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.

在这项工作中研究了具有非恒定密度的不可压缩流体的经典 Kelvin-Voigt 方程。对于在假定的 Lipschitz 连续边界上赋予零狄利克雷条件的相关初值问题，我们证明了弱解的存在：速度和密度。我们还证明了唯一压力的存在。这些结果对d ∈ {2, 3, 4} 有效。特别是，如果d ∈ {2, 3}，则速度和密度的规律性得到改善，从而可以显示它们的唯一性。特别地，建立了解的规律性对问题给定数据的平滑度的依赖性。