超音速绕流中产生的跨音速激波的稳定性研究
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摘要
本文研究超音速流绕楔形物体流动中产生的跨音速激波的稳定性。
当超音速流遇到障碍物时,会在障碍物之前产生激波,在[19]中,Courant和Friedrichs证明了当直楔的顶角小于某一临界值时,在直楔之前产生的激波与其顶点相连。所有可能出现的连体激波可以分为两类:超音速激波(激波后为超音速流)及跨音速激波(激波后为亚音速流)。当直楔的顶角小于一个临界值时,有两个可能出现的连体激波,它们都满足Rankine-Hugoniot条件与熵条件,其中强度较弱的一个可能是超音速激波,而较强的一个是跨音速激波。由于不稳定的激波实际并不出现,因此关于激波稳定性的研究是十分重要的。关于超音速激波的稳定性已较清楚,但对跨音速激波稳定性的了解尚不多。本文对跨音速激波的稳定性进行研究,分别对描述定常可压缩流体的位势流方程与Euler方程组进行了分析和讨论,证明了跨音速激波关于上游流场及直楔边界的扰动是条件稳定的,也就是说,如果上游流场或直楔边界有一个小的扰动,且在无穷远处满足一定的控制条件,那么存在新的激波位置与下游流场,它们也是原始状态的一个扰动。
下面对全文的结构安排作一简单介绍。
第一章是绪论。在这一章中,我们简单介绍了超音速绕流问题的物理、数学背景,并说明了论文中的主要结果与证明方法。
第二章是准备工作。我们引入了加权H(¨|o)lder空间及加权Sobolev空间,并在这类函数空间中求解无界扇形区域上的二阶线性椭圆型方程边值问题与一阶常系数椭圆型方程组边值问题,给出了解的存在性、唯一性及关于解的先验估计。
第三章以位势流方程为模型研究跨音速激波的稳定性。它可以归结为一个无界区域上二阶非线性椭圆-双曲复合型方程的自由边界问题。我们利用第二章中的结果通过部分速度图变换及非线性迭代得到自由边界问题的解,从而证明了激波的条件稳定性。
The present Ph.D. dissertation is concerned with stability of transonic shocks in supersonic flow past a wedge.It is well known that when a supersonic flow attacks a given body, a shock will be generated ahead of it. R.Courant and K.O.Friedrichs indicated in their famous book [19] that the generated shock will attach its edge if the given body is a straight wedge with its vertex angle less than the criticle value. The attached shocks can be supersonic( with supersonic flow behind the shock fronts) or transonic( with subsonic flow behind the shock fronts). For a given straight wedge with a vertex angle less than a critical value, these two kinds of shocks are both possible and satisfy the Rankine-Hugoniot conditions and the entropy condition. The weaker one