Solvability conditions of the Cauchy problem for a system of first-order quasi-linear equations, where ${f_1}(t,x), {f_2}(t,x), {S_1}, {S_2}$ are given functions

M. V. Dontsova

Čebyševskij sbornik, Tome 24 (2023) no. 2, pp. 165-178

Résumé

We consider a Cauchy problem for a system of two quasilinear first order partial differential equations with continuous and bounded free terms. Theorems on the local and nonlocal existence and uniqueness of solutions to the Cauchy problem are formulated and proved. The sufficient conditions for the existence and uniqueness of a local solution of the Cauchy problem in the initial coordinates at which the solution has the same smoothness with respect to $ x $ as the initial functions of the Cauchy problem are determined. The sufficient conditions for the existence and uniqueness of a nonlocal solution of the Cauchy problem in the initial coordinates (for a given finite interval $t\in[0,T]$) are determined. Local existence and uniqueness theorem of the solution of the Cauchy problem for a system of quasilinear first order partial differential equations with continuous and bounded free terms is proved with the method of an additional argument. The investigation of a nonlocal solvability of the Cauchy problem is based on the method of an additional argument. The proof of the nonlocal solvability of the Cauchy problem for a system of quasilinear first order partial differential equations with continuous and bounded free terms relies on global estimates.

Détail

Keywords: a system of quasilinear equations, the method of an additional argument, Cauchy problem, global estimates.

Bibliographie

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

[2] Alekseenko, S. N., Dontsova, M. V., “The local existence of a bounded solution of the system of equations, describing a distribution of electrons in low-pressure plasma in an electric field of sprite”, Matematicheskij vestnik pedvuzov i universitetov Volgo-Vyatskogo regiona, 2013, no. 15, 52–59

[3] Alekseenko, S. N., Dontsova, M. V., “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 | Zbl

[4] Alekseenko, S. N., Shemyakina, T. A., Dontsova, M. V., “Nonlocal solvability conditions for systems of first order partial differential equations”, Nauchno-tekhnicheskie vedomosti SPbGPU. Fiziko - matematicheskie nauki, 2013, no. 3(177), 190–201

[5] Dontsova, M. V., “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, 2014, no. 3, 21–36

[6] Dontsova, M. V., “Nonlocal solvability conditions of the Cauchy problem for a system of first order partial differential equations with continuous and bounded right-hand sides”, Vestnik VGU. Seriya: Fizika. Matematika, 2014, no. 4, 116–130 | MR | Zbl

[7] Dontsova, M. V., “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), 71–82 | MR | Zbl

[8] Dontsova, M. V., “Study of solvability for a system of first order partial differential equations with free terms”, Materialy Mezhdunarodnogo molodezhnogo nauchnogo foruma «LOMONOSOV-2014» (Moscow, 2014) (1 electron. wholesale. disc (DVD-ROM))

[9] Dontsova, M. V., “Investigation of the solvability of a system of quasilinear equations of the first order”, Mezhdunarodnaya konferenciya po differencial'nym uravneniyam i dinamicheskim sistemam, Tezisy dokladov (Suzdal, 2016), 68 | MR

[10] Dontsova, M. V., “The nonlocal solvability conditions for a system of quasilinear equations of the first order special right-hand sides”, Zhurnal Srednevolzhskogo matematicheskogo obshchestva, 20:4 (2018), 384–394

[11] Imanaliev, M. I., Alekseenko, S. N., “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 | MR | Zbl

[12] Imanaliev, M. I., Pankov, P. S., Alekseenko, S. N., “Method of an additional argument”, Vestnik KazNU. Ceriya matematika, mekhanika, informatika, 2006, no. 1, Spec. vypusk, 60–64

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

[14] Shemyakina, T. A., “The theorem on existence of a bounded solution of the Cauchy problem for the Frankl system of hyperbolic type”, Nauchno-tekhnicheskie vedomosti SPbGPU. Fiziko - matematicheskie nauki, 2012, no. 2(146), 130–140

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