Geodesic
On properties of solutions of the Cauchy problem for two-dimensional transport equations on a rotating plane
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 1 (2021), pp. 3-10
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The limiting case of the system of equations of two-dimensional gas dynamics in the presence of the Coriolis force, which can be obtained under the assumption of small pressure, is considered. With this approach, the equation for the velocity vector (transport equation) is split off from the system and can be solved separately. An explicit asymptotic representation of a smooth solution to transport equations is obtained with the use of the method of stochastic perturbation along characteristics and the process of formation of singularities of solution is analysed on a specific example. It is concluded that the presence of the Coriolis force prevents formation of singularities. 
@article{VMUMM_2021_1_a0,
     author = {O. S. Rozanova and O. V. Uspenskaya},
     title = {On properties of solutions of the {Cauchy} problem for two-dimensional transport equations on a rotating plane},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {3--10},
     year = {2021},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2021_1_a0/}
}
TY  - JOUR
AU  - O. S. Rozanova
AU  - O. V. Uspenskaya
TI  - On properties of solutions of the Cauchy problem for two-dimensional transport equations on a rotating plane
JO  - Vestnik Moskovskogo universiteta. Matematika, mehanika
PY  - 2021
SP  - 3
EP  - 10
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VMUMM_2021_1_a0/
LA  - ru
ID  - VMUMM_2021_1_a0
ER  - 
%0 Journal Article
%A O. S. Rozanova
%A O. V. Uspenskaya
%T On properties of solutions of the Cauchy problem for two-dimensional transport equations on a rotating plane
%J Vestnik Moskovskogo universiteta. Matematika, mehanika
%D 2021
%P 3-10
%N 1
%U http://geodesic.mathdoc.fr/item/VMUMM_2021_1_a0/
%G ru
%F VMUMM_2021_1_a0
O. S. Rozanova; O. V. Uspenskaya. On properties of solutions of the Cauchy problem for two-dimensional transport equations on a rotating plane. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 1 (2021), pp. 3-10. http://geodesic.mathdoc.fr/item/VMUMM_2021_1_a0/

[1] Vallis G.K., Atmospheric and oceanic fluid dynamics. Fundamentals and large-scale circulation, Cambridge University Press, Cambridge, 2006

[2] Obukhov A.M., “On the geostrophical wind”, Izv. Acad. Nauk (Izvestiya of Academie of Science of URSS). Ser. Geography and Geophysics, XIII (1949), 281–306

[3] Dafermos S.M., Hyperbolic conservation laws in continuum physics, 4th ed., Springer, Berlin–Heidelberg, 2016

[4] Rozanova O.S., “Obrazovanie osobennostei reshenii s kompaktnym nositelem uravnenii Eilera na vraschayuscheisya ploskosti”, Differents. uravneniya, 34:8 (1998), 1114–1118

[5] Rozanova O.S., “Classes of smooth solutions to multidimensional balance laws of gas dynamic type on Riemannian manifolds”, Trends in mathematical physics research, Nova Sci. Publ., Hauppauge, NY, 2004, 155–204

[6] Cheng B., Tadmor E., “Long time existence of smooth solutions for the rapidly rotating shallow-water and Euler equations”, SIAM J. Math. Anal., 39 (2008), 1668–1685

[7] Rozanova O.S., “Singularity formation for rotational gas dynamics”, J. Math. Anal. and Appl., 492 (2020), 124405

[8] Shandarin S.F., Zeldovich Y.B., “Large-scale structure of the universe: turbulence, intermittency, structures in a selfgravitating medium”, Rev. Modern Phys., 61 (1989), 185–220

[9] Chen G.-Q., Liu H., “Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids”, Physica D: Nonlinear Phenomena, 189 (2004), 141–165

[10] Liu H., Tadmor E., “Rotation prevents finite-time breakdown”, Physica D: Nonlinear Phenomena, 188 (2004), 262–276

[11] Korshunova A., Rozanova O., “On effects of stochastic regularization for the pressureless gas dynamics”, Contemp. Appl. Math., 17 (2012), 486–493

[12] Albeverio S., Korshunova A., Rozanova O., “A probabilistic model associated with the pressureless gas dynamics”, Bull. Sci. Math., 137 (2013), 902–922

[13] Egorov A.I., Uravneniya Rikkati, 2-e izd., SOLON-Press, M., 2017

[14] Freidlin M., Markov processes and differential equations: asymptotic problems, Lect. Math. ETH Zürich, Birkhaüser, Basel, 1996

[15] Rozanova O.S., Yu J.-L., Hu C.-K., “On the position of vortex in two-dimensional model of atmosphere”, Nonlinear Analysis: Real World Applications, 13 (2012), 1941–1954

[16] Rozanova O.S., “O svyazi uravneniya Gamiltona–Yakobi c nekotorymi sistemami kvazilineinykh uravnenii”, Tr. In-ta matematiki UrO RAN, 21, no. 2, 2015, 206–219