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Applications of variational inequalities to a moving boundary problem for Hele Shaw flows. (English) Zbl 0605.76043
We consider a class of two-dimensional moving boundary problems, originating from a Hele Shaw flow problem. Concepts of classical and weak solutions are introduced. We show that a classical solution also is a weak solution and, by using variational inequalities, that given arbitrary initial $$(t=0)$$ data there exists a unique weak solution defined on the time interval $$0\leq t<\infty$$. We also prove some monotonicity properties of weak solutions and that, under reasonable hypotheses, the moving boundaries consist of analytic curves for $$t>0$$.

##### MSC:
 76D10 Boundary-layer theory, separation and reattachment, higher-order effects 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35Q30 Navier-Stokes equations
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