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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Time-dependent coating flows in a strip, Part I: The linearized problem


Authors: Avner Friedman and Juan J. L. Velázquez
Journal: Trans. Amer. Math. Soc. 349 (1997), 2981-3074
MSC (1991): Primary 35R35, 76D05.; Secondary 35J40
MathSciNet review: 1422605
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Abstract: This work is concerned with time-dependent coating flow in a strip $0 < y < 1$. The Navier-Stokes equations are satisfied in the fluid region, the bottom substrate $y = 0$ is moving with fixed velocity $(U,0)$, and fluid is entering the strip through the upper boundary $y = 1$. The free boundary has the form $y = f(x,t)$ for $-\infty < x < R(t)$, where $R(t)$ is the moving contact point. Our objective is to prove that if the initial data are close to those of a stationary solution (the existence of such a solution was established by the authors in an earlier paper) then the time-dependent problem has a unique solution with smooth free boundary, at least for a small time interval. In this Part I we study the linearized problem, about the stationary solution, and obtain sharp estimates for the solution and its derivatives. These estimates will be used in Part II to establish existence and uniqueness for the full nonlinear problem.


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Additional Information

Avner Friedman
Affiliation: Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota 55455

Juan J. L. Velázquez
Affiliation: Departamento de Matematica Aplicada, Universidad Complutense, Facultad de Matematicas, 28040 Madrid, Spain

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01956-9
PII: S 0002-9947(97)01956-9
Keywords: Coating flow, elliptic equations, boundary value problem, Navier-Stokes equations, free boundary problems
Received by editor(s): November 17, 1995
Article copyright: © Copyright 1997 American Mathematical Society