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X СИБИРСКИЙ КОНГРЕСС ЖЕНЩИН-МАТЕМАТИКОВ
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Сведения об авторах
Sreelatha Chandragiri
PhD student
Siberian Federal University Krasnoyarsk, Russia Email: srilathasami66@gmail. com E.K Leinartas Scientific adviser, professor Siberian Federal University Krasnoyarsk, Russia Email: [email protected]
Information about authors
Срелатаха Чандрагири
Аспирант
Сибирский федеральный университет Красноярск, Россия Эл. почта: [email protected] Е. К. Лейнартас
Научный руководитель, профессор Сибирский федеральный университет Красноярск, Россия Эл. почта: [email protected]
UDK 517.956.8 Yu.Yu. Klevtsova
Siberian Regional Hydrometeorological Research Institute Siberian State University of Telecommunications and Information Sciences
ON THE RATE OF CONVERGENCE AS t ^ +<x> OF THE DISTRIBUTIONS OF SOLUTIONS TO THE STATIONARY MEASURE FOR THE STOCHASTIC SYSTEM OF THE QUASI-SOLENOIDAL LORENZ MODEL FOR A BAROCLINIC ATMOSPHERE
It was obtained the sufficient conditions on the right-hand side and the parameters of Lorenz model for a baroclinic atmosphere with white noise perturbation for existence of a unique stationary measure of Markov semigroup defined by solutions of the Cauchy problem ^ for this system and^ for the exponential convergence of the distributions of solutions to the stationary measure as t ^ +<x>. Keywords: Lorenz model, white noise perturbation, stationary measure, rate of convergence
We consider the system of equations for the quasi-solenoidal Lorenz model for a baroclinic atmosphere
d
— Aiu + vA2u + A3u + B(u) = g , t > 0, (1)
dt
on the two-dimensional unit sphere S centered at the origin of the spherical polar coordinates (X, 9), n n
Xe [0,2п), фе
2 2
ц = sin ф. Here v > 0 is the kinematic viscosity,
u(t, x, ra) = (ui (t, x, ra), u 2(t, x, ra)) is an unknown vector function and
g(t,x,ra) = (g1(t,x,ra),g2(t,x,ra))T is a given vector function, x = (X,raeQ, (Q,P,F) is a complete probability space,
ЛТ СИБИРСКИЙ КОНГРЕСС ЖЕНЩИН-МАТЕМАТИКОВ
Ai =
Г-А 0
0 - А + у/
A 2 =
ГА2
о
0 А
A з =
г- k о А
2k0 А
к о А - (2к о + ki + vy)A + pI
B(u) = (J(Aui + 2|^,ui) + J(Au2,u2),J(Au2— Yu2,ui) + J(Aui + 2^,u2))T . Also, y, p, k0, ki > 0 are numerical parameters, I is the identity operator, J(y, 9) = — V^x is
2 2 — 1
the Jacobi operator and Ay = ((i — ^ + (i — ^ ) vxx is the Laplace-Beltrami operator on the
sphere S. A random vector function g = f + n is taken as the right-hand side of (1); here
T T
f(x)=(fi(x),f2(x)) and n(t,x,ro) = (r|i(t,x,ro),n2(t,x,ro)) is a white noise in t. In [1] and in the
present work it was obtained for existence of a unique stationary measure of Markov semigroup defined
by solutions of the Cauchy problem for (1) and for the exponential convergence of the distributions of
solutions to the stationary measure as t ^ +<x> the sufficient conditions on the right-hand side of (1) and
the parameters v, y , p, ko, ki:
k0 < inf çOX Ç(0 =
2
i=1,2,..., i*
(KO -Y )2
(3v j2(i)(j(i) + Y) + x(j(i))
+
^3v j2(i)(j(i
2 (i)(j(i) + Y) + X(j(i))f + (j(i) - Y)2 (v2 j3 (i)(j(i) + Y) + v j(i)x(j(i)))),
2
X(y) = (kl + vy)(y +Yy) + p(Y + y), j(y) = y(y +1), y > 0; u =
.-1
c* 2v
1+c*+1
> 1,
k(i), if y ^ 2, r, .
c* = < r - the integer part of r .
U(2), if y = 2, M
A similar result is obtained for the equation of a barotropic atmosphere and the two-dimensional Navier-Stokes equation. A comparative analysis with some of the available related results is given for the latter.
The author was supported by the Russian Foundation for Basic Research (Grant 14-01-31110).
References
1. Klevtsova Yu. Yu. On the rate of convergence as t ^ +<x> of the distributions of solutions to the stationary measure for the stochastic system of the Lorenz model describing a baroclinic atmosphere // Sb. Math., 208, No. 7, 929-976 (2017).
About the author
Yulia Yur'evna Klevtsova
PhD in Physics and Mathematics, Leading Scientific Researcher
Siberian Regional Hydrometeorological Research Institute Associate Professor
Siberian State University of Telecommunications and Information Sciences Email: [email protected]
Сведения об авторе Юлия Юрьевна Клевцова
канд. физ.-мат. наук, ведущий научный сотрудник Сибирский региональный гидрометеорологический научно-исследовательский институт Доцент
Сибирский государственный университет телекоммуникаций и информатики Эл. почта: [email protected]
