Математические модели и методы интегрирования

We consider the Cauchy problem for the Navier-Stokes equations over R^3 × [0, T] with a positive time T over specially constructed scale offunction spaces of Bochner-Sobolev type. We prove that the problem induces anopen both injective and surjective mapping of each space of the scale. In particular,intersection of these classes gives a uniqueness and existence theorem for smoothsolutions to the Navier-Stokes equations for smooth data with a prescribed asymp-totic behaviour at the infinity with respect to the time and the space variables.Actually, we propose the following modified scheme of the proof of the existencetheorem, based on apriori estimates and operator approach in Banach spaces:

1. We prove that the Navier-Stokes equations induce continuous injective OPEN mapping between the chosen Banach spaces.

2. Next, the standard topological arguments immediately imply that a nonemptyopen connected set in a topological vector space coincides with the space itself ifand only if the set is closed. This reduces the proof of the existence theorem to an L^s([0, T], L^r(R^3)) a priori estimate for the INVERSE IMAGE OF PRECOMPACTSETS in the target Banach space wheres,rare Ladyzhenskaya-Prodi-Serrin num-bers satisfying 2/s + 3/r= 1 and r > 3. In this way we avoid proving a GLOBAL L^s([0, T], L^r(R^3)) a priori estimate.

3. To prove the weak L^s([0, T], L^r(R^3)) a priori estimate with r > 3 we calculateprecisely the excess between the left hand side and the right hand side of thecorresponding energy inequality, that equals to 2r when expressed in terms of theLebesgue integrability index r. Then we operate with absolutely convergent seriesinvolving Lebesgue norms that gives the possibility to group together summandsin a suitable way, using the energy type inequalities, interpolation inequalities andmatching the asymptotic behaviour in order to exclude the unbounded sequencesin the inverse image of a precompact set.

An early version of the paper is uploaded on arxiv.org:https://arxiv.org/abs/2009.10530

A similar approach can be used for investigation of the Navier-Stokes equationsin the periodic setting:https://arxiv.org/abs/2007.14911

Methods of construction and analysis of finite-difference mathematical models based on symmetry are discussed.
The group analysis methods allow one to construct invariant finite-difference schemes preserving the basic geometric and qualitative physical properties of the original continuous models.
The authors propose a number of methods for constructing invariant schemes possessing conservation laws. Examples of invariant schemes for partial differential equations (PDEs) and ordinary differential equations (ODEs) are provided.
Among the family of invariant schemes for ODEs, exact schemes are found, that is, schemes whose solutions coincide with the corresponding set of ODEs' solutions at the nodes of the finite-difference mesh of an arbitrary density.
Invariant conservative schemes are constructed for various PDEs (wave equations, one-dimensional shallow water equations and Green-Naghdi equations). The constructed schemes possess difference analogues of the local conservation laws of the original differential models.

References

1. Dorodnitsyn V. A.. Applications of Lie Groups to Difference Equations. CRC Press, Boca Raton, 2011.

2. Dorodnitsyn V. A., Kaptsov E. I., Shallow water equations in Lagrangian coordinates: Symmetries, conservation laws and its preservation in difference models, Communications in Nonlinear Science and Numerical Simulation 89 (2020) P. 105343

3. Cheviakov A. F., Dorodnitsyn V. A., Kaptsov E. I., Invariant conservation law-preserving discretizations of linear and nonlinear wave equations, Journal of Mathematical Physics 61 (2020) P. 081504.

4. Dorodnitsyn V. A., Kaptsov E. I., Meleshko S. V., Symmetries, conservation laws, invariant solutions and difference schemes of the one-dimensional Green-Naghdi equations, Journal of Nonlinear Mathematical Physics (2020), accepted.

5. Dorodnitsyn V. A., Kaptsov, E. I., Discretization of second-order ordinary differential equations with symmetries, 2013, Computational Mathematics and Mathematical Physics. Vol. 53, No. 8, pp. 1153–1178.

В докладе будет рассказано о задаче построения коммутативных колец разностных операторов. С помощью одноточечных коммутирующих разностных операторов ранга один будет построена дискретизация оператора Ламе.

В докладе будет показано, что на двумерном торе существуют метрики сколь угодно близкие к метрикам Лиувилля и слабые магнитные поля
с интегрируемыми магнитными геодезическими потоками на одном уровне энергии. Доклад основан на совместной работе с Сергеем Агаповым (Новосибирск) и Михаилом Бялым (Тель-Авив).

