Existence of Weak Solutions for a Nonlinear Elliptic System
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Research Article Existence of Weak Solutions for a Nonlinear Elliptic System Ming Fang1 and Robert P. Gilbert2 1 2
Department of Mathematics, Norfolk State University, Norfolk, VA 23504, USA Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA
Correspondence should be addressed to Ming Fang, [email protected] Received 3 April 2009; Accepted 31 July 2009 Recommended by Kanishka Perera We investigate the existence of weak solutions to the following Dirichlet boundary value problem, which occurs when modeling an injection molding process with a partial slip condition on the boundary. We have −Δθ kθ|∇p|r qx in Ω; −div{kθ|∇p|r−2 βx|∇p|r0 −2 ∇p} 0 in Ω; θ θ0 , and p p0 on ∂Ω. Copyright q 2009 M. Fang and R. P. Gilbert. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction Injection molding is a manufacturing process for producing parts from both thermoplastic and thermosetting plastic materials. When the material is in contact with the mold wall surface, one has three choices: i no slip which implies that the material sticks to the surface ii partial slip, and iii complete slip 1–5 . Navier 6 in 1827 first proposed a partial slip condition for rough surfaces, relating the tangential velocity vα to the local tangential shear stress τα3
vα −βτα3 ,
1.1
where β indicates the amount of slip. When β 0, 1.1 reduces to the no-slip boundary condition. A nonzero β implies partial slip. As β → ∞, the solid surface tends to full slip. There is a full description of the injection molding process in 3 and in our paper 7 . The formulation of this process as an elliptic system is given here in after.
2
Boundary Value Problems
Problem 1. Find functions θ and p defined in Ω such that r −Δθ kθ∇p qx − div
r−2 r −2 kθ∇p βx∇p 0 ∇p 0 θ θ0 ,
p p0
1.2
in Ω, in Ω,
on ∂Ω.
1.3 1.4
Here we assume that Ω is a bounded domain in RN with a C1 boundary. We assume also that q, θ0 , p0 , β, and k are given functions, while r is a given positive constant related to the power law index; p is the pressure of the flow, and θ is the temperature. The leading order term βx|∇p|r0 −2 of the PDE 1.3 is derived from a nonlinear slip condition of Navier type. Similar derivations based on the Navier slip condition occur elsewhere, for example, 8, 9 , 10, equation 2.4 . The mathematical model for this system was established in 7 . Some related papers, both rigorous and formal, are 3, 11–13 . In 11, 13 , existence results in no-slip surface, β 0, are obtained, while in 3, 7 , Navier’s slip conditions, β / 0 and r0 0, are investigated, and numerical, existence, uniqueness, and regularity results are given. Although the physical models are two dimensional, we shall carry out our proofs in the case of N dimension. In Section 2, we introduce some notat
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