Geodesic
Unique weak solvability of a nonlinear initial boundary value problem with distributed parameters in a netlike domain
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 13 (2017) no. 3, pp. 264-277 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This paper considers a fairly wide range of issues related to the solvability of the initial boundary value problem of the Navier–Stokes equations with distributed parameters on the net like region of the space ${\rm {\mathcal R}}^{n} $ ($n\ge 2$). The authors here develop an idea, advanced in their work for the case of $n=1$ (the problems with distributed parameters on the graph), in the direction of the dimension increase $n$ and in forming the correct Hadamard conditions for the studied initial boundary value problem. The general scheme of the study is classical: the problem is solved in the functional space which is selected (the space of feasible solutions) and a special basis is formed for it, the problem of approximate solutions is settled by the Faedo–Galerkin method, for which a priori estimates of the energy inequalities type are set and the weak compactness of the family of these solutions is shown based on these estimates. Using non-burdensome conditions, the smoothness of the solution to the time variable is demonstrated. The uniqueness of the weak solution is shown in the particular case $n=2$, a feature quite often encountered in practice. The estimate for the norm of weak solution makes it possible to establish the continuous dependence of the weak solution from the initial data of the problem. The results obtained in this way are of interest to applications in the field of fluid mechanics and related sections of continuum mechanics, namely for the analysis of optimum control dynamics problems of multiphase media. It should be noted that the methods and approaches can be broadly generalized and are applicable to a wide class of nonlinear problems. Refs 20.
Keywords: boundary value problem, distributed parameters on the netlike domain, the existence of a weak solution, the uniqueness conditions, Hadamard correctness. 
@article{VSPUI_2017_13_3_a3,
     author = {V. V. Provotorov and V. I. Ryazhskikh and Yu. A. Gnilitskaya},
     title = {Unique weak solvability of a nonlinear initial boundary value problem with distributed parameters in a netlike domain},
     journal = {Vestnik Sankt-Peterburgskogo universiteta. Prikladna\^a matematika, informatika, processy upravleni\^a},
     pages = {264--277},
     year = {2017},
     volume = {13},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VSPUI_2017_13_3_a3/}
}
TY  - JOUR
AU  - V. V. Provotorov
AU  - V. I. Ryazhskikh
AU  - Yu. A. Gnilitskaya
TI  - Unique weak solvability of a nonlinear initial boundary value problem with distributed parameters in a netlike domain
JO  - Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
PY  - 2017
SP  - 264
EP  - 277
VL  - 13
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VSPUI_2017_13_3_a3/
LA  - en
ID  - VSPUI_2017_13_3_a3
ER  - 
%0 Journal Article
%A V. V. Provotorov
%A V. I. Ryazhskikh
%A Yu. A. Gnilitskaya
%T Unique weak solvability of a nonlinear initial boundary value problem with distributed parameters in a netlike domain
%J Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
%D 2017
%P 264-277
%V 13
%N 3
%U http://geodesic.mathdoc.fr/item/VSPUI_2017_13_3_a3/
%G en
%F VSPUI_2017_13_3_a3
V. V. Provotorov; V. I. Ryazhskikh; Yu. A. Gnilitskaya. Unique weak solvability of a nonlinear initial boundary value problem with distributed parameters in a netlike domain. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 13 (2017) no. 3, pp. 264-277. http://geodesic.mathdoc.fr/item/VSPUI_2017_13_3_a3/

[1] Gnilitskaya Yu. A., Mathematical modeling and numerical study of processes in netlike objects described by evolutionary equations, Cand. Diss. of physical and mathematical sciences, Voronezh State Technical University, Voronezh, 2015, 224 pp. (in Russian)

[2] Volkova A. S., Provotorov V. V., “Generalized solutions and generalized eigenfunctions of boundary-value problems on a geometric graph”, Russian Mathematics. Proceeding of Higher Educatianal institutions, 58:3 (2014), 1–13 (in Russian) | MR | Zbl

[3] Volkova A. S., Gnilitskaya Yu. A., Provotorov V. V., “On the solvability of boundary-value problems for parabolic and hyperbolic equations on geometrical graphs”, Automation and Remote Control, 75:2 (2014), 405–412 | DOI | MR | Zbl

[4] Provotorov V. V., Gnilitskaya Yu. A., “Boundary control of a wave system in the space of generalized solutions on a graph”, Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2013, no. 3, 112–120 (in Russian)

[5] Mir Publ., M., 1972, 587 pp. | MR

[6] Mir Publ., M., 1971, 371 pp. | MR | Zbl

[7] Ladyzhenskaya O. A., Boundary value problems of mathematical physics, Nauka Publ., M., 1973, 407 pp. (in Russian) | MR

[8] Podvalny S. L., Provotorov V. V., “Starting control of a parabolic system with distributed parameters on a graph”, Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2015, no. 3, 126–142 (in Russian) | MR

[9] Kutateladze S. S., Styrikovich M. A., Hydrodynamics of gas-liquid systems, Energiya Publ., M., 1976, 296 pp. (in Russian) | MR

[10] Provotorov V. V., “Boundary control of a parabolic system with distributed parameters on a graph in the class of summable functions”, Automation and Remote Control, 76:2 (2015), 318–322 | DOI | MR | Zbl

[11] Podvalny S. L., Provotorov V. V., “The questions of controllability of a parabolic systems with distributed parameters on the graph”, Intern. Conference “Stability and Control Processes” in memory of V. I. Zubov (SCP) (2015), 117–119

[12] Provotorov V. V., “Optimal control of a parabolic system with distributed parameters on a graph”, Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2014, no. 3, 154–163 (in Russian) | MR

[13] Potapov D. K., “Optimal control of higher order elliptic distributed systems with a spectral parameter and discontinuous nonlinearity”, Journal of Computer and System Sciences International, 52:2 (2009), 180–185 (in Russian) | DOI | MR

[14] Aleksandrov A. Yu., Zhabko A. P., “On stability of solutions to one class of nonlinear difference systems”, Siberian Mathematical Journal, 44:6 (2003), 951–958 (in Russian) | DOI | MR | Zbl

[15] Aleksandrov A., Aleksandrova E., Zhabko A., “Asymptotic stability conditions for certain classes of mechanical systems with time delay”, WSEAS Transactions on Systems and Control, 9 (2014), 388–397 | MR

[16] Aleksandrov A. Yu., Platonov A. V., “On stability and dissipativity of some classes of complex systems”, Automation and Remote Control, 70:8 (2009), 1265–1280 (in Russian) | DOI | MR | Zbl

[17] Veremey E. I., Korchanov V. M., “Multiobjective stabilization of the dynamic systems from a certain class”, Automation and Remote Control, 1988, no. 9, 126–137 (in Russian) | MR

[18] Veremey E. I., Sotnikova M. V., “Plasma stabilization by prediction with stable linear approximation”, Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2011, no. 1, 116–133 (in Russian)

[19] Karelin V. V., “Penalty functions in the control problem of an observation process”, Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2010, no. 4, 109–114 (in Russian)

[20] Kamachkin A. M., Yevstafyeva V. V., “Oscillations in a relay control system at an external disturbance”, Control Applications of Optimization 2000, Proceedings of the 11th IFAC Workshop, v. 2, 2000, 459–462