An investigation of the singular perturbation problems for the compressible Reynolds equation.
The author studies the problem \[ \nabla \cdot \bigl [ (H^3p+\lambda H^2)\, \nabla p\bigr ]\;=\frac {\partial }{\partial x_1}\bigl [ Hp\bigr ]\; \text{ in } \; \Omega , \] \(p=1\) on \(\partial \Omega . \) The problem is a mathematical model of a flow of air in a thin layer between two surfaces, that are both close to the \(x_1,\, x_2\)-plane. \(H(x)\) denotes the standardized gap thickness between the surfaces at the point \(x=(x_1,x_2)\), \(p\) is the pressure, \(\lambda >0\) is related to the Knudsen number (at atmospheric pressure) and \(\Omega \) is a domain in \(\mathbb {R}^2\), where the flow between the two surfaces takes place. Function \(H\) is assumed to be in \(L^{\infty }(\Omega )\) and decreasing in variable \(x_1\) in \(\Omega \). The considered problem corresponds to the physical situation, when the air flows in a layer between the read--write head and the surface of a rotating hard disk, where the magnetic information is stored. The author deals with function \(H\) in the form \(H(x)=h_0(x)+\varepsilon \), where \(\varepsilon >0\) and \(h_0\) is assumed to satisfy \(\inf _{\overline {\Omega }}\, h_0=0\), \(h_0>0\) in \(\Omega \). If \(F\) is the prescribed vertical force, acting on the upper surface, then the equilibrium position is a situation when \(p\) satisfies the condition \[ \int _{\Omega }p\, dx =F . \] The author solves the problem for \(\varepsilon >0\) and proves a series of results on the convergence of solution \(p\) for \(\varepsilon \) tending to zero in the case of function \(h_0\), satisfying various additional conditions, and on properties of a limit function \(p_0\). These results are finally applied to show the existence of an equilibrium position.
scientific article; zbMATH DE number 5944811