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AIMS Mathematics

2018, Volume 3, Issue 4: 539-553. doi: 10.3934/Math.2018.4.539
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Lp-solutions of the Navier-Stokes equation with fractional Brownian noise

  • 1.
    Dipartimento di Matematica, Università di 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
  • Received: 07 September 2018 Accepted: 06 November 2018 Published: 14 November 2018

  • 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 $p>d$.

    Citation: Benedetta Ferrario, Christian Olivera. Lp-solutions of the Navier-Stokes equation with fractional Brownian noise[J]. AIMS Mathematics, 2018, 3(4): 539-553. doi: 10.3934/Math.2018.4.539

    Related Papers:

  • 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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    [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, γ-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.
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