Existence of weak solutions for the generalized Navier–Stokes equations with damping

Abstract

Abstract. In this work we consider the generalized Navier-Stokes equations with the presence of a damping term in the momentum equation. The problem studied here derives from the set of equations which govern isothermal flows of incompressible and homogeneous non-Newtonian fluids. For the generalized Navier-Stokes problem with damping, we prove the existence of weak solutions by using regularization techniques, the theory of monotone operators and compactness arguments together with the local decomposition of the pressure and the Lipschitz-truncation method. The existence result proved here holds for any q > 2N N +2 and any σ > 1, where q is the exponent of the diffusion term and σ is the exponent which characterizes the damping term. Mathematics Subject Classification (2010

Superconvergence analysis of the bilinear‐constant scheme for two‐dimensional incompressible convective Brinkman–Forchheimer equations

“…Much work has been devoted to theoretical analysis of partial differential equations with the term $$$ \alpha {\left|u\right|}^{r-2}u $$$ $$$ \left(r\ge 2\right) $$$, which was called the damping term or source term in [5‐8]. As for the Stokes/Navier–Stokes equations with damping, we refer to [1, 9‐18] and the references therein.…”

“…Much work has been devoted to theoretical analysis of partial differential equations with the term 𝛼|u| r−2 u (r ≥ 2), which was called the damping term or source term in [5][6][7][8]. As for the Stokes/Navier-Stokes equations with damping, we refer to [1,[9][10][11][12][13][14][15][16][17][18] and the references therein.…”

Stochastic 3D Navier-Stokes equations with nonlinear damping: martingale solution, strong solution and small time large deviation principles

“…In [35], for α = 1, Zhou proved that the strong solution exists globally for β ≥ 3 and strong-weak uniqueness for β ≥ 1, and established two regularity criteria as 1 ≤ β ≤ 3. Oliveira has studied the existence of weak solutions for the generalized Navier-Stokes equations with damping [24] for non-Newtonian fluids.…”

Global attractors of the 3D micropolar equations with damping term

“…When w = 0, k = 0, f 1 = 0 and f 2 = 0, the system (1) is Navier-Stokes equations with damping. In the past several years, there were many authors who were devoted to research three-dimensional Navier-Stokes equations with damping [1,6,9,10,12,19]. In [1,10], the well-posedness of solutions for threedimensional Navier-Stokes equations with damping was showed.…”

“…Zhou [19] considered the regularity and uniqueness of the three-dimensional Navier-Stokes equations with damping. In [12], Oliveira showed the existence of weak solution. In [6], the existence of strong global attractors for three-dimensional Navier-Stokes equations with damping was proved.…”