Andrei Faminskii
Doctor of Physical and Mathematical Sciences
Professor,
s.M. Nikol’skii Mathematical Institute
Development of theoretical mathematics
1975-1980
Student of the Faculty of Mechanics and Mathematics, M.V. Lomonosov Moscow State University.
1980-1983
Postgraduate student at the Department of Differential Equations, Faculty of Mechanics and Mathematics, M.V. Lomonosov Moscow State University.
1984
Defended his Postgraduate diploma. PhD thesis: “The Cauchy problem and the mixed problem in a half-strip for equations of the Korteweg-de Vries type”.
1983-1988
Assistant of the Mathematics Department of the Moscow Institute of Radio Engineering, Electronics and Automation (MIREA).
1988-1990
Assistant professor at the Mathematics Department of MIREA.
1990-2002
Associate Professor of the Department of Differential Equations and Functional Analysis of RUDN University.
2001
Defended his Doctoral thesis. Dr.Sci. thesis: “Boundary value problems for the Korteweg-de Vries equation and its generalizations”.
2002-2018
Professor at the Department of Differential Equations and Functional Analysis (Department of Nonlinear Analysis and Optimization after 2005), RUDN University.
2012
Received the academic title of Professor.
Since 2018
Professor of the S. Nikolsky Mathematical Institute.
2014
Winner of the RUDN prize in the field of science and innovations.
Teaching
- Made training courses, the most significant are the following:
- “Evolutionary type functional spaces” - Moscow: RUDN University, 2011, 2016 - 2nd edition.
- "Selected chapters of the theory of evolutionary equations" - Moscow: RUDN University, 2014.
- Conducts the following lecture courses for bachelor and master’s degree students in RUDN University:
- "Complex analysis" ("Mathematics" direction, bachelor degree),
- "Partial differential equations" ("Mathematics" direction, bachelor degree),
- "Nonlinear evolution equations" ("Mathematics" direction, magistracy)
Science
- Results on the global solvability and correctness of the Cauchy problem and initial boundary value problems for nonlinear evolution equations of odd order were obtained. These equations include the Korteweg-de Vries, Kawahary, Kadomtsev-Petviashvili, Zakharov-Kuznetsov equations. These equations describe nonlinear wave processes in dispersion media. These results have been established both for wide classes of equations of the type under consideration and for individual equations. For the considered boundary problems, the problems of internal regularity of solutions, the behavior of solutions at large times and controllability are also studied.
- The studies were started in the 1980s together with Professor Stanislav N. Kruzhkov for solving the Cauchy problem for the Korteweg-de Vries equation - the most famous representative of this class of equations. In particular, the so-called local smoothing effect was discovered (at the same time with the famous American mathematician T. Kato), which allowed us to establish the results of the existence of global time solutions with irregular initial data. In addition, the property was found to increase the internal smoothness of solutions depending on the decay rate at infinity of the initial data. The research method essentially used the properties of the corresponding linearized equation, namely, the Airy equation. In particular, the idea of inversion of the linear part of the Korteweg-de Vries equation was applied in order to establish the results on the uniqueness of the considered solutions.
- Further study of the properties of the Airy equation allowed us to construct special solutions of this equation of the type of boundary potentials. The use of these boundary potentials made it possible to obtain results on the global correctness of initial boundary value problems for the Korteweg-de Vries equation with natural or close to natural conditions for the smoothness of boundary data.
- In the following, these methods, developed for the Korteweg-de Vries equation, were applied to a wider class of equations of odd order. Results were obtained on the existence and uniqueness of global solutions. Assumptions on the class of the equations considered allowed us to use such analogs of the properties of the Korteweg-de Vries equation, such as the existence of conservation laws and the effect of local smoothing. Special solutions of the type of boundary potentials of the corresponding linearized equations were also constructed, which were used in the study of initial-boundary value problems.
- In the process of studying the generalizations of the Korteweg-de Vries equation, special attention was paid to those that were model for describing the propagation of nonlinear waves. In particular, for the Kadomtsev-Petviashvili equation in the case of two spatial variables, the first results on the existence of global solutions to the Cauchy problem were obtained.
