Abstract
This contribution is concerned with the effective viscosity problem, that is, the homogenization of the steady Stokes system with a random array of rigid particles, for which the main difficulty is the treatment of close particles. Standard approaches in the literature have addressed this issue by making moment assumptions on interparticle distances. Such assumptions, however, prevent clustering of particles, which is not compatible with physically-relevant particle distributions. In this contribution, we take a different perspective and consider moment bounds on the size of clusters of close particles. On the one hand, assuming such bounds, we construct correctors and prove homogenization. On the other hand, based on subcritical percolation techniques, these bounds are shown to hold for various mixing particle distributions with nontrivial clustering. As a by-product of the analysis, we also obtain similar homogenization results for compressible and incompressible linear elasticity with unbounded random stiffness.
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Notes
This minimization problem is clearly well-posed: due to the rigidity constraint the functional is equivalent to \(\int _U |\!{\text {D}}(v)|^2 - f\cdot v\,\mathbbm {1}_{U{\setminus }\mathcal {I}_\varepsilon (U)}\), the latter is weakly lower semicontinuous and coercive on \(H^1_0(U)^d\) by Korn’s inequality, and the incompressibility and rigidity constraints are feasible and weakly closed.
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Acknowledgements
The authors thank Hugo Duminil-Copin for his feedback on the subcritical percolation estimates in Sect. 3. MD acknowledges financial support from the CNRS-Momentum program and from F.R.S.-FNRS, and AG from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 864066).
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Communicated by A. Braides.
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Duerinckx, M., Gloria, A. Continuum Percolation in Stochastic Homogenization and the Effective Viscosity Problem. Arch Rational Mech Anal 247, 26 (2023). https://doi.org/10.1007/s00205-023-01857-w
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DOI: https://doi.org/10.1007/s00205-023-01857-w