Functional-analytical methods for the study of boundary value problems for differential and functional differential equations with partial derivatives
Functional-analytical methods for the study of boundary value problems for differential and functional differential equations with partial derivatives
Year 2020
The main idea of the project: research of new classes of differential and functional-differential equations, inequalities and systems and the application of the obtained results to interdisciplinary research in mathematical models of physical and biological processes.
- A.L. Skubachevskii, A.Sh. Adkhamova. Damping Problem for a Neutral Control System with Delay. Doklady Mathematics (Q2) DOI: 10.31857/S2686954320010038
- A. Savin, E. Schrohe. Analytic and algebraic indices of elliptic operators associated with discrete groups of quantized canonical transformations. J. . Funct. Anal. 278 (2020), no. 5, 108400, 45 pp. (Q1) https://doi.org/10.1016/j.jfa.2019.108400
- N. Bessonov, G. Bocharov, A. Meyerhans, V. Popov, V. Volpert. Nonlocal reaction-diffusion model of viral evolution: emergence of virus strains. Mathematics (Q1) doi:10.3390/math8010117
- V. A. Derkach, S. Hassi, M.M. Malamud. Generalized boundary triples, I. Some classes of isometric and unitary boundary pairs and realization problems for subclasses of Nevanlinna functions. Math. Nachr. (Q1) DOI: 10.1002/mana.201800300
Project goals
- The aim of this project relate to the study of nonlocal equations and reaction equations with delay arising in biomedicine, including immunology and neurology.
- It is supposed to obtain new solutions by energy substitution under conditions of quasineutrality for the Vlasov-Poisson and Vlasov-Maxwell equations.
- One of the goals of the project is to construct a theory of boundary value problems for elliptic functional differential equations with affine transformations of independent variables.
- It is planned to study the analytical and topological aspects of the theory of nonlocal elliptic operators.
- To investigate the relationship between the scattering matrix of a pair of self-adjoint operators, whose resolvent difference is of finite order but is not nuclear. Obtain an analogue of the invariance principle of M. S. Birman and T. Kato in the considered situation. To explore the connection of this formula with trace formulas.
- It is proposed to study initial-boundary value problems for various classes of semilinear and quasilinear parabolic equations of the structure of linear and nonlinear diffusion - nonlinear absorption degenerating on various manifolds with singular boundary or initial data.
Project leader
All participants
Vitalii Volpert
Doctor of Sciences, professor
Project results
Local and nonlocal reaction-diffusion equations are used to simulate virus mutations in the genotype space. The existence of various types of solutions that characterize the resistance and evolution of virus strains has been investigated. The dynamics of such decisions determines the emergence of new strains, resistance to treatment, and other properties.
The reduction of the Vlasov-Poisson equation to elliptic equations by energy substitution is carried out and the general solution is obtained under the conditions of quasineutrality. Various forms of the Vlasov-Maxwell Einstein equations and their hydrodynamic consequences are derived. A theorem on coincidence of time averages with Boltzmann extremals is proved and its consequences are studied.
The Dirichlet problem is considered for an elliptic and strongly elliptic second-order equation with affine transformations of the arguments of the highest derivatives of an unknown function. Algebraic conditions for the unique and Fredholm solvability are obtained, expressed using the characteristic functions of the measures present in the equation.
Smoothness theorems for operators associated with Morse-Smale diffeomorphisms, smoothness theorems, and index formulas for operators associated with a metaplectic group are proved.
A formula is found that expresses the Weyl function of the degree of a symmetric operator with infinite defect indices in terms of the Weyl function of the symmetric operator itself. Based on this formula, an analogue of the Birman-Kato invariance principle for scattering matrices is proved.
Trace formulas for normal operators, trace formulas for non-operator-Lipschitz functions, and characterization of the class of operator-Lipschitz functions in terms of divided difference in each variable are obtained.
Existence theorems for solutions in Sobolev spaces for linear and quasilinear stationary and non-stationary nondivergence equations with the Wentzell boundary condition are proved. It is assumed that the leading coefficients in both the equation and the boundary condition are discontinuous.