- Another example of a multidimensional generalization of the Korteweg-de Vries equation is the Zakharov-Kuznetsov equation. For this equation, global correctness classes were constructed, both in the case of the Cauchy problem and in the initial-boundary value problems. In this case, cases of both two-dimensional and three-dimensional equations are considered. The use of boundary potentials made it possible to obtain these results under the natural conditions of smoothness of the boundary data. Results on the internal regularity of solutions were also established.
- Another important equation that generalizes the Korteweg-de Vries equation is the Kawahara equation, which models the propagation of waves in media with a higher order of dispersion. Here, results are also obtained on the global correctness of initial-boundary problems under the natural conditions of smoothness of boundary data and on the internal regularity of solutions.
- In recent years, issues of the behavior of solutions at large times, as well as some control problems, have also been studied for the equations above.
Scientific interests
- solvability and correctness of boundary problems for nonlinear evolution equations of odd order
- internal regularity of solutions of boundary value problems for nonlinear evolution equations of odd order
- behavior at large times of solutions of boundary value problems for nonlinear evolution equations of odd order
- controllability of boundary value problems for nonlinear evolution equations of odd order
An initial boundary-value problem in a half-strip with one boundary condition for the Korteweg-de Vries equation is considered and results on global well--posedness of this problem are established in Sobolev spaces of various orders, including fractional. Initial and boundary data satisfy natural (or close to natural) conditions, originating from properties of solutions of a corresponding initial-value problem for a linearized KdV equation. An essential part of the study is the investigation of special solutions of a "boundary potential" type for this linearized KdV equation.
In the present paper we establish results concerning the decay of the energy related to the damped Korteweg–de Vries equation posed on infinite domains. We prove the exponential decay rates of the energy when an initial value problem and a localized dissipative mechanism are in place. If this mechanism is effective in the whole line, we get a similar result in H k -level, k∈ℕ. In addition, the decay of the energy regarding an initial boundary value problem posed on the right half-line, is obtained considering convenient a smallness condition on the initial data but a more general dissipative effect.
An initial-boundary value problem in a strip with homogeneous Dirichlet boundary conditions for two-dimensional generalized Zakharov--Kuznetsov equation is considered. In particular, dissipative and absorbing degenerate terms can be supplemented to the original Zakharov-Kuznetsov equation. Results on global existence, uniqueness and long-time decay of weak solutions are established.
An initial–boundary value problem in a strip with homogeneous Dirichlet boundary conditions for two-dimensional Zakharov–Kuznetsov–Burgers equation is considered. Results on global well-posedness and large-time decay of solutions in the spaces Hs for s∈[0,2] are established.
An initial–boundary value problem with homogeneous Dirichlet boundary conditions for three-dimensional Zakharov–Kuznetsov equation is considered. Results on global existence, uniqueness and large-time decay of weak solutions in certain weighted spaces are established.
Initial-boundary value problems in a bounded rectangle with different types of boundary conditions for two-dimensional Zakharov-Kuznetsov equation are considered. Results on global well-posedness in the classes of weak and regular solutions are established. As applications of the developed technique results on boundary controllability and long-time decay of weak solutions are also obtained.
In this paper, we consider initial-boundary value problem on semiaxis for generalized Kawahara equation with higher-order nonlinearity. We obtain the result on existence and uniqueness of the global solution. Also, if the equation contains the absorbing term vanishing at infinity, we prove that the solution decays at large time values.
Initial–boundary value problems in a half-strip with different types of boundary conditions for two-dimensional Zakharov–Kuznetsov equation are considered. Results on global well-posedness in classes of regular solutions in the cases of periodic and Neumann boundary conditions, as well as on internal regularity of solutions for all types of boundary conditions are established. Also in the case of Dirichlet boundary conditions one result on long-time decay of regular solutions is obtained.
In this paper, we consider questions of inner regularity of weak solutions of initial-boundary value problems for the Zakharov-Kuznetsov equation with two spatial variables. The initial function is assumed to be irregular, and the main parameter governing the regularity is the decay rate of the initial function at infinity. The main results of the paper are obtained for the problem on a semistrip. In this problem, different types of initial conditions (e. g., Dirichlet or Neumann conditions) influence the inner regularity. We also give a survey of earlier results for other types of areas: a plane, a half-plane, and a strip.
