JournalsrsmupVol. 125pp. 51–70

Global Weak Solutions of the Navier-Stokes Equations with Nonhomogeneous Boundary Data and Divergence

  • Reinhard Farwig

    Technische Hochschule Darmstadt, Germany
  • H. Kozono

    Tohoku University, Sendai, Japan
  • H. Sohr

    Universität Paderborn, Germany
Global Weak Solutions of the Navier-Stokes Equations with Nonhomogeneous Boundary Data and Divergence cover
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Abstract

Consider a smooth bounded domain ΩR3\Omega\subseteq\mathbb R^3 with boundary Ω\partial\Omega, a time interval [0,T)[0,T), with T(0,]T\in(0,\infty], and the Navier-Stokes system in [0,T)×Ω[0,T) \times \Omega, with initial value u0Lσ2(Ω)u_0 \in L^2_{\sigma} (\Omega) and external force f=divFf= {\mathrm{div}}\,F, FL2(0,T;L2(Ω))F \in L^2 (0,T;L^2(\Omega)). Our aim is to extend the well-known class of Leray-Hopf weak solutions uu satisfying uΩ=0u_{\vert{\partial \Omega}}=0, divu=0{\mathrm{div}}\,u=0 to the more general class of Leray-Hopf type weak solutions uu with general data uΩ=gu_{\vert{\partial \Omega}} =g, divu=k{\mathrm{div}}\,u=k satisfying a certain energy inequality. Our method rests on a perturbation argument writing uu in the form u=v+Eu=v+E with some vector field EE in [0,T)×Ω[0,T)\times \Omega satisfying the (linear) Stokes system with f=0f=0 and nonhomogeneous data. This reduces the general system to a perturbed Navier-Stokes system with homogeneous data, containing an additional perturbation term. Using arguments as for the usual Navier-Stokes system we get the existence of global weak solutions for the more general system.

Cite this article

Reinhard Farwig, H. Kozono, H. Sohr, Global Weak Solutions of the Navier-Stokes Equations with Nonhomogeneous Boundary Data and Divergence. Rend. Sem. Mat. Univ. Padova 125 (2011), pp. 51–70

DOI 10.4171/RSMUP/125-4