Abstract. The viscous flow induced by a shrinking sheet is studied. Existence and (non)uniqueness are proved. Exact solutions, both numerical and in closed form, are found.

Von Kármán's problem of a rotating disk in an infinite viscous fluid is extended to the case where the disk surface admits partial slip. The nonlinear similarity equations are integrated accurately for the full range of slip coefficients. The effects of slip are discussed. An existence proof is also given. (2000). 76D05, 76D03, 65L10. Mathematics Subject Classification

It is shown that a perturbation argument that guarantees persistence of inertial (invariant and exponentially attracting) manifolds for linear perturbations of linear evolution equations applies also when the perturbation is nonlinear. This gives a simple but sharp condition for existence of inertial manifolds for semilinear parabolic as well as for some nonlinear hyperbolic equations. Fourier transform of the explicitly given equation for the tracking solution together with the Plancherel's theorem for Banach valued functions are used.

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