研究了具有含尘气体状态方程的二维轴对称Euler方程组整体经典解的存在性.导出了具有含尘气体状态方程的二维轴对称Euler方程组的特征分解.通过该特征分解式,给出了关于初值的一个充分条件,使得二维轴对称Euler方程组的Cauchy问题在决定区域内存在整体经典解.
We studied the global existence of classical solutions of two-dimensional axisymmetric Euler equations for dusty gas. We derived a group of characteristic decompositions for a two-dimensional axisymmetric Euler system. Using these characteristic decompositions, we determined a sufficient condition for the initial data to ensure the global existence of classical solutions to the Cauchy problem.
[1] John F. Formation of singularities in one-dimensional nonlinear waves propagation [J]. Communications on Pure and Applied Mathematics, 1974, 27: 377-405.
[2] Lax P. Development of singularities of solutions on nonlinear hyperbolic partial differential equations [J]. Journal of Mathematical Physics, 1964, 5: 611-613.
[3] Li T P. Development of singularities in the nonlinear waves for quasilinear hyperbolic partial differential equations [J]. Journal of Differential Equations, 1979, 30: 92-111.
[4] Pan R H, Zhu Y. Singularity formation for one dimensional full Euler equations [J]. Journal of Differential Equations, 2016, 261: 7132-7144.
[5] Sideris T C. Formation of singularities in three-dimensional compressible fluids [J]. Communications in Mathematical Physics, 1985, 101: 475-485.
[6] Chen G, Pan R H, Zhu S G. Singularity formation for the compressible Euler equations [J]. SIAM Journal on Mathematical Analysis, 2017, 49: 2591-2614.
[7] Chen G. Formation of singularity and smooth wave propagation for the non-isentropic compressible Euler equations [J]. Journal of Hyperbolic Differential Equations, 2011, 8: 671-690.
[8] Lai G, Zhao Q. Existence of global bounded smooth solutions for the one-dimensional nonisentropic Euler system [J]. Mathematical Methods in the Applied Sciences, 2021, 44: 2226-2236.
[9] Chen G, Young R. Shock-free solutions of the compressible Euler equations [J]. Archive for Rational Mechanics and Analysis, 2015, 217: 1265-1293.
[10] Li T T. Global classical solutions for quasilinear hyperbolic systems [M]. Paris: John Wiley and Sons, 1994.
[11] Lin L W, Liu H X, Yang T. Existence of globally bounded continuous solutions for nonisentropic gas dynamics equations [J]. Journal of Mathematical Analysis and Applications, 1997, 209: 492-506.
[12] Liu F G. Global smooth resolvability for one-dimensional gas dynamics systems [J]. Nonlinear Analysis, 1999, 36: 25-34.
[13] Zhu C J. Global smooth solution of the nonisentropic gas dynamics system [J]. Proceedings of the Royal Society of Edinburgh Section A, 1996, 126: 769-775.
[14] Serre D. Solutions classiques globales des équations d’Euler pour un fluide parfait compressible [J]. Annales de Institut Fourier, 1997, 47: 139-153.
[15] Grassin M. Global smooth solutions to Euler equations for a perfect gas [J]. Indiana University Mathematics Journal, 1998, 47: 1397-1432.
[16] Magali L M. Global smooth solutions of Euler equations for van der Waals gases [J]. SIAM Journal on Mathematical Analysis, 2011, 43: 877-903.
[17] Godin P. Global existence of a class of smooth 3D spherically symmetric flows of Chaplygin gases with variable entropy [J]. Journal de Mathematiques Pures et Appliquees, 2007, 87: 91-117.
[18] Hou F, Yin H C. Global smooth axisymmetric solutions to 2D compressible Euler equations of Chaplygin gases with non-zero vorticity [J]. Journal of Differential Equations, 2019, 267: 3114-3161.
[19] Hou F, Yin H C. On global axisymmetric solutions to 2D compressible full Euler equations of Chaplygin gases [J]. Discrete and Continuous Dynamical Systems, 2010, 40: 1435-1492.
[20] Kong D X, Liu K, Wang Y. Global existence of smooth solutions to two-dimensional compressible isentropic Euler equations for Chaplygin gases [J]. Science China Mathematics, 2010, 53: 719-738.
[21] Saffmann P G. The stability of laminar flow of a dusty gas [J]. Journal of Fluid Mechanics, 1962, 13: 120-128.
[22] Steiner H, Hirschler T. A self-similar solution of a shock propagation in a dusty gas [J]. European Journal of Mechanics B/Fluids, 2002, 21: 371-380.
[23] Li T T, Yu W C. Boundary value problems for quasilinear hyperbolic systems [M]. Durham: Duke University, 1985.
[24] Li J Q, Zhang T, Zheng Y X. Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations [J]. Communications in Mathematical Physics, 2006, 267: 1-12.
[25] Hu Y, Li J Q, Sheng W C. Degenerate Goursat-type boundary value problems arising from the study of two-dimensional isothermal Euler equations [J]. Zeitschrift fũr Angewandte Mathematik und Physik, 2012, 63: 1021-1046.
[26] Lai G. On the expansion of a wedge of van der Waals gas into a vacuum [J]. Journal of Differential Equations, 2015,, 259: 1181-1202.
[27] Lai G. On the expansion of a wedge of van der Waals gas into a vacuum Ⅱ [J]. Journal of Differential Equations, 2016, 260: 3538-3575.
[28] Lai G. Interaction of composite waves of the two-dimensional full Euler equations for van der Waals gases [J]. SIAM Journal on Mathematical Analysis, 2018, 50: 3535-3597.
[29] Lai G. Global solutions to a class of two-dimensional Riemann problems for the isentropic Euler equations with general equations of state [J]. Indiana University Mathematics Journal, 2019, 68: 1409-1464.
[30] Li J Q, Yang Z C, Zheng Y X. Characteristic decompositions and interactions of rarefaction waves of 2-D Euler equations [J]. Journal of Differential Equations, 2011, 250: 782-798.
[31] Li J Q, Zheng Y X. Interaction of rarefaction waves of the two-dimensional self-similar Euler equations [J]. Archive for Rational Mechanics and Analysis, 2009, 193: 623-657.
[32] Li J Q, Zheng Y X. Interaction of four rarefaction waves in the bi-symmetric class of the two-dimensional Euler equations [J]. Communications in Mathematical Physics, 2010, 296: 303-321.
