Vol 66, No 2 (2020): Proceedings of the Crimean Autumn Mathematical School-Symposium

In this paper, we study the scalar conjugation problem, which models the problem of small oscillations of two viscoelastic fluids filling a fixed vessel. An initial-boundary value problem is investigated and a theorem on its unique solvability on the positive semiaxis is proven with semigroup theory methods. The spectral problem that arises in this case for normal oscillations of the system is studied by the methods of the spectral theory of operator functions (operator pencils). The resulting operator pencil generalizes both the well-known S. G. Kreyn’s operator pencil (oscillations of a viscous fluid in an open vessel) and the pencil arising in the problem of small motions of a viscoelastic fluid in a partially filled vessel. An example of a two-dimensional problem allowing separation of variables is considered, all points of the essential spectrum and branches of eigenvalues are found. Based on this two-dimensional problem, a hypothesis on the structure of the essential spectrum in the scalar conjugation problem is formulated and a theorem on the multiple basis property of the system of root elements of the main operator pencil is proved.

The article is an extensive review in the theory of symmetric spaces of measurable functions. It contains a number of new (recent) and old (known) results in this field. For the most of the results, we give their proofs or exact references, where they can be found. The symmetric spaces under consideration are Banach (or quasi-Banach) latices of measurable functions equipped with symmetric (rearrangement invariant) norm (or quasinorm). We consider symmetric spaces E = E(Ω, Fμ, μ) ⊂ L0 (Ω, Fμ, μ) on general measure spaces (Ω, Fμ, μ), where the measures μ are assumed to be finite or infinite σ-finite and nonatomic, while there are no assumptions that (Ω, Fμ, μ) is separable or Lebesgue space. In the first section of the review, we describe main classes and basic properties of symmetric spaces, consider minimal, maximal, and associate spaces, the properties (A), (B), and (C), and Fatou’s property. The list of specific symmetric spaces we use includes Orlicz LΦ(Ω, Fμ, μ), Lorentz ΛW (Ω, Fμ, μ), Marcinkiewicz MV (Ω, Fμ, μ), and Orlicz-Lorentz LW,Φ (Ω, Fμ, μ) spaces, and, in particular, the spaces Lp (w), Mp(w), Lp,q, and L∞(U ). In the second section, we deal with the dilation (Boyd) indexes of symmetric spaces and some applications of classical Hardy-Littlewood operator H. One of the main problems here is: when H acts as a bounded operator on a given symmetric space E(Ω, Fμ, μ)? A spacial attention is paid to symmetric spaces, which have Hardy-Littlewood property (HLP) or weak Hardy-Littlewood property (WHLP). In the third section, we consider some interpolation theorems for the pair of spaces (L1 , L∞) including the classical Calderon-Mityagin theorem. As an application of general theory, we prove in the last section of review Ergodic Theorems for Cesaro averages of positive contractions in symmetric spaces. Studying various types of convergence, we are interested in Dominant Ergodic Theorem (DET ), Individual (Pointwise) Ergodic Theorem (IET ), Order Ergodic Theorem (OET ), and also Mean (Statistical) Ergodic Theorem (MET ).

We consider a second-order nonlinear degenerate parabolic equation in the case when the flux vector and the nonstrictly increasing diffusion function are merely continuous. In the case of zero diffusion, this equation degenerates into a first order quasilinear equation (conservation law). It is known that in the general case under consideration an entropy solution (in the sense of Kruzhkov-Carrillo) of the Cauchy problem can be non-unique. Therefore, it is important to study special entropy solutions of the Cauchy problem and to find additional conditions on the input data of the problem that are sufficient for uniqueness. In this paper, we obtain some new results in this direction. Namely, the existence of the largest and the smallest entropy solutions of the Cauchy problem is proved. With the help of this result, the uniqueness of the entropy solution with periodic initial data is established. More generally, the comparison principle is proved for entropy suband super-solutions, in the case when at least one of the initial functions is periodic. The obtained results are generalization of the results known for conservation laws to the parabolic case.

On the base of boundary-value, spectral and initial-boundary value problems studied earlier for the case of single domain, we consider corresponding problems generated by sesquilinear form for two domains. Arising operator pencils with corresponding operator coefficients acting in a Hilbert space and depending on two parameters are studied in detail. In the perturbed and unperturbed cases, we consider two situations when one of the parameters is spectral and the other is fixed. In this paper, we use the superposition principle that allow us to present the solution of the original problem as a sum of solutions of auxiliary boundary-value problems containing inhomogeneity either in the equation or in one of the boundary conditions. The necessary and sufficient conditions for the correct solvability of boundary-value problems on given time interval are obtained. The theorems on properties of the spectrum and on the completeness and basicity of the system of root elements are proved.