The classical Kelvin–Voigt problem for incompressible fluids with unknown non-constant density: existence, uniqueness and regularity

Nonlinearity - Tập 34 Số 5 - Trang 3083-3111 - 2021
S. N. Antont︠s︡ev1,2, Hermenegildo Borges de Oliveira1,3, Kh. Khompysh4
1CMAF CIO—Universidade de Lisboa, Portugal
2Lavrentyev Institute of Hydrodynamics SB RAS, Novosibirsk, Russia
3FCT-Universidade do Algarve, Faro, Portugal
4Al-Farabi Kazakh National University, Almaty, Kazakhstan

Tóm tắt

Abstract The classical Kelvin–Voigt equations for incompressible fluids with non-constant density are investigated in this work. To the associated initial-value problem endowed with zero Dirichlet conditions on the assumed Lipschitz-continuous boundary, we prove the existence of weak solutions: velocity and density. We also prove the existence of a unique pressure. These results are valid for d ∈ {2, 3, 4}. In particular, if d ∈ {2, 3}, the regularity of the velocity and density is improved so that their uniqueness can be shown. In particular, the dependence of the regularity of the solutions on the smoothness of the given data of the problem is established.

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