In this paper, we consider the problem of formal iteration. We construct an area preserving mapping which does not have any square root. This leads to a counterexample to Moser’s existence theorem for an interpolation problem. We give examples of formal transformation groups such that the iteration problem has a solution for every element of the groups.

In this paper, we consider two Boussinesq models that describe propagation of small-amplitude long water waves. Exact solutions of the classical Boussinesq equation that represent the interaction of wave packets and waves on solitons are found. We use the Hirota representation and computer algebra methods. Moreover, we find various solutions for one of the variants of the Boussinesq system. In particular, these solutions can be interpreted as the fusion and decay of solitary waves, as well as the interaction of more complex structures.

We consider the initial problem for the Navier–Stokes equations over R^3 × [0,T] with a positive time T in the spatially periodic setting. Identifying periodic vector-valued functions on R^3 with functions on the 3-dimensional torus T^3, we prove that the problem induces an open both injective and surjective mapping of specially constructed scale of function spaces of Bochner–Sobolev type parametrised with the smoothness index s ∈ N. The intersection of these classes with respect s gives a uniqueness and existence theorem for smooth solutions to the Navier–Stokes equations for each finite T > 0. Then additional intersection with respect to T ∈ (0, +∞) leads to a uniqueness and existence theorem for smooth solutions and data having prescribed asymptotic behaviour at the infinity with respect to the time variable. Actually, we propose the following modified scheme of the proof of the existence theorem, based on apriori estimates and operator approach in Banach spaces:
1. We prove that the Navier–Stokes equations induce continuous injective OPEN mapping between the chosen Banach spaces.
2. Next, the standard topological arguments immediately imply that a nonempty open connected set in a topological vector space coincides with the space itself if and only if the set is closed. This reduces the proof of the existence theorem to an L^s ([0,T], L^r(T^3)) a priori estimate for the INVERSE IMAGE OF PRECOMPACT SETS in the target Banach space where s, r are Ladyzhenskaya–Prodi–Serrin numbers satisfying 2/s + 3/r = 1 and r > 3. In this way we avoid proving a GLOBAL L^s ([0,T], L^r(T^3)) a priori estimate.
3. To prove the weak L^s ([0,T], L^r(T^3)) a priori estimate with r > 3 we calculate precisely the excess between the left hand side and the right hand side of the corresponding energy inequality, that equals to 2r when expressed in terms of the Lebesgue integrability index r. Then we operate with absolutely convergent series involving Lebesgue norms that gives the possibility to group together summands in a suitable way, using the energy type inequalities, interpolation inequalities and matching the asymptotic behaviour in order to exclude the unbounded sequences in the inverse image of a precompact set.

In this report a theoretical approach will be presented for intersatellite communications (ISC) between two satellites (belonging to satellite configurations GPS or GLONASS), moving on (one-plane) elliptical orbits. The new approach is based on the introduction of two null cones with origins at the emitting-signal and receiving-signal satellites. The two null cones (intersected also with a hyperplane) account for the variable distance between the satellites. This intersection of the two null cones gives the space-time interval in GRT. Applying some theorems from higher algebra, it was proved that this space-time distance can become zero, consequently it can be also negative and positive. But in order to represent the geodesic distance travelled by the signal, the space-time interval has to be «compatible» with the Euclidean distance. So this «compatibility condition», conditionally called «condition for ISC», is the most important consequence of the theory. The other important consequence is that the geodesic distance turns out to be the space-time interval, but with account also of the «condition for ISC». The geodesic distance turns out to be greater than the Euclidean distance — a result, entirely based on the «two null cones approach» and moreover, without any use of the Shapiro delay formulae. Application of the same higher algebra theorems shows that the geodesic distance cannot have any zeroes, in accord with being greater than the Euclidean distance. The theory also puts a restriction on the eccentric anomaly angle E=45.00251 [deg], which is surprisingly close to the angle of disposition of the satellites in the GLONASS satellite constellation (the Russian analogue of the American GPS) — 8 satellites within one and the same plane equally spaced at 45 deg. Under some specific restrictions and for the case of plane motion of the satellites, an analytical formula was derived for the propagation time of the signal, emitted by a moving along an elliptical orbit satellite. The formula can be represented as a sum of elliptic integrals of the first, second and the third kind.

