Weak solutions to lubrication equations in the presence of strong slippage

Abstract

The existence of global weak solutions is proved for one-dimensional lubrication models that describe the dewetting process of nanoscopic thin polymer films on hydrophobyzed substrates and take account of large slippage at the polymer-substrate interface. The convergence of these solutions as either the Reynolds number or the capillarity goes to zero, as well as their limiting behaviour as the slip length goes to zero or infinity are investigated. *

Asymptotic decay and non-rupture of viscous sheets

“…For compatibility with boundary conditions (19) and conservation of Lagrange function (18) one has to impose the following integral condition onto initial data:…”

“…h(x, 0) = h 0 (x) > 0, v(x, 0) = v 0 (x) for x ∈ [0, 1], was shown by Kitavtsev et al [18] via an additional introduction of the regularising Lennard-Jones potential in (1a). Observe, that the boundary conditions (2) for v guarantee the mass conservation…”

On the Existence of Global Weak Solutions to 1D Pollutant Transport Model

“…The authors in (Bresch, Desjardins & Grard-Varet, 2007;Bresch, Desjardins & Lin, 2003;Marche, 2007;Mellet & Vasseur, 2007) use the BD entropy inequality to get the existence results of global weak solutions for Shallow-Water and viscous compressible Navier-Stokes equations. In (Kitavtsev, Laurençot & Niethammer, 2011) the authors proved the existence of global weak solutions for one-dimensional lubrication models that describe the dewetting process of nanoscopic thin polymer films on hy-drophobyzed substrates and take into account a large slippage at the polymer-substrate interface. In their work, the authors have used an intermolecular force that is very crucial in ours.…”

“…Proof: We follow the lines performed in (Kitavtsev, Laurençot & Niethammer, 2011). Using the bound on ∂ x h 1 we obtain:…”

“…(α > 0), see (Kitavtsev, Laurençot & Niethammer, 2011;Seemann, Herminghaus & Jacobs, 2001). This force of Van Der Waals allows us to lower the height of water which allows us to get around hypothesis made in (Roamba, Zabsonré& Zongo, 2017).…”