Page 640 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 640
MATHEMATICAL PHYSICS
Asymptotics of eigenvalue fluctuations for random Schroedinger
operators
Yoel Grinshpon, yoel.grinshpon@mail.huji.ac.il
Hebrew University in Jerusalem, Israel
Random Schroedinger operators arise in mathematical physics as models of quantum particles
in disordered media. Thus, the spectral properties of such operators have attracted a consider-
able amount of attention. In this talk, we will discuss some recent results regarding eigenvalue
asymptotics on various scales, both in the random decaying case and for the celebrated Ander-
son model.
Some of the results are joint work with Jonathan Breuer and Moshe White.
Constructive Method to Solving 3D Navier - Stokes Equations
Alexander Koptev, Koptevav@gumrf.ru
Admiral Makarov State University of Maritime and Inland Shipping,
Saint-Petersburg, Russian Federation
We consider 3D Navier – Stokes equations for motiom of incompressible media. This equations
are of mathematical interest and have a lot of applications to practical problems. For today
many aspects connected with the Navier – Stokes equations have been studied not enough and
need more profound investigation [1-2]. Main unresolved problem is the lack of a constructive
method of solution. How to resolve the Navier – Stokes equations while preserving all nonlinear
terms is the question that needs to be addressed.
The author proposes an approach to this problem the essence of which is to reduce the
basic problem of solving the Navier – Stokes equations to a set of simple tasks. We face to
five more simple tasks that you need to consistently allow. Each of the individual Navier –
Stokes equations must be reduced to a free divergent form and integrated. Resulting equality
can be converted so as to exclude some nonlinear and non-divergent terms. As the result we
arrive to nine equations which link main unknowns u, v, w, p, associated unknowns Ψi where
i = 1, 2, . . . , 15 and an arbitrary additive functions in three variables αi, βi, γi, δi. Considered
together these nine ratios provide the first integral of 3D Navier – Stokes equations [3]. The
well-known integrals of Bernoulli, Euler - Bernoulli and Lagrange - Cauchy are its special cases
[4]. Four of the nine obtained equations represent expressions for basic unknowns. So they
determine the overall structure of the solutions. The remaining five equations can be resolved
relative to six associated unknowns Ψj where j = 10, 11, . . . , 15 only if two conditions of
compatibility are hold. They reduce to two fifth order equations with respect to nine unknown
Ψk where k = 1, 2, . . . , 9. Each set of functions Ψk satisfying a given system determines exact
solution of the Navier - Stokes equations. To complete the solution you need to find the six
remaining associated unknowns Ψj, where k = 10, 11, . . . , 15. Three of the last functions can
be set arbitrarily whereas the remaining three are defined as solution of linear inhomogeneous
equations.
As a result all values presents of the structure formula for main unknowns are defined and
these unknowns are easy to find. Some exact solutions constructed in this way are given in
[5-6]. A similar approach can be applied to construct solutions of the Euler equations for case
of motion for inviscid incompressible media. In all relationships it is enough to put 1 = 0.
Re
638
Asymptotics of eigenvalue fluctuations for random Schroedinger
operators
Yoel Grinshpon, yoel.grinshpon@mail.huji.ac.il
Hebrew University in Jerusalem, Israel
Random Schroedinger operators arise in mathematical physics as models of quantum particles
in disordered media. Thus, the spectral properties of such operators have attracted a consider-
able amount of attention. In this talk, we will discuss some recent results regarding eigenvalue
asymptotics on various scales, both in the random decaying case and for the celebrated Ander-
son model.
Some of the results are joint work with Jonathan Breuer and Moshe White.
Constructive Method to Solving 3D Navier - Stokes Equations
Alexander Koptev, Koptevav@gumrf.ru
Admiral Makarov State University of Maritime and Inland Shipping,
Saint-Petersburg, Russian Federation
We consider 3D Navier – Stokes equations for motiom of incompressible media. This equations
are of mathematical interest and have a lot of applications to practical problems. For today
many aspects connected with the Navier – Stokes equations have been studied not enough and
need more profound investigation [1-2]. Main unresolved problem is the lack of a constructive
method of solution. How to resolve the Navier – Stokes equations while preserving all nonlinear
terms is the question that needs to be addressed.
The author proposes an approach to this problem the essence of which is to reduce the
basic problem of solving the Navier – Stokes equations to a set of simple tasks. We face to
five more simple tasks that you need to consistently allow. Each of the individual Navier –
Stokes equations must be reduced to a free divergent form and integrated. Resulting equality
can be converted so as to exclude some nonlinear and non-divergent terms. As the result we
arrive to nine equations which link main unknowns u, v, w, p, associated unknowns Ψi where
i = 1, 2, . . . , 15 and an arbitrary additive functions in three variables αi, βi, γi, δi. Considered
together these nine ratios provide the first integral of 3D Navier – Stokes equations [3]. The
well-known integrals of Bernoulli, Euler - Bernoulli and Lagrange - Cauchy are its special cases
[4]. Four of the nine obtained equations represent expressions for basic unknowns. So they
determine the overall structure of the solutions. The remaining five equations can be resolved
relative to six associated unknowns Ψj where j = 10, 11, . . . , 15 only if two conditions of
compatibility are hold. They reduce to two fifth order equations with respect to nine unknown
Ψk where k = 1, 2, . . . , 9. Each set of functions Ψk satisfying a given system determines exact
solution of the Navier - Stokes equations. To complete the solution you need to find the six
remaining associated unknowns Ψj, where k = 10, 11, . . . , 15. Three of the last functions can
be set arbitrarily whereas the remaining three are defined as solution of linear inhomogeneous
equations.
As a result all values presents of the structure formula for main unknowns are defined and
these unknowns are easy to find. Some exact solutions constructed in this way are given in
[5-6]. A similar approach can be applied to construct solutions of the Euler equations for case
of motion for inviscid incompressible media. In all relationships it is enough to put 1 = 0.
Re
638
