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On square summability and uniqueness questions concerning nonstationary stokes flow in an exterior domain Ma, Chun-Ming


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.

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