Affiliation:
1. School of Mathematical Sciences, Peking University , Beijing , People’s Republic of China
Abstract
Abstract
The Chaplygin gas model is both interesting and important in the theory of gas dynamics and conservation laws, all the characteristic families of which are linearly degenerate. Majda conjectured that the shock formation never happens for smooth data. In this article, we prove the conjecture for the two space dimensional axisymmetric case. Different from previous approaches to study wave equations with different speeds, we reformulate the problem in the Lagrangian coordinates and consider a single wave equation with variable coefficients. This not only gives a simpler proof but also enables us to treat the case with large angular velocities.
Subject
General Mathematics,Statistical and Nonlinear Physics
Reference32 articles.
1. S. Alinhac, Temps de vie des solutions régulières des équations d’Euler compressibles axisymétriques en dimension deux, Invent. Math. 111 (1993), no. 3, 627–670.
2. A. Arnold, S. Geevers, I. Perugia, and D. Ponomarev, On the exponential time-decay for the one-dimensional wave equation with variable coefficients, 2022, arXiv e-print.
3. H. B. Benaoum, Modified Chaplygin gas cosmology, Adv. High Energy Phys. 2012 (2012), 357802.
4. D. Christodoulou, The formation of shocks in 3-dimensional fluids, EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, 2007.
5. D. Christodoulou and S. Miao, Compressible flow and Euler’s equations, volume 9 of Surveys of Modern Mathematics, International Press, Somerville, MA; Higher Education Press, Beijing, 2014.