1. Bogdan G. Dimitrov, Two null gravitational cones in the theory of GPS-intersatellite communications between two moving satellites. I. Physical and mathematical theory of the space-time interval and the geodesic distance on intersecting null cones, (third) extended version of https://arxiv.org/abs/1712.01101v3 [gr-qc], 162 pages.

2. Bogdan G. Dimitrov, New Mathematical Models of GPS Intersatellite Communications in the Gravitational Field of the Near-Earth Space, AIP Confer. Proc. 2075, 040007 (2019); https://doi.org/10.1063/1.5091167, 9 pages.

A general knowledge about the fundamental physical principles of the Global Positioning System (GPS) will be presented. One of these principles is related to the fundamental fact (the Michelson-Morley experiment) about the independence of the velocity of light from the velocity of the source of light and the non-existence of “ether”, which was the starting point for the creation of the Special Theory of Relativity by Albert Einstein. Particular attention will be paid to some (elementary) model examples, resulting in important relations, concerning the frequency change of the signal between the stations on the Earth’s surface and the rotating satellites around the Earth. This frequency change depends on the rotation of the Earth, as well as on the variation of the gravitational potential. The amazing relation of these dependencies to the approach of Special Theory of Relativity will be demonstrated, also the further extension of the approach in the framework of the General Theory of Relativity, which is being applied in the theory of the Global Positioning System since 2003.
The Geocentric Relativistic Reference System will be briefly reviewed, also the determination of the atomic clock times with respect to an attached to the Earth rotating coordinate system, which is important for taking into account the General Relativity Theory effects during the satellite motion in the near-Earth space.

REFERENCES
1. Neil Ashby, Relativistic effects in the Global Positioning System, in Gravitation and Relativity at the Turn of the Millenium, Proceedings of the 15th International Conference on General Relativity and Gravitation, edited by N. Dadhich and J. Narlikar (International University Centre for Astronomy and Astrophysics, 1998).
2. N. Ashby, Relativity in the Global Positioning System, Living Reviews in Relativity 6, 1-42 (2003), https://link.springer.com/content/pdf/10.12942%2Flrr-2003-1.pdf.
3. N. Ashby, and R. A. Nelson, in Relativity in Fundamental Astronomy: Dynamics, Reference Frames, and Data Analysis, Proceedings of the IAU Symposium 261 2009, edited by S. A. Klioner, P. K. Seidelmann, and M. H. Soffel (Cambridge University Press, Cambridge, 2010).
4. J. — F. Pascual Sanchez, Introducing Relativity in Global Navigation Satellite System, Ann. Phys. (Leipzig) 16, 258–273 (2007).
5. Michael H. Soffel, and Wen-Biao Han, Applied General Relativity. Theory and Applications in Astronomy, Celestial Mechanics and Metrology, Springer Nature, Switzerland AG 2019.
6. Michael H. Soffel, and R. Langhans, Space-Time Reference Systems (Springer-Verlag, Berlin Heidelberg, 2013 ).
7. Sergei M. Kopeikin, Michael Efroimsky, and George Kaplan, Relativistic Celestial Mechanics of the Solar System (Wiley-VCH, New York, 2011).
8. L. Duchayne, Transfert de temps de haute performance: le Lien Micro-Onde de la mission ACES. Physique mathematique [math-ph]. PhD Thesis, Observatoire de Paris, 2008. Francais, HAL Id: tel-00349882, https://tel.archives-ouvertes.fr/tel-00349882/document.
9. M. Gulklett, Relativistic effects in GPS and LEO, October 8 2003, PhD Thesis, University of Copenhagen, Denmark, Department of Geophysics, The Niels Bohr Institute for Physics, Astronomy and Geophysics, available at https://www.yumpu.com/en/document/view/4706552/relativistic-e_ects-in-gps-and-leo-niels-bohr-institutet.
10. B. Hofmann-Wellenhof, and H. Moritz, Physical Geodesy (Springer-Verlag, Wien-New York, 2005).
11. Slava G. Turyshev, Viktor T. Toth, and Mikhail V. Sazhin, General relativistic observables of the GRAIL mission, Phys. Rev. D87, 024020 (2013), arXiv:1212.0232v4 [gr-qc].
12. Slava G. Turyshev, Mikhail V. Sazhin, and Viktor T. Toth, General relativistic laser interferometric observables of the GRACE-Follow-On mission, Phys. Rev. D89, 105029 (2014), arXiv: 1402.7111v1 [qr-qc].
13. Slava G. Turyshev, Nan Yu, and Viktor T. Toth, General relativistic observables for the ACES experiment, Phys. Rev. D93, 045027 (2016), arXiv: 1512.09019v2 [gr-qc].
14. R. A. Nelson, Relativistic time transfer in the vicinity of the Earth and in the Solar system, Metrologia 48, S171 (2011).
15. Bogdan G. Dimitrov, the (third) extended version of arXiv:1712.01101 [gr-qc] (contains a lot of references).
16. Bogdan G. Dimitrov, New Mathematical Models of GPS Intersatellite Communications in the Gravitational Field of the Near-Earth Space, AIP Confer. Proc. 2075, 040007 (2019); https://doi.org/10.1063/1.5091167.
Web pages: http://iaps.institute/mathematical-physics/general-relativity/
http://iaps.institute/mathematical-physics/theory-of-gps/

Three-dimensional far turbulent wake in a passive stratified medium,
axisymmrtric submerged turbulent jet and far swirling turbulent wake are considered using RANS approach. We use methods of a group-theoretical analisys to reduce corresponding semi-emprirical models of turbulence to systems of ordinary differential equations (ODEs). Modified shooting method and asymptotic expansion are used to solve boundary-value problems for obtained systems of ODEs. The constructed solutions are in good agreement with experimental data. Moreover, a detailed comparison with numerical solutions obtained by G. G. Chernykh with co-authors on the basis of the full models of turbulence were conducted.

Kaptsov O. V., Shmidt A. V. A three-dimensional semi-empirical model of a far turbulent wake // J. Appl. Math. Mech., 2015, V. 79, № 5, P. 459–466
Shmidt A. V. Self-Similar solution of the problem of a turbulent flow in a round submerged jet // J. of Appl. Mech. and Tech. Phys., 2015, V. 56, № 3, P. 414–419
Shmidt A. V. Similarity in the far swirling momentumless turbulent wake // J. SFU. Math. & Phys., 2020, V. 13, № 1, P. 79-86

На основе подхода RANS рассмотрены трехмерный дальний турбулентный след в пассивно-стратифицированной среде, осесимметричная затопленная турбулентная струя и дальний закрученный турбулентный след. С помощью методов теоретико-группового анализа соответствующие полуэмпирические модели турбулентности редуцируются к системам обыкновенных дифференциальных уравнений. Поставленные краевые задачи для систем обыкновенных дифференциальных уравнений решены с использованием модифицированного метода стрельбы и асимптотического разложения решения в окрестности особой точки. Построенные решения находятся в хорошем согласии с экспериментальными данными. Кроме того, было проведено детальное сопоставление с численными решениями, полученными Г. Г. Черных с соавторами на основе полных моделей турбулентности.

This talk will give a totally different view to the problem addressed in my previous talk «Discrete orthogonal polynomials: anomalies of time series and boundary effects of polynomial filters».

Using a sort of adaptive filtering we solve the problem of boundary attenuation effects of polynomial filters. The techniques we use may be classified as (elementary) machine learning.

Another facet of the GNSS (Global Navigation Satellite Systems) theory and practice exposed in this talk is the problem of interpolation of positions of GNSS satellites.

Using the data from IGS (International GNSS Service) as an example, we demonstrate a simple but unexpectedly effective technique that allows interpolation of the positions of GPS and GLONASS satellites with an accuracy of a few millimeters. It is natural to call the described interpolation technique «free» since it is not related to polynomials, nor trigonometric and other functions commonly used in standard interpolation techniques.

The free interpolation technique also allows developing much more accurate (nevertheless very simple) models of media that are important in the operation of space navigation systems: the ionosphere, troposphere, etc.

The basis for the development of this method is Big Data, accumulated over many years of operation of satellite navigation systems. We will discuss some common problems of the Big Data we use. The following conclusion turned out to be paradoxical, but real: the main problem with working with big data is that there are too few of them...

This talk is a modified version of my Russian language talk given in 2018: http://www.mathnet.ru/php/presentation.phtml?&presentid=24129&option_lang=eng

Paper references:

1. Pustoshilov, A. S., & Tsarev, S. P. (2017). Universal coefficients for precise interpolation of GNSS orbits from final IGS SP3 data. In 2017 International Siberian Conference on Control and Communications (SIBCON) (pp. 1-6). IEEE. https://ieeexplore.ieee.org/abstract/document/7998463

2. Pustoshilov, A. S., & Tsarev, S. P. (2018). Two-point free nonlinear interpolation of coordinates and velocities of navigation satellites from SP3 data. (in Russian) Achievements of Modern Radioelectronics / No. 12 — 2018 http://www.radiotec.ru/article/22602#english

3. Tsarev, S. P., Denisenko, V. V., & Valikhanov, M. M. (2018). Multidimensional free interpolation framework for high-precision modeling of slant total electron contents in mid-latitude and equatorial regions. http://elib.sfu-kras.ru/handle/2311/109067?locale-attribute=en

In this talk I will first overview some general results concerning the effects of the parallel shear flow on long weakly-nonlinear surface and internal ring waves in a stratified fluid (e.g., oceanic internal waves generated in narrow straits and river-sea interaction zones), generalising the results for surface waves in a homogeneous fluid [1]. We showed that despite the clashing geometries of the waves and the shear flow, there exists a linear modal decomposition (separation of variables) in the far-field set of Euler equations describing the waves in a stratified fluid, more complicated than the known decomposition for plane waves [2,3]. We used it to describe the wavefronts of surface and internal waves, and to derive a 2D cylindrical Korteweg — de Vries (cKdV)-type model for the amplitudes of the waves. The distortion of the wavefronts is described explicitly by constructing the singular solution (envelope of the general solution) of a respective nonlinear first-order differential equation.

Next, we consider a two-layer fluid with a rather general depth-dependent upper-layer current (e.g. a river inflow, or a wind-generated current). In the rigid-lid approximation, we find the necessary singular solution of the nonlinear first-order ordinary differential equation responsible for the adjustment of the speed of the long interfacial ring wave in different directions in terms of the hypergeometric function [4]. This allows us to obtain an analytical description of the wavefronts and vertical structure of the ring waves for a large family of the current profiles and to illustrate their dependence on the density jump and the type and the strength of the current. We will also discuss a 2D generalisation of the long-wave instability criterion for plane interfacial waves on a piecewise-constant current [4], which on physical level manifests itself in the counter-intuitive squeezing of the wavefront of the interfacial ring wave.

Unstationary and stationary two-dimensional flows of incompressible viscoelastic Maxwell medium with upper, low and corotational convective derivatives in the theological constitutive law are considered. A class of partially invariant solutions is analyzed. Using transition to Lagrangian coordinates, an exact solution of the problem of unsteady flow near free-stagnation point was constructed. For the model with Johnson-Segalman convected derivative and special linear dependence of the vertical component of velocity, the general solution was derived. Analysis of the analytical unstationary solution provides a new class of stationary solutions. The solutions found comprise both already known as well as substantially new solutions. Nonsingular solutions of the stress tensor at the critical point and bounded at infinity are constructed. Exact analytical formulae for the stress tensor with the Weissenberg number Wi=1/2 are obtained.

В докладе рассказано об одном новом результате в давно ставшей классикой теории дискретных ортогональных многочленов одной переменной: аномально быстрое убывание их значений вблизи границы для полиномов достаточно большой степени. Данный эффект резко отличает поведение дискретных ортогональных полиномов от поведения их непрерывных аналогов. Практическое значение этого результата для дискретных полиномиальных фильтров, широко применяемых для нахождения аномалий временных рядов, продемонстрировано на примере нахождения разрывов и аномальных выбросов в траекториях спутников GPS и ГЛОНАСС. Дискретные полиномиальные фильтры, с одной стороны, могут обнаруживать очень малые аномалии в разреженных временных рядах (с амплитудой порядка 10^(-11) относительно типичных значений исходных данных). С другой стороны, полученный нами общий результат ограничивает чувствительность таких фильтров вблизи границы анализируемого временного ряда. Основной проблемой при практическом применении метода было преодоление эффекта численной неустойчивости при построении соответствующих дискретных ортогональных полиномов высоких степеней.

Работа поддержана Красноярским математическим центром, финансируемым Минобрнауки РФ в рамках мероприятий по созданию и развитию региональных НОМЦ (Соглашение 075–02–2020–1631).