Uniqueness Theorem for the Basic Nonstationary Problem in the Dynamics of an Ideal Incompressible Fluid

The initial boundary value problem is considered for the Euler equations for an incompressible fluid in a bounded domain $D \subset R^n$. 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 $p \to \infty$. 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.

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Published 1 January 1995