The aim of the work is to prove the existence of a weak solution to the initial-boundary value problem for an inhomogeneous incompressible Kelvin–Voigt fluid motion model of an arbitrary finite order. To do this, the paper derives a system of equations corresponding to the considered model. After that, the initial-boundary value problem for the considered model in a bounded domain in 2D and 3D cases is formulated. Then a definition of a weak solution to this problem is given and its existence is proved. For the proof, it is considered some auxiliary problem and its solvability using the Leray–Schauder Theorem is proved. After that, the passage to the limit is carried out as the approximation parameter tends to zero and it is shown that the solutions of the auxiliary problem weakly converge to the solution of the original problem.

这项工作的目的是证明对于任意有限阶的非均匀不可压缩 Kelvin-Voigt 流体运动模型的初始边界值问题存在弱解。为此，本文导出了与所考虑模型相对应的方程组。之后，制定了 2D 和 3D 情况下所考虑模型在有界域中的初始边界值问题。然后给出了该问题弱解的定义并证明了其存在性。为了证明，它被认为是一些辅助问题，并且使用 Leray-Schauder 定理证明了它的可解性。之后，随着近似参数趋于零，进行到极限的传递，表明辅助问题的解弱收敛于原问题的解。