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On square summability and uniqueness questions concerning nonstationary stokes flow in an exterior domain

University of British Columbia

In this thesis we investigate the square summability, uniqueness, and convergence to steady state of solutions of the nonstationary Stokes equations in an exterior domain. A class of generalized solutions (which will be called class H[sub o] solutions), whose members are required a priori to have finite Dirichlet integrals but not necessarily to have finite L² norms, has been introduced by J.G. Heywood for the purpose of studying the convergence of nonstationary solutions to stationary ones as time t → ∞. In our present work, we prove that, in the case of an exterior domain Ω, of R[sup n](n > 2) , such solutions are necessarily square-summable if both the initial data and the force are square-summable. 2 We give a partial result for Ω in R². Furthermore, we prove that if Ω = R³ the unique class H[sub o] solution is identical with the unique finite energy solution (i.e. L²(Ω)) of various classes when the data permits existence of both types of solutions. This has enabled us to show that the finite energy solutions of a particular nonstationary Stokes problem converge to solutions of steady state as t → ∞. We have also succeeded in extending the definition of class H[sub o] solutions to nonstationary Stokes problems with general nonhomogeneous boundary values in such, a way that the uniqueness theorem for such solutions is preserved.

Text

2010-02-05

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http://hdl.handle.net/2429/19662

Mathematics

University of British Columbia

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