Subdifferential boundary value problems for the nonstationary navier-stokes equations

Solvability of Navier-Stokes equations with leak boundary conditions

“…From Lemma 5 and Theorem 2, we have (11). Then there admits T < T such that u ε is also a strong solution in [0, T ) and satisfies (24).…”

“…u ε is a weak solution to the variational inequality (11) if and only if u ε is a weak solution to the variational problem (12).…”

“…For given u 0 ∈ H τ , f ∈ L 2 (0, T, V ) and g ∈ L 2 (0, T, L 2 (S)), there exists at least a weak solution u ε of the variational inequality (11) satisfying (13).…”

“…Some hydrodynamics problems with the leak boundary conditions (3) have been discussed by many scholars, such as [3][4][5][6]11,14,15] and the references therein. Fujita and Saito in [4][5][6]14,15] mainly discussed the stationary and non-stationary Stokes equations and obtained the regularized solutions.…”

“…Chebotarev in [3] dealt with the nonlinear stationary Navier-Stokes equations and obtained the existence of weak solutions via a limited tangential component of velocity. For time-dependent nonlinear Navier-Stokes equations, Konovalova in [11] proved the weak solvability under the assumption that…”