Solvability of Navier-Stokes equations with leak boundary conditions

Abstract

The time-dependent Navier-Stokes equations with leak boundary conditions are investigated in this paper. The equivalent variational inequality is derived, and the weak and strong solvabilities of this variational inequality are obtained by the Galerkin approximation method and the regularized method. In addition, the continuous dependence property of solutions on given initial data is established, from which the strong solution is unique.

Some properties on the surfaces of vector fields and its application to the Stokes and Navier–Stokes problems with mixed boundary conditions

“…In this way different problems are studied (cf. [7,8,10,34,36,[38][39][40][41][42]55,61,62,66]). In [36] another equivalent variational formulation, where strain, pressure, velocity and rotation are unknown functions, also is given.…”

Two-level variational multiscale finite element methods for Navier–Stokes type variational inequality problem

“…This Uzawa iteration method is based on the following equivalence relationship. It can be shown that Navier-Stokes type variational inequality problem (2.4) is equivalent to the following variational equation: 2 OUT OF MEMORY Table 2 Convergence of two-level Newton iteration VMS method with Re = 1000. From the view of computational cost, we can obviously observe by comparing Tables 1-2 that two-level Newton iteration VMS method saves CPU time than one-level VMS method, and obtains nearly the same approximation results.…”

“…The boundary conditions (1.2) are introduced by H. Fujita in [1]. Subsequently, some well-posedness problems for the steady and nonstationary problems are studied by many scholars, such as R. An, Y. Li and K. Li [2] H. Fujita [3][4][5], T. Kashiwabara [6], Y. Li and K. Li [7,8], Le Roux [9,10], F. Saidi [11], N. Saito [12] and references cited therein. Meanwhile, there also exist some works about the finite element methods for solving the numerical solution to the problem (1.1) and (1.2).…”

Numerical methods for the Stokes and Navier–Stokes equations driven by threshold slip boundary conditions

“…Subsequently, many studies have focused on the properties of the solution of the resulting boundary value problem, for example, existence, uniqueness, regularity, and continuous dependence on data, for Stokes, Navier-Stokes and Brinkman-Forchheimer equations under such boundary conditions. Details can be found in [1,3,11,12,13,14,15,16,17,43,44,45,46] among others.…”