zbMATH — the first resource for mathematics

On the one-dimensional motion of the polytropic ideal gas non-fixed on the boundary. (English) Zbl 0598.34021
Several initial-boundary value problems of equations describing the one- dimensional motion of the ideal polytropic gas are discussed. In our problems the gas is put into the region where the ends are not fixed, so the (specific) volume of the gas may grow infinitely as time increases.
In the article the above conjecture is shown, that is, the temporally global existence of the unique and classical solutions and the growth of the (specific) volume of the gas under some assumptions are proved. To be precise, at least under the assumption guaranteeing the global existence of the solution, the upper limit of the volume of the gas tends to infinity as time increases. Moreover, adding some (not so strong) assumptions, the (specific) volume tends to infinity, and we can find the upper and lower bounds of the order of growth.

MSC:
 34B99 Boundary value problems for ordinary differential equations 35B40 Asymptotic behavior of solutions to PDEs 35B45 A priori estimates in context of PDEs 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 76N15 Gas dynamics, general
Full Text:
References:
 [1] Greenberg, J.M; Li, T.T, The effect of boundary damping for the quasilinear wave equation, J. differential equations, 52, 66-75, (1984) · Zbl 0576.35080 [2] Itaya, N, On the Cauchy problem for the system of fundamental equations describing the movement of compressible viscous fluid, (), 60-120 · Zbl 0219.76080 [3] Itaya, N, On the fundamental system of equations for compressible viscous fluid, Sûgaku, 28, 121-136, (1976), [Japanese] [4] Kawashima, S; Nishida, T, Global solutions to the initial value problem for the equations of one-dimensional motion of viscous polytropic gases, J. math. Kyoto univ., 21, 825-837, (1981) · Zbl 0478.76097 [5] Kazhykhov (Kazhikhov), A.V, Sur la solubilité globale des problèmes monodimensionnels aux valeurs initiales-limitées pour LES équations d’un gaz visquex et calorifère, C. R. acad. sci. Paris Sér. A, 284, 317-320, (1977) · Zbl 0355.35071 [6] КaзичoV, a.β, К тeorии кraevыч зaдaч для uravnenий oдnoмernoгo necтaциonarnoгo дVижenvя Vязкoгo тeплoпrovoдnoгo гaзa, V cб.: кraevыe зaдaчи для uravnenий гидroдиnaмики, Novocибиrcк: изд. иN-тa гидroдиnaмики, 50, 37-62, (1981) [7] КaзичoV, A.V; чeлuчиN, V.V, Oдnoзnaчnaя raзreчимocть “V цeлoм” пo vreмenи naчaльno-кraevыч зaдaч для oдnoмernыч uravnenий Vязкoгo гaзa, Пrиикл. mam. MAX., 41, 282-291, (1977) [8] Nash, J, Le problème de Cauchy pour LES équations differentielles d’un fluide général, Bull. soc. math. France, 90, 487-497, (1962) · Zbl 0113.19405 [9] Tani, A, On the first initial-boundary value problem of compressible viscous fluid motion, Publ. res. inst. math. sci., 13, 193-253, (1977) · Zbl 0366.35070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.