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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.

MSC:

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)
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References:

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