Wagner, David H. Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions. (English) Zbl 0647.76049 J. Differ. Equations 68, 118-136 (1987). (Author‘s summary.) This paper demonstrates the equivalence of the Euler and the Lagrangian equations of gas dynamics in one space dimension for weak solutions which are bounded and measurable in Eulerian coordinates. The precise hypotheses include all known global solutions on \({\mathbb{R}}\times {\mathbb{R}}\) \(+\). In particular, solutions containing vacuum states (zero mass density) are included. Furthermore, there is a one-to- one correspondence between the convex extensions of the two systems, and the corresponding admissibility criteria are equivalent. In the presence of a vacuum, the definition of weak solution for the Lagrangian equations must be strengthened to admit test functions which are discontinuous at the vacuum. As an application, we translate a large-data existence result of DiPerna for the Euler equations for isentropic gas dynamics into a similar theorem for the Lagrangian equations. Reviewer: B.Sleeman Cited in 1 ReviewCited in 80 Documents MSC: 76N15 Gas dynamics (general theory) 35Q99 Partial differential equations of mathematical physics and other areas of application Keywords:Lagrangian equations; gas dynamics; Eulerian coordinates; global solutions; vacuum states; zero mass density; weak solution; isentropic gas dynamics PDFBibTeX XMLCite \textit{D. H. Wagner}, J. Differ. Equations 68, 118--136 (1987; Zbl 0647.76049) Full Text: DOI References: [1] Antman, S. S.; Osborne, J. E., The principle of virtual work and integral laws of motion, Arch. Rational Mech. Anal., 69, 231-261 (1979) · Zbl 0403.73003 [2] Courant, R.; Friedrichs, K. O., Supersonic Flow and Shock Waves (1948), Interscience: Interscience New York · Zbl 0041.11302 [3] DiPerna, R. J., Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys., 91, 1-30 (1983) · Zbl 0533.76071 [4] Donoghue, W. F., Distributions and Fourier Transforms (1969), Academic Press: Academic Press New York/London · Zbl 0188.18102 [5] Federer, H., Geometric Measure Theory (1969), Springer-Verlag: Springer-Verlag New York · Zbl 0176.00801 [6] Friedrichs, K. O.; Lax, P. D., Systems of conservation equations with a convex extension, (Proc. Nat. Acad. Sci. U.S.A., 68 (1971)), 1686-1688 · Zbl 0229.35061 [7] P. D. Lax, Shock waves and entropy, in “Contributions to Nonlinear Functional Analysis” (E. A. Zarantonello, Ed.), pp. 603-634, Academic Press, New York.; P. D. Lax, Shock waves and entropy, in “Contributions to Nonlinear Functional Analysis” (E. A. Zarantonello, Ed.), pp. 603-634, Academic Press, New York. [8] Liu, T. P., Initial-boundary-value problems for gas dynamics, Arch. Rational Mech. Anal., 64, 137-168 (1977) · Zbl 0357.35016 [9] Liu, T. P.; Smoller, J., The vacuum state in isentropic gas dynamics, Advan. Appl. Math., 1, 345-359 (1980) · Zbl 0461.76055 [10] Nishida, T.; Smoller, J., Mixed problems for nonlinear conservation laws, J. Differential Equations, 23, 244-269 (1977) · Zbl 0303.35052 [11] Vol’pert, A. I., The spaces BV and quasilinear equations, Math. USSR-Sb., 2, 225-267 (1967) · Zbl 0168.07402 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.