Mamontov A.E.  

On existence and uniqueness of solutions in several boundary-value problems for the Euler equations

We consider several boundary value problems for the Euler equations describing flows of an ideal incompressible fluid in a bounded domain: the problem NP with nonpenetration condition at the boundary as well as so called through flow problems describing flows of the fluid through the domain, with different types of boundary conditions (then three problems appear, we call them TF.I, TF.II and TF.III). We are interested in global (i.e., "in the whole" in time and input data) existence and uniqueness theorems for these problems formulated in the widest possible classes of solutions. Such interest if stimulated by well-known global existence problem for three-dimensional Euler equations, where solutions become nonsmooth even if they are smooth at the initial moment of time, so the named problem seems to meet its solution only in the classes of extremely irregular functions. In the other words, we have to study nonsmooth solutions of the Euler equations and prove their existence and uniqueness. We present two main results. The first result consists in uniqueness of solutions to the problems NP, TF.II and TF.III (for any dimensions of the flow) in the classes with unbounded vorticity. These classes are presented using the Orlicz classes and seem to be easy to verify in applications since they are formulated in a rather clear form as against well-known results. The second result consists in global existence theorem for the two-dimensional problem TF.I in the classes of solutions with unbounded vorticity that belongs to the Lebesgue spaces Lp with p>4/3. Our methods discover curious relations of the named problems with the theory of integral transforms.

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