5, 27100 Pavia, Italy
2 Departamento de Matemática, Universidade Estadual de Campinas, Rua Sergio Buarque de Holanda 651, 13083-859 Campinas, SP, Brazil

Topical Section: Mathematical Analysis in Fluid Dynamics

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We study the Navier-Stokes equations on a smooth bounded domain $D\subset \mathbb R^d$ ($d=2$ or 3), under the effect of an additive fractional Brownian noise. We show local existence and uniqueness of a mild $L^p$-solution for $p>d$.
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Citation: Benedetta Ferrario, Christian Olivera. Lp-solutions of the Navier-Stokes equation with fractional Brownian noise. AIMS Mathematics, 2018, 3(4): 539-553. doi: 10.3934/Math.2018.4.539

References

• 1. S. Albeverio, B. Ferrario, Some Methods of Infinite Dimensional Analysis in Hydrodynamics: An Introduction, In: SPDE in Hydrodynamic: Recent Progress and Prospects, Lecture Notes in Mathematics 1942, Berlin: Springer, 1–50, 2008.
• 2. A. Bensoussan, R. Temam, Équations stochastiques du type Navier-Stokes, J. Funct. Anal., 13 (1973), 195–222.
• 3. Z. BrzÉzniak, J. van Neerven, D. Salopek, Stochastic evolution equations driven by Liouville fractional Brownian motion, Czech. Math. J., 62 (2012), 1–27.
• 4. P. Čoupek, B. Maslowski, M. Ondreját, Lp-valued stochastic convolution integral driven by Volterra noise, Stoch. Dynam., (2018), 1850048.
• 5. T. E. Duncan, B. Pasik-Duncan, B. Maslowski, Fractional Brownian motion and stochastic equations in Hilbert spaces, Stoch. Dynam., 2 (2002), 225–250.
• 6. L. Fang, P. Sundar, F. Viens, Two-dimensional stochastic Navier-Stokes equations with fractional Brownian noise, Random Operators and Stochastic Equations, 21 (2013), 135–158.
• 7. F. Flandoli, Dissipativity and invariant measures for stochastic Navier-Stokes equations, NoDEANonlinear Differential Equations Appl., 1 (1994), 403–423.
• 8. F. Flandoli, An introduction to 3D stochastic fluid dynamics, In: SPDE in Hydrodynamic: Recent Progress and Prospects, Lecture Notes in Math. 1942, Berlin: Springer, 51–150, 2008.
• 9. H. Fujita, T. Kato, On the Navier-Stokes initial value problem. I, Arch. Ration. Mech. Anal., 16 (1964), 269–315.
• 10. Y. Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system, J. Differ. Equations, 62 (1986), 186–212.
• 11. Y. Giga, T. Miyakawa, Solutions in Lr of the Navier-Stokes initial value problem, Arch. Ration. Mech. An., 89 (1985), 267–281.
• 12. T. Kato, H. Fujita, On the nonstationary Navier-Stokes system, Rend. Semin. Mat. U. Pad., 32 (1962), 243–260.
• 13. T. Kato, Strong Lp-solutions of the Navier-Stokes equation in Rm, with applications to weak solutions, Math. Z., 187 (1984), 471–480.
• 14. S. B. Kuksin, A. Shirikyan, Mathematics of two-dimensional turbulence, Vol. 194, Cambridge: Cambridge University Press, 2012.
• 15. P. G. Lemarie-Rieusset, Recent developments in the Navier-Stokes problem, CRC Press, 2002.
• 16. J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193–248.
• 17. P-L Lions, Mathematical topics in fluid mechanics, Vol. I: incompressible models, Oxford Lecture Series in Mathematics and its applications, Oxford University Press, 1996.
• 18. D. Nualart, Malliavin Calculus and Related Topics, Second Edition, New York: Springer, 2006.
• 19. B. Pasik-Duncan, T. E. Duncan, B. Maslowski, Linear stochastic equations in a Hilbert space with a fractional Brownian motion, In: Stochastic processes, optimization, and control theory: applications in financial engineering, queueing networks, and manufacturing systems, 201–221, Springer, New York, 2006.
• 20. H. Sohr, The Navier-Stokes equations. An elementary functional analytic approach, Birkhäuser Advanced Texts, Basel: Birkhäuser Verlag, 2001.
• 21. J. van Neerven, $\gamma$-radonifying operators – a survey, In: Proc. Centre Math. Appl. Austral. Nat. Univ. 44, Austral. Nat. Univ., The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis, Canberra, 1–61, 2010.
• 22. M. J. Vishik, A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Dordrecht: Kluver, 1980.
• 23. F. B. Weissler, The Navier-Stokes initial value problem in Lp, Arch. Ration. Mech. An., 74 (1980), 219–230.