YU Kangning, GUO Lihui. Limits of Riemann Solutions for Generalized Chaplygin Gas Magnetohydrodynamic Euler Equations With Source Terms[J]. Applied Mathematics and Mechanics, 2020, 41(4): 420-437. doi: 10.21656/1000-0887.400122
Citation: YU Kangning, GUO Lihui. Limits of Riemann Solutions for Generalized Chaplygin Gas Magnetohydrodynamic Euler Equations With Source Terms[J]. Applied Mathematics and Mechanics, 2020, 41(4): 420-437. doi: 10.21656/1000-0887.400122

Limits of Riemann Solutions for Generalized Chaplygin Gas Magnetohydrodynamic Euler Equations With Source Terms

doi: 10.21656/1000-0887.400122
Funds:  The National Natural Science Foundation of China(11761068;11401508;11461066)
  • Received Date: 2019-03-25
  • Rev Recd Date: 2019-07-13
  • Publish Date: 2020-04-01
  • The asymptotic behaviors of Riemann solutions for generalized Chaplygin gas magnetohydrodynamic Euler equations with source terms were considered. The self-similarity of the solutions is no longer true due to the inhomogeneous term. They will converge to Riemann solutions for zero-pressure flow transport equations when pressure and magnetic induction disappear at the same time, and δ-shock wave and vacuum will appear in the solutions. The solutions will converge to Riemann solutions for generalized Chaplygin gas Euler equations with inhomogeneous terms in the case of vanishment of magnetic induction, additionally only δ-shock wave appears in the solutions.
  • loading
  • [1]
    LARMOR J. How could a rotating body such as the sun become a magnet[J]. Reports of the British Association,1919,〖STHZ〗 87: 159-160.
    [2]
    COWLING T G. Thestability of gaseous stars[J]. Monthly Notices of the Royal Astronomical Society,1934,94: 768-782.
    [3]
    FERRARO V C A. The non-uniform rotation of the sun and its magnetic fiel[J]. Monthly Notices of the Royal Astronomical Society,1937,97: 458-472.
    [4]
    HARTMANN J. Theory of the laminar flow of an electrically conductive liquid in a homogeneous magnetic field[J]. Mathematisk-Fysiske Meddelelser, 1937,6: 1-28.
    [5]
    ALFVN H. On the cosmogony of the solar system[J]. Stockholms Observatoriums Annaler,1942,14(2): 85-100.
    [6]
    SHERCLIFF J A. Steady motion of conducting fluids in pipes under transverse magneticfields[J]. Mathematical Proceedings of the Cambridge Philosophical Society,1953,49: 136-144.
    [7]
    SHERCLIFF J A. The flow of conducting fluids in circular pipes under transversemagnetic fields[J]. Journal of Fluid Mechanics,1956,1(6): 644-666.
    [8]
    RDLER K H. Mean-field approach to spherical dynamo models[J]. Astronomische Nachrichten,1980,301(3): 101-129.
    [9]
    XU B, LI B Q, STOCK D E. An experimental study of thermally induced convection of molten gallium in magnetic fields[J]. International Journal of Heat and Mass Transfer,2006,49(13/14): 2009-2019.
    [10]
    SMITH D L, PARK J H, LYUBLINSKI I. Progress in coating development for fusion systems[J]. Fusion Engineering and Design,2002,61/62: 629-641.
    [11]
    YING A Y, GAIZER A A. The effects of imperfect insulator coatings on MHD and heat transfer in rectangular ducts[J]. Fusion Engineering and Design,1994,27: 634-641.
    [12]
    LIU Y J, SUN W H. Elementary wave interactions in magnetogasdynamics[J]. Indian Journal of Pure and Applied Mathematics,2016,47(1): 33-57.
    [13]
    LIU Y J, SUN W H. Riemann problem and wave interactions in magnetogasdynamics[J]. Journal of Mathematical Analysis and Applications,2013,397(2): 454-466.
    [14]
    SHEN C. The Riemann problem for the pressureless Euler system with the Coulomb-like friction term[J]. IMA Journal of Applied Mathematics,2016,81(1): 76-99.
    [15]
    SHENG W C, ZHANG T. The Riemann problem for the transportation equations in gas dynamics[J]. Memoirs of the American Mathematical Society,1999,137: 654.
    [16]
    WEINAN E, RYKOV Y G, SINAI Y G. Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics[J]. Communications in Mathematical Physics,1996,〖STHZ〗 177(2): 349-380.
    [17]
    SHANDARIN S F, ZELDOVICH Y B. Large-scale structure of the universe: turbulence, intermittency, structures in a self-gravitating medium[J]. Review of Modern Physics,1989,61(2): 185-220.
    [18]
    SUN M N. The exact Riemann solutions to the generalized Chaplygin gas equations with friction[J]. Communications in Nonlinear Science and Numerical Simulation,2016,36: 342-353.
    [19]
    BRENIER Y. Solutions with concentration to the Riemann problem for the one-dimensional Chaplygin gas equations[J]. Journal of Mathematical Fluid Mechanics,2005,7(3): 326-331.
    [20]
    CHEN S X, QU A F. Two-dimensional Riemann problems for Chaplygin gas[J]. SIAM Journal on Mathematical Analysis,2012,44(3): 2146-2178.
    [21]
    GUO L H, LI T, PAN L J,et al. The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations with a source term[J]. Nonlinear Analysis: Real World Applications,2018,41: 588-606.
    [22]
    SHEN C. The Riemann problem for the Chaplygin gas equations with a source term[J]. Zeitschrift fǜr Angewandte Mathematik und Mechik,2016,96(6): 681-695.
    [23]
    GUO L H, SHENG W C, ZHANG T. The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system[J]. Communications on Pure & Applied Analysis,2010,9(2): 431-458.
    [24]
    WANG G D. The Riemann problem for one dimensional generalized Chaplygin gas dynamics[J]. Journal of Mathematical Analysis and Applications,2013,403(2): 434-450.
    [25]
    FACCANONI G, MANGENEY A. Exact solution for granular flows[J]. International Journal for Numerical and Analytical Methods,2012,37: 1408-1433.
    [26]
    CHEN G Q, LIU H L. Formation of δ -shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids[J]. SIAM Journal on Mathematical Analysis,2003,34(4): 925-938.
    [27]
    LI J Q. Note on the compressible Euler equations with zero temperature[J]. Applied Mathematics Letters,2001,14(4): 519-523.
    [28]
    SHEN C. The limits of Riemann solutions to the isentropic magnetogasdynamics[J]. Applied Mathematics Letters,2011,24(7): 1124-1129.
    [29]
    CHEN J J, SHENG W C. The Riemann problem and the limit solutions as magnetic field vanishes to magnetogasdynamics for generalized Chaplygin gas[J]. Communications on Pure & Applied Analysis,2018,17(1): 127-142.
    [30]
    SHENG W C, WANG G J, YIN G. Delta wave and vacuum state for generalized Chaplygin gas dynamics system as pressure vanishes[J]. Nonlinear Analysis: Real World Applications,2015,22: 115-128.
    [31]
    YANG H C, WANG J H. Delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations for modified Chaplygin gas[J]. Journal of Mathematical Analysis and Applications,2014,413(2): 800-820.
    [32]
    SHEN C, SUN M N. Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model[J]. Journal of Differential Equations,2010,249(12): 3024-3051.
    [33]
    GUO L H, LI T, YIN G. The limit behavior of the Riemann solutions to the generalized Chaplygin gas equations with a source term[J]. Journal of Mathematical Analysis and Applications,2017,455(1): 127-140.
    [34]
    尹淦, 谢娇艳. 广义Chaplygin气体磁流体力学方程组的Riemann问题[J]. 应用数学与计算数学学报, 2013,〖STHZ〗 27(4): 508-516.(YIN Gan, XIE Jiaoyan. Riemann problem for generalized Chaplygin magnetogasdynamics equations[J]. Communication on Applied Mathematics Computation,2013,27(4): 508-516.(in Chinese))
  • 加载中

Catalog

    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Article Metrics

    Article views (1351) PDF downloads(312) Cited by()
    Proportional views
    Related

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return