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A blow-up phenomenon in the Hamilton-Jacobi equation in an unbounded domain
P. 161–179.
We construct an example of blow-up in a ”ow of min-plus linear operators arising as solution operators for a Hamilton…Jacobi equation @S/@t+|∇S| / + U(x, t) = 0, where > 1 and the potential U(x, t) is uniformly bounded together with its gradient. The construction is based on the fact that, for a suitable potential de“ned on a time interval of length T, the absolute value of velocity for a Lagrangian minimizer can be as large as O(log T)2−2/ . We also show that this growth estimate cannot be surpassed. Implications of this example for existence of global generalized solutions to randomly forced Hamilton…Jacobi or Burgers equations are discussed.
In book
Providence: American Mathematical Society, 2005.
Maslov V., Теоретическая и математическая физика 2021 Т. 206 № 3 С. 448–452
We consider the construction of asymptotic solutions of linear equations related to equations of classical mechanics: the Hamilton–Jacobi equation and the transport equation. We show that these methods and also the theory of the mechanics of an infinitely narrow beam as a whole can be applied to some objects in bioenergy if the thin organic ...
Added: July 14, 2021
Saakian D., Vardanyan E., Yakushkina T., Physica A: Statistical Mechanics and its Applications 2020 Vol. 541 Article 123091
We investigate the evolutionary model with recombination and random switches in the fitness function due to change in a special gene. The dynamical behaviour of the fitness landscape induced by the specific mutations is closely related to the mutator phenomenon, which, together with recombination, plays an important role in modern evolutionary studies. It is of ...
Added: December 10, 2019
Yakushkina T., Saakian D., Chinese Journal of Physics 2018 Vol. 56 No. 5 P. 2235–2240
In this study, we considered the model by Beckman and Loeb [Proc. Natl. Acad. Sci. U.S.A. 103 (2006) 14140] for the mutator phenomena. We construct an infinite population Crow-Kimura model with a mutator gene, directed mutations, a linear fitness function, and a finite genome length. We solved analytically the dynamics of the model using the ...
Added: October 25, 2018
Afanasiev V., Presnova A., Мехатроника, автоматизация, управление 2018 № 3 С. 153–159
The method of forming optimization algorithms for non-stationary control systems is developed in the article, based on the application of the Hamilton-Jacobi equation and the Pontryagin minimum principle. In this article, the original nonlinear differential equation that describes the original control system is transformed into a system with a linear structure, but with State Dependent ...
Added: October 1, 2018
Yakushkina T., Saakian D., Journal of the Physical Society of Japan 2017 Vol. 86 P. 084801-1–084801-6
To describe virus evolution, it is necessary to define a fitness landscape. In this article, we consider the microscopic models with the advanced version of neutral network fitness landscapes. In this problem setting, we suppose a fitness difference between one-point mutation neighbors to be small. We construct a modification of the Wright–Fisher model, which is ...
Added: July 19, 2017
Saakian D., Yakushkina T., Hu C., Scientific Reports 2016 Vol. 6
We propose a modification of the Crow-Kimura and Eigen models of biological molecular evolution to include a mutator gene that causes both an increase in the mutation rate and a change in the fitness landscape. This mutator effect relates to a wide range of biomedical problems. There are three possible phases: mutator phase, mixed phase ...
Added: September 16, 2016
Presnova A., Качество. Инновации. Образование 2016 № 2 С. 31–40
In this paper, using the theory of differential games developed the algorithm of constructing guaranteed control for nonlinear systems. Through the transition from the nonlinear model to model the linear form with parameters that depend on the state, will move from finding solutions to equations of Hamilton-Jacobi-Bellman equation to the Riccati equation. Moreover, the appointment ...
Added: May 3, 2016
Zlotnik A., Lapukhina A. V., Journal of Mathematical Sciences 2010 Vol. 169 No. 1 P. 84–97
We consider an initial-boundary value problem for the one-dimensional nonstationary Schrödinger equation on the half-axis and study a two-level symmetric finite-difference scheme of Numerov type with higher approximation order. This scheme is constructed on a finite mesh, which is uniform with respect to space, with a nonlocal approximate transparent boundary condition of a general form ...
Added: December 23, 2015
Злотник А.А., Лапухина А., Проблемы математического анализа 2010 № 47 С. 77–88
Нестационарное уравнение Шрёдингера относится к основным уравнениям математической физики и находит многочисленные приложения. Очень часто его приходится численно решать в неограниченных по пространству областях. Для этой цели разработан ряд подходов, связанных с постановкой искусственных или приближенных прозрачных граничных условий (ПГУ) на искусственных границах. Среди них следует выделить подход, использующий так называемые дискретные ПГУ. Серьезный практический ...
Added: December 22, 2015
Boldyrev I., Kragh M., Journal of the History of Economic Thought 2015
We study the system of Euler equations with the so-called Ekman damping in the whole 2D space. The global well-posedness and dissipativity for the weak infinite energy solutions of this problem in the uniformly local spaces is verified based on the further development of the weighted energy theory for the Navier-Stokes and Euler type problems. ...
Added: October 9, 2015
Yakushkina T., Saakian D., Hu C., Chinese Journal of Physics 2015 Vol. 53 No. 5-I P. 100904-1–100904-13
We investigate the dynamics of a molecular evolution model related to the mutator gene phenomenon. Here mutation in one gene drastically changes the properties of the whole genome. We investigated the Crow-Kimura version of the model, which can be mapped into a Hamilton-Jacobi equation. For the symmetric fitness landscape, we calculated the dynamics of the ...
Added: September 15, 2015
Chepyzhov V. V., Zelik S., Journal of Mathematical Fluid Mechanics 2015 Vol. 17 No. 3 P. 513–532
We study the system of Euler equations with the so-called Ekman damping in the whole 2D space. The global well-posedness and dissipativity for the weak infinite energy solutions of this problem in the uniformly local spaces is verified based on the further development of the weighted energy theory for the Navier–Stokes and Euler type problems. ...
Added: September 3, 2015
Khanin K., Sobolevski A., Archive for Rational Mechanics and Analysis 2016 Vol. 219 No. 2 P. 861–885
Characteristic curves of a Hamilton–Jacobi equation can be seen as action minimizing trajectories of fluid particles. However this description is valid only for smooth solutions. For nonsmooth “viscosity” solutions, which give rise to discontinuous velocity fields, this picture holds only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian ...
Added: August 18, 2015
Zlotnik Alexander, / Series math "arxiv.org". 2015.
We deal with an initial-boundary value problem for the generalized time-dependent Schr\"odinger equation with variable coefficients in an unbounded $n$--dimensional parallelepiped ($n\geq 1$). To solve it, the Crank-Nicolson in time and the polylinear finite element in space method with the discrete transpa\-rent boundary conditions is considered. We present its stability properties and derive new error ...
Added: March 27, 2015
Zlotnik A., , in: Finite Difference Methods, Theory and Applications 6th International Conference, FDM 2014, Lozenetz, Bulgaria, June 18-23, 2014, Revised Selected PapersVol. 9045.: Zürich: Springer, 2015. P. 129–141.
We deal with an initial-boundary value problem for the generalized time-dependent Schrödinger equation with variable coefficients in an unbounded $n$-dimensional parallelepiped ($n\geq 1$). To solve it, the Crank-Nicolson in time and the polylinear finite element in space method with the discrete transparent boundary conditions is considered. We present its stability properties and derive new error ...
Added: March 23, 2015