No abstract

A b s t r a c t . The initial boundary value problem is considered for the Euler equations for an incompressible fluid in a bounded domain D ⊂ R n . It is known [Y1] that uniqueness holds for those flows with bounded vorticity. We present here a uniqueness theorem in some classes (B-spaces) of incompressible flows with vorticity which is unbounded but belongs to any L p (D). The regularity of the flow is characterized by restrictions on the growth rate of the L p -norms as p → ∞. Roughly speaking, logarithmic singularities are forbidden but iterated logarithm singularities are permissible. It is notable that the uniqueness conditions for the Euler equations and for the motions of fluid particles are the same. The result is obtained by the energy method and a counterexample is constructed to demonstrate that it is impossible to weaken the restrictions still using the energy method. IntroductionThe basic initial boundary value problem for the Euler equations for the case of an incompressible ideal fluid is considered. The first theorems on existence and uniqueness were obtained by N. Gunter and L. Lichtenstein in two extensive series of articles [Gu, L]. They investigated the classical solutions (velocity is C 1,λ -smooth, 0 < λ < 1) and the results were local in time. The global solvability for the 2-dimensional problem was proved by W. Wolibner [W] for the classical solutions; T. Kato [K] presented this result in modern form and included external forces.The generalized solutions in the 2-dimensional case were introduced in [Y1] and the global existence theorem was proved for flows with vorticity in L p for any given p > 1. However, the uniqueness theorem was obtained only for the class of flows with essentially bounded vorticity. This is the strongest known result on uniqueness whereas the results on existence were extended by A. Morgulis [M] (vorticity in Orlič spaces between L 1 and any L p , p > 1) and J.-M. Delort [D1, D2]. It is interesting to mention also

No abstract

No abstract

A complete description is given for the unstable spectrum of the Euler equation linearized about shear flows with sinusoidal profiles. The spectral problem is treated analytically and numerically. The spectral asymptotics are determined using the method of averaging for general rapidly oscillating profiles. Such flows are always unstable.