Stochastic shear thickening fluids: strong convergence of the Galerkin approximation and the energy equality. (English) Zbl 1242.76146

Summary: We consider a stochastic partial differential equation (SPDE) which describes the velocity field of a viscous, incompressible non-Newtonian fluid subject to a random force. Here, the extra stress tensor of the fluid is given by a polynomial of degree \(p - 1\) of the rate of strain tensor, while the colored noise is considered as a random force. We focus on the shear thickening case, more precisely, on the case \(p \in [1 + d/2, 2d/(d - 2))\), where \(d\) is the dimension of the space. We prove that the Galerkin scheme approximates the velocity field in a strong sense. As a consequence, we establish the energy equality for the velocity field.


76M10 Finite element methods applied to problems in fluid mechanics
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
35Q35 PDEs in connection with fluid mechanics
76A05 Non-Newtonian fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
Full Text: DOI arXiv Euclid


[1] Da Prato, G. and Debussche, A. (2002). Two-dimensional Navier-Stokes equations driven by a space-time white noise. J. Funct. Anal. 196 180-210. · Zbl 1013.60051 · doi:10.1006/jfan.2002.3919
[2] Flandoli, F. (2008). An introduction to 3D stochastic fluid dynamics. In SPDE in Hydrodynamic : Recent Progress and Prospects. Lecture Notes in Math. 1942 51-150. Springer, Berlin. · Zbl 1426.76001 · doi:10.1007/978-3-540-78493-7_2
[3] Flandoli, F. and Gatarek, D. (1995). Martingale and stationary solutions for stochastic Navier-Stokes equations. Probab. Theory Related Fields 102 367-391. · Zbl 0831.60072 · doi:10.1007/BF01192467
[4] Flandoli, F., Gubinelli, M., Hairer, M. and Romito, M. (2008). Rigorous remarks about scaling laws in turbulent fluids. Comm. Math. Phys. 278 1-29. · Zbl 1140.76011 · doi:10.1007/s00220-007-0398-9
[5] Grafakos, L. (2004). Classical and Modern Fourier Analysis . Pearson Education, Upper Saddle River, NJ. · Zbl 1148.42001
[6] Kuksin, S. B. (2006). Randomly Forced Nonlinear PDEs and Statistical Hydrodynamics in 2 Space Dimensions . Eur. Math. Soc., Zürich. · Zbl 1099.35083
[7] Málek, J., Nečas, J., Rokyta, M. and Ružička, M. (1996). Weak and Measure-Valued Solutions to Evolutionary PDEs. Applied Mathematics and Mathematical Computation 13 . Chapman and Hall, London. · Zbl 0851.35002
[8] Taylor, M. E. (1996). Partial Differential Equations. III . Springer, New York. · Zbl 1206.35003 · doi:10.1007/978-1-4419-7052-7
[9] Terasawa, Y. and Yoshida, N. (2011). Stochastic power law fluids: Existence and uniqueness of weak solutions. Ann. Appl. Probab. 21 1827-1859. · Zbl 1230.60068 · doi:10.1214/10-AAP741
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.