Exact Beltrami flows in a spherical shell

Exact flows of an incompressible fluid satisfying the Beltrami equation inside a spherical shell are constructed in the Cartesian coordinates in terms of elementary functions. Two scale-invariant equations defining two infinite series of eigenvalues λ n and λ ̃ m ${\tilde {\lambda }}_{m}$ of the operator curl in the shell with the nonpenetration boundary conditions on the boundary spheres are derived. The corresponding eigenfields are presented in explicit form and their symmetries are investigated. Asymptotics of the eigenvalues λ n and λ ̃ m ${\tilde {\lambda }}_{m}$ at n, m → ∞ are obtained.

Asymptotics of the Spectrum of a Periodic Boundary ValueProblem for an Odd-Order Differential Operator

The spectrum of a differential operator of high odd order with periodic boundary conditions is studied. The asymptotics of the fundamental system of solutions of the differential equation defining the operator are obtained by the method of successive Picard approximations. With the help of this fundamental system of solutions the periodic boundary conditions are studied. As a result, the equation for the eigenvalues of the differential operator under study is obtained, which is a quasi-polynomial. The indicator diagram of this equation, which is a regular polygon, is investigated. In each of the sectors of the complex plane, defined by the indicator diagram, the asymptotics of the eigenvalues of the operator under study is found. An equation for the eigenvalues of the differential operator under study is derived. The indicator diagram of this equation has been studied. The asymptotics of the eigenvalues of the studied operator in different sectors of the indicator diagram is found.

On the convexity of the reachable set with respect to a part of coordinates at small time intervals

We investigate the convexity of the reachable sets for some of the coordinates of nonlinear systems with integral constraints on the control at small time intervals. We have proved sufficient convexity conditions in the form of constraints on the asymptotics of the eigenvalues of the Gramian of the controllability of a linearized system for some of the coordinates. There are two nonlinear third-order systems under study as examples. The system linearized along a trajectory generated by zero control is uncontrollable, and the system in the other example is completely controllable. We investigate the sufficient conditions for convexity of projection of reachable sets. Numerical modeling has been carried out, demonstrating the non-convexity of some projections even for small time intervals.

Scalar problems in junctions of rods and a plate

In this work we deal with a scalar spectral mixed boundary value problem in a spacial junction of thin rods and a plate. Constructing asymptotics of the eigenvalues, we employ two equipollent asymptotic models posed on the skeleton of the junction, that is, a hybrid domain. We, first, use the technique of self-adjoint extensions and, second, we impose algebraic conditions at the junction points in order to compile a problem in a function space with detached asymptotics. The latter problem is involved into a symmetric generalized Green formula and, therefore, admits the variational formulation. In comparison with a primordial asymptotic procedure, these two models provide much better proximity of the spectra of the problems in the spacial junction and in its skeleton. However, they exhibit the negative spectrum of finite multiplicity and for these “parasitic” eigenvalues we derive asymptotic formulas to demonstrate that they do not belong to the service area of the developed asymptotic models.