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

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.

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Correspondence to H. B. de Oliveira.

Additional information

The author’s work was supported by the grant SFRH/BSAB/1058/2010, MCTES, Portugal, and by the research project PTDC/MAT/110613/2010, FCT, Portugal.

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de Oliveira, H.B. Existence of weak solutions for the generalized Navier–Stokes equations with damping. Nonlinear Differ. Equ. Appl. 20, 797–824 (2013). https://doi.org/10.1007/s00030-012-0180-3

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Mathematics Subject Classification (2010)

  • 35D05
  • 35K55
  • 35Q30
  • 76D03
  • 76D05

Keywords

  • Generalized Navier–Stokes
  • Damping
  • Existence of weak solutions
  • Decomposition of the pressure
  • Lipschitz truncation