This study examines the multidimensional Navier–Stokes equations in conjunction with the \(k-\varepsilon \) turbulence model to establish the existence of weak solutions for the velocity vector, turbulent kinetic energy, and dissipation rate, denoted as \((\mathbf {\overline{u}}, k, \varepsilon )\) . Under certain conditions in external forces, initial, and boundary data, we prove that weak solutions exist within the \(L^2\) space. The provided result implies the existence and boundedness of solutions, which can be of help for further computational investigations under the scope of the \(k-\varepsilon \) turbulence model.