The initial boundary value problem on a rectangle for the Zakharov-Kuznetsov equation is considered. The right-hand side of the equation includes an unknown function, which is the control. An additional condition for integral redefinition is set. Under certain conditions of smallness, an unambiguous solvability of this problem is established. In the case of the corresponding linearized equation, a similar result is obtained without the assumption of smallness.
We establish results on the unique solvability of control problems for the Korteweg–de Vries equation and its linearized analog in a bounded domain under an integral overdetermination condition. For the Korteweg–de Vries equation itself, we impose smallness conditions on either the input data or the time interval. These restrictions are absent in the linear case. For the control we take either the value of the derivative of the solution on one of the boundaries or the right-hand side of the equation, which has a special form.
The initial and initial-boundary value problems, posed on infinite domains for Korteweg–de Vries equation, are considered. The right-hand side of the equation contains an unknown function, regarded as a control, which must be chosen such that the corresponding solution should satisfy certain additional integral condition. Results on existence and uniqueness are established in the cases of small input data or small time interval.
Initial-boundary value problems in a half-strip with different types of boundary conditions for two-dimensional Zakharov–Kuznetsov equation are considered. Results on global existence, uniqueness and long-time decay of weak and regular solutions are established.
Classical Strichartz type argument is applied for an abstract one-parameter set of linear continuous operators and a Strichartz type estimate in a non-endpoint case is rigorously justified.
Faminskii A. V. Functional spaces of the evolutionary type. Textbook, 2nd edition, revised and expanded. RUDN University, 2016, 144 p.
The manual contains a systematic presentation of the theory of functional spaces used in the study of evolutionary partial differential equations. The elements of such spaces are functions that map the interval of a real line to a Banach space.
Stationary solutions on a bounded interval for an initial-boundary value problem to Korteweg–de Vries and modified Korteweg–de Vries equation (for the last one both in focusing and defocusing cases) are constructed. The method of the study is based on the theory of conservative systems with one degree of freedom. The obtained solutions turn out to be periodic. Exact relations between the length of the interval and coefficients of the equations which are necessary and sufficient for existence of nontrivial solutions are established.
We consider global weak solutions of the Cauchy problem for the generalized Kawahara equation. We prove theorems on the existence, uniqueness, and interior regularity as well as the exponential decay for large time.
Results on the internal regularity of solutions of the initial boundary value problem given on the half-plane for the Zakharov-Kuznetsov equation are established.
We consider an initial boundary value problem on the semi-axis for the generalized Kawahara equation containing an absorbing term that can degenerate on a finite segment. The result about decreasing at large times of weak solutions is established.
Antonova A. P., Faminskii A. V. On the internal regularity of solutions to the Cauchy problem for the Zakharov-Kuznetsov equation // Bulletin of the Tambov State University, Vol. 20, issue 5, 2015, pp. 999-1006.
The problem of the interior regularity of generalized solutions of the Cauchy problem for the Zakharov-Kuznetsov equation is studied. The existence of Hölder-continuous derivatives of given solutions is established. The study is based on the properties of the fundamental solution of the corresponding linearized equation.
In this work, we study at the L_2-level global well-posedness as well as long-time stability of an initial-boundary value problem, posed on a bounded interval, for a generalized higher order nonlinear Schrödinger equation, modeling the propagation of pulses in optical fiber, with a localized damping term. In addition, we implement a precise and efficient code to study the energy decay of the higher order nonlinear Schrödinger equation and we prove its convergence and exponential stability of the discrete energy.
An initial-boundary value problem posed on a bounded interval is considered for a class of odd-order (more than one) quasilinear evolution equations with general nonlinearity. Assumptions on the equations do not provide global a priori estimates for solutions of an arbitrary size. For small initial and boundary data, small right-hand side function results on global existence and uniqueness of small weak solutions, as well as on their large-time exponential decay are established.
Initial-boundary value problems on a half-strip with different types of boundary conditions for the modified Zakharov-Kuznetsov equation are considered. Results on local and global well-posedness in classes of mild and regular solutions, internal regularity of mild solutions and long-time decay of both mild and regular solutions are established. The solutions are considered in weighted at infinity Sobolev spaces.