Geodesic
Asymptotic equations of gas dynamics: qualitative analysis, construction of solutions, and applications
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the IV International Scientific Conference "Actual Problems of Applied Mathematics". Kabardino-Balkar Republic, Nalchik, Elbrus Region, May 22–26, 2018. Part I, Tome 165 (2019), pp. 47-62

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In this paper, we propose asymptotic expansions for the velocity potential and obtain asymptotic equations of gas dynamics for irrotational isentropic flows of an ideal gas: an equation of linear theory, a nonlinear equation for supersonic flows, and a nonlinear transonic equation. We construct some exact particular solutions for the asymptotic nonlinear transonic equation, which takes into account transverse perturbations. Based on a linear asymptotic equation, we examine the dynamic stability of an elastic deformable element of a channel at a subsonic flow rate of a gas or liquid. The study of stability is carried out in a statement corresponding to small perturbations of a homogeneous flow and small deformations of an elastic element, and is based on the construction of a positive definite functional. Sufficient stability conditions are obtained.
Keywords: aerodynamics, partial differential equation, asymptotic expansion, transonic gas flow, channel, de Laval nozzle, aerohydroelasticity, dynamic stability. 
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     title = {Asymptotic equations of gas dynamics: qualitative analysis, construction of solutions, and applications},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
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     volume = {165},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2019_165_a4/}
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P. A. Vel'misov; J. А. Tamarova; E. P. Semеnova. Asymptotic equations of gas dynamics: qualitative analysis, construction of solutions, and applications. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the IV International Scientific Conference "Actual Problems of Applied Mathematics". Kabardino-Balkar Republic, Nalchik, Elbrus Region, May 22–26, 2018. Part I, Tome 165 (2019), pp. 47-62. http://geodesic.mathdoc.fr/item/INTO_2019_165_a4/