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 \({q > \frac{2N}{N+2}}\) and any σ > 1, where q is the exponent of the diffusion term and σ is the exponent which characterizes the damping term.
Highlights
We shall study the existence of weak solutions for the generalized Navier–Stokes equations with damping: div u = 0 in QT
There, we have proved the existence of weak solutions, its uniqueness and some asymptotic properties
We introduce the main result of this work, where it is established the existence of weak solutions to the problem (1.1)–(1.4) under the minor possible assumptions on q and σ
Summary
We shall study the existence of weak solutions for the generalized Navier–Stokes equations with damping: div u = 0 in QT ,. The consideration of damping terms in the generalized Navier–Stokes equations it is useful as a regularization procedure to prove the existence of weak solutions (see [13,14]). In [18] we proved the weak solutions of (1.1)–(1.4) extinct in a finite time for q ≥ 2, provided 1 < σ < 2 This property is well known for the generalized Navier–Stokes problem (1.1)–(1.4) with α = 0 in the case 1 < q < 2. Vol 20 (2013) Existence for the generalized Navier–Stokes equations solution to the same problem.
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