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Báo cáo hóa học: "SOME ELEMENTARY INEQUALITIES IN GAS DYNAMICS EQUATION"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: SOME ELEMENTARY INEQUALITIES IN GAS DYNAMICS EQUATION | SOME ELEMENTARY INEQUALITIES IN GAS DYNAMICS EQUATION V. A. KLYACHIN A. V. KOCHETOV AND V. M. MIKLYUKOV Received 12 January 2005 Accepted 25 August 2005 We describe the sets on which difference of solutions of the gas dynamics equation satisfy some special conditions. By virtue of nonlinearity of the equation the sets depend on the solution gradient quantity. We show double-ended estimates of the given sets and some properties of these estimates. Copyright 2006 V. A. Klyachin et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Main results Consider the gas dynamics equation ỈÉ ơ f D fxi 0 I Oxi C 1 1 1.1 where Ơ t 1 - 1 td1 1 -1 1.2 Here is a constant -TO TO. This equation describes the velocity potential of a steady-state flow of ideal gas in the adiabatic process. In the case n 2 the parameter characterizes the flow of substance. For different values it can be a flow of gas fluid plastic electric or chemical field in different mediums and so forth see e.g. 1 Section 2 2 Section 15 Chapter IV . For 1 0 we assume ơ t exp I - 2t2 1.3 Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006 Article ID 21693 Pages 1-29 DOI 10.1155 JIA 2006 21693 2 Some elementary inequalities in gas dynamics equation The case of y -1 is known as the minimal surface equation Chaplygin s gas Vf div J 0. 1 IV f 2 1.4 For y -TO 1.1 becomes the Laplace equation. In general a solution of 1.1 with a function Ơ of variables x1 . xn is called Ơ -harmonic function. Such functions were studied in many works see. e.g. 3 4 and literature quoted therein . We set Qy Rn for y 1 Qy j k G Rn IkI for y 1. 1.5 The following inequalities were crucial in previous analysis of solutions to 1.1 for y -1 see 5-9 n n C1 X ki - hi 2 E ơ IkI ki - Ơ InDndtki- nò k n G Qy 1.6 i 1 i 1 nn z ơ IkDki - ơ InI ni 2 C2 ơ IkDki -