Geodesic
The nonlocal solvability conditions for a system of quasilinear equations of the first order special right-hand sides
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 20 (2018) no. 4, pp. 384-394.

Voir la notice de l'article provenant de la source Math-Net.Ru

The Cauchy problem for a system of first-order quasilinear equations with special right-hand sides is considered. The study of solvability of this system in the original coordinates is based on the method of additional argument. It is proved that the local solution of such system exists and that its smoothness is not lower than the smoothness of the initial conditions. For system of two equations non-local solutions are considered that are continued by finite number of steps from the local solution. Sufficient conditions for the existence of such non-local solution are derived. The proof of the non-local resolvability of the system relies on original global estimates.
Keywords: method of additional argument, global estimates, global estimates, Cauchy problem, first-order partial differential equations. 
@article{SVMO_2018_20_4_a2,
     author = {M. V. Dontsova},
     title = {The nonlocal solvability conditions  for a  system  of   quasilinear  equations of the first order  special right-hand sides},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {384--394},
     publisher = {mathdoc},
     volume = {20},
     number = {4},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2018_20_4_a2/}
}
TY  - JOUR
AU  - M. V. Dontsova
TI  - The nonlocal solvability conditions  for a  system  of   quasilinear  equations of the first order  special right-hand sides
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2018
SP  - 384
EP  - 394
VL  - 20
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SVMO_2018_20_4_a2/
LA  - ru
ID  - SVMO_2018_20_4_a2
ER  - 
%0 Journal Article
%A M. V. Dontsova
%T The nonlocal solvability conditions  for a  system  of   quasilinear  equations of the first order  special right-hand sides
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2018
%P 384-394
%V 20
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SVMO_2018_20_4_a2/
%G ru
%F SVMO_2018_20_4_a2
M. V. Dontsova. The nonlocal solvability conditions  for a  system  of   quasilinear  equations of the first order  special right-hand sides. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 20 (2018) no. 4, pp. 384-394. http://geodesic.mathdoc.fr/item/SVMO_2018_20_4_a2/

[1] M. V. Dontsova, “Nonlocal solvability conditions for Cauchy problem for a system of first order partial differential equations with special right-hand sides”, Ufa Mathematical Journal, 6:4 (2014), 68–80 | MR

[2] M. I. Imanaliev, S. N. Alekseenko, “To the question of the existence of a smooth bounded solution for a system of two first-order nonlinear partial differential equations”, Doklady RAN, 379:1 (2001), 16–21 (In Russ.) | MR | Zbl

[3] M. I. Imanaliev, P. S. Pankov, S. N. Alekseenko, “Method of an additional argument”, Vestnik KazNU. Series: Mathematics, mechanics, informatics, 1, Spec. issue (2006), 60—64 (In Russ.)

[4] S. N. Alekseenko, M. V. Dontsova, “The investigation of a so lvability of the system of equations, describing a distribution of electrons in an electric field of sprite”, Matem. vestnik pedvuzov, universitetov Volgo-Vyatskogo regiona, 14 (2012), 34–41 (In Russ.)

[5] S. N. Alekseenko, T. A. Shemyakina, M. V. Dontsova, “Nonlocal solvability conditions for systems of first order partial differential equations”, St. Petersburg State Polytechnical University Journal. Physics and Mathematics, 177:3 (2013), 190–201 (In Russ.)

[6] S. N. Alekseenko, M. V. Dontsova, “The solvability conditions of the system of long waves in a water rectangular channel, the depth of which varies along the axis”, Zhurnal Srednevolzhskogo matematicheskogo obshchestva, 18:2 (2016), 115–124 (In Russ.) | Zbl

[7] M. V. Dontsova, “Nonlocal solvability conditions of the Cauchy problem for a system of first order partial differential equations with continuous and bounded right-hand sides”, Vestnik of VSU. Series: Physics. Mathematics, 4 (2014), 116–130 (In Russ.) | Zbl

[8] M. V. Dontsova, “The nonlocal existence of a bounded solution of the Cauchy problem for a system of two first order partial differential equations with continuous and bounded right-hand sides”, Vestnik TvGU. Seriya: Prikladnaya matematika, 3 (2014), 21–36 (In Russ.)

[9] T. A. Shemyakina, “Conditions for the existence and diferentiability of solutions of Frankl in the hyperbolic case”, Zhurnal Srednevolzhskogo matematicheskogo obshchestva, 13:2 (2011), 127–131 (In Russ.) | Zbl

[10] S. N. Alekseenko, M. V. Dontsova, D. E. Pelinovsky, “Global solutions to the shallow-water system with a method of an additional argument”, Applicable Analysis, 96:9 (2017), 1444–1465 | DOI | MR | Zbl

[11] T. A. Shemyakina, “The theorem on existence of a bounded solution of the Cauchy problem for the Frankl system of hyperbolic type”, St. Petersburg State Polytechnical University Journal. Physics and Mathematics, 146:2 (2012), 130–140

[12] S. N. Alekseenko, M. V. Dontsova, D. E. Pelinovsky, “Global solutions to the shallow-water system with a method of an additional argument”, Applicable Analysis, 96:9 (2017), 1444–1465 | DOI | MR | Zbl