Articles

GLOBAL INSTABILITY OF MULTI-DIMENSIONAL PLANE SHOCKS FOR ISOTHERMAL FLOW

  • Ning-An LAI ,
  • Wei XIANG ,
  • Yi ZHOU
Expand
  • 1. College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua, 321004, China;
    2. Department of Mathematics, Lishui University, Lishui, 323000, China;
    3. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, 999077, China;
    4. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

Received date: 2020-07-04

  Online published: 2022-06-24

Supported by

N. A. Lai and Y. Zhou were supported by NSFC (12171097). W. Xiang was supported in part by the Research Grants Council of the HKSAR, China (Project No.CityU 11303518, Project CityU 11304820 and Project CityU 11300021).

Abstract

In this paper, we are concerned with the long time behavior of the piecewise smooth solutions to the generalized Riemann problem governed by the compressible isothermal Euler equations in two and three dimensions. A non-existence result is established for the fan-shaped wave structure solution, including two shocks and one contact discontinuity which is a perturbation of plane waves. Therefore, unlike in the one-dimensional case, the multi-dimensional plane shocks are not stable globally. Moreover, a sharp lifespan estimate is established which is the same as the lifespan estimate for the nonlinear wave equations in both two and three space dimensions.

Cite this article

Ning-An LAI , Wei XIANG , Yi ZHOU . GLOBAL INSTABILITY OF MULTI-DIMENSIONAL PLANE SHOCKS FOR ISOTHERMAL FLOW[J]. Acta mathematica scientia, Series B, 2022 , 42(3) : 887 -902 . DOI: 10.1007/s10473-022-0305-7

References

[1] Christodoulou D, Miao S. Compressible flow and Euler's equations[M]. Beijing:Higher Education Press, 2014
[2] Li T T, Zhou Y, Kong D X. Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems[J]. Communications in Partial Differential Equations, 1994, 19(7/8):1263-1317
[3] Sideris T C. Formation of singularities in three-dimensional compressible fluids[J]. Communications in Mathematical Physics, 1985, 101(4):475-485
[4] Bressan A. Hyperbolic Systems of Conservation Laws[M]//Hyperbolic systems of conservation laws. Oxford:Oxford University Press, 2000
[5] Dafermos C. Hyperbolic conservation laws in continuum physics[M]. Third edition. Heidelberg:Springer-Verlag, 2010
[6] Xin Z P. Some current topics in nonlinear conservation laws[M]//AMS/IP Stud Adv Math, 15. Providence, RI:Amer Math Soc, 2000
[7] Bae M, Chen G Q, Feldman M. Regularity of solutions to regular shock reflection for potential flow[J]. Inventiones Mathematicae, 2009, 175(3):505-543
[8] Cao G W, Xiang W, Yang X Z. Global structure of admissible solutions of multi-dimensional non-homogeneous scalar conservation law with Riemann-type data[J]. Journal of Differential Equations, 2017, 263(2):1055-1078
[9] Chen G Q, Feldman M. Global Solutions of Shock Reflection by Large-Angle Wedges for Potential Flow[J]. Annals of Mathematics, 2010, 171(2):1067-1182
[10] Chen G Q, Feldman M. Mathematics of Shock Reflection-Diffraction and von Neumann Conjectures[M]. Princeton:Princeton University Press, 2018
[11] Chen G Q, Feldman M, Hu J C, et al. Loss of Regularity of Solutions of the Lighthill Problem for Shock Diffraction for Potential Flow[J]. SIAM J Math Anal, 2020, 52(2):1096-1114
[12] Chen G Q, Feldman M, Xiang W. Convexity of Self-Similar Transonic Shocks and Free Boundaries for Potential Flow. arXiv:1803.02431, 2018
[13] Li J Q, Zheng Y X. Interaction of Rarefaction Waves of the Two-Dimensional Self-Similar Euler Equations[J]. Archive for Rational Mechanics Analysis, 2009, 193(3):623-657
[14] Zheng Y X. Systems of conservation laws, Two-dimensional Riemann Problems[M]. Progress in Nonlinear Differential Equations and their Applications, 38. Boston, MA:Birkhäuser Boston, Inc, 2001
[15] Yang C L, Cao G W, Yang X Z. Uniform formula for the Riemann solutions of a scalar combustion model[J]. Acta Mathematica Scientia, 2016, 36B(5):1405-1418
[16] Cao G W, Hu K, Yang X Z. Formula of global smooth solution for non-homogeneous M-D conservation law with unbounded initial value[J]. Acta Mathematica Scientia, 2015, 35B(2):508-526
[17] Courant R, Friedrichs K. Supersonic flow and shock waves[M]. New York:Wiley Interscience, 1948
[18] Bae M, Chen G Q, Feldman M. Prandtl-Meyer reflection for supersonic flow past a solid ramp[J]. Quart Appl Math, 2013, 71(3):583-600
[19] Chen S X. Existence of Stationary Supersonic Flows Past a Pointed Body[J]. Archive for Rational Mechanics and Analysis, 2001, 156(2):141-181
[20] Chen S X. Mach Configuration in Pseudo-Stationary Compressible Flow[J]. Journal of the American Mathematical Society, 2008, 21(1):63-100
[21] Chen S X, Xin Z P, Yin H C. Global Shock Waves for the Supersonic Flow Past a Perturbed Cone[J]. Communications in Mathematical Physics, 2002, 228(1):47-84
[22] Elling V, Liu T P. Supersonic flow onto a solid wedge[J]. Communications on Pure Applied Mathematics, 2008, 61(10):1347-1448
[23] Fang B X, Liu L, Yuan H R. Global Uniqueness of Steady Transonic Shocks in Two-Dimensional Compressible Euler Flows[J]. Archive for Rational Mechanics Analysis, 2010, 207(1):317-345
[24] Fang B X, Xiang W. The uniqueness of transonic shocks in supersonic flow past a 2-D wedge[J]. Journal of Mathematical Analysis Applications, 2016, 437(1):194-213
[25] Li J, Ingo W, Yin H C. On the Global Existence and Stability of a Multi-Dimensional Supersonic Conic Shock Wave[J]. Communications in Mathematical Physics, 2014, 329(2):609-640
[26] Li J, Witt I, Yin H C. Global multidimensional shock waves for 2-D and 3-D unsteady potential flow equations[J]. SIAM J Math Anal, 2018, 50:933-1009
[27] Qu A F, Xiang W. Three-dimensional steady supersonic Euler flow past a concave cornered wedge with lower pressure at the downstream[J]. Arch Ration Mech Anal, 2018, 228(2):431-476
[28] Wang Z, Yin H C. Local Structural Stability of a Multidimensional Centered Rarefaction Wave for the Three-Dimensional Steady Supersonic Euler Flow around a Sharp Corner[J]. Siam Journal on Mathematical Analysis, 2010, 42(4):1639-1687
[29] Wang Y G, Yu F. Structural stability of supersonic contact discontinuities in three-dimensional compressible steady flows[J]. SIAM J Math Anal, 2015, 47:1291-1329
[30] Wang Y G, Yuan H R. Weak stability of transonic contact discontinuities in three-dimensional steady non-isentropic compressible Euler flows[J]. Zeitschrift Fur Angewandte Mathematik Und Physik, 2015, 66(2):341-388
[31] Xin Z P, Yin H C. Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone[J]. Analysis Applications, 2006, 4(2):101-132
[32] Zhang Y Q. Steady supersonic flow past an almost straight wedge with large vertex angle[J]. J Differential Equations, 2003, 192:1-46
[33] Blokhin A M. Estimation of the energy integral of a mixed problem for gas dynamics equations with boundary conditions on the shock wave (in Russian)[J]. Sibirsk Mat Zh, 1981, 22(4):23-51
[34] Blokhin A, Trakhinin Y. Stability of Strong Discontinuities in Fluids and MHD[J]. Handbook of Mathematical Fluid Dynamics, 2002, 1(2):545-652
[35] Majda A. The stability of multi-dimensional shock fronts[J]. Memoirs of the American Mathematical Society, 1983, 41 (275):iv+95 pp
[36] Majda A. The existence of multidimensional shock fronts[J]. Memoirs of the American Mathematical Society, 1983, 43 (281):v+93 pp
[37] Métivier G. Stability of Multi-Dimensional Weak Shocks[J]. Comm Partial Differential Equations, 1990, 15:983-1028
[38] Alinhac S. Existence d'ondes de rarefaction pour des systémes quasi-linéaires hyperboliques multidimensionnels[J]. Communications in Partial Differential Equations, 1989, 14(2):173-230
[39] Coulombel J, Secchi P. Nonlinear compressible vortex sheets in two-dimensions[J]. Ann Sci Ec Norm Super, 2008, 41(1):85-139
[40] Chen S X, Li D N. Cauchy problem with general discontinuous initial data along a smooth curve for 2-d Euler system[J]. Journal of Differential Equations, 2014, 257(6):1939-1988
[41] Godin P. Long time existence of a class of perturbations of planar shock fronts for second order hyperbolic conservation laws[J]. Duke Mathematical Journal, 1990, 60(2):425-463
[42] Li T T, Zhao Y C. Global shock solutions to a class of piston problems for the system of one-dimensional isentropic flow[J]. Chinese Ann Math Ser B, 1991, 12(5):495-499
[43] Li T T, Wang L B. Global propagation of regular nonlinear hyperbolic waves[M]. Progress in Nonlinear Differential Equations and their Applications, 76. Birkhäuser Boston, 2009
[44] Li T T, Zhou Y. Nonlinear Wave Equations (in Chinese)[M]//Series in Contemporary Mathematics. Shanghai:Shanghai Scientific Technical Publishers, 2016
Options
Outlines

/