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) |
Keywords:
stochastic partial differential equation; power law fluids; Galerkin approximation; energy equalityReferences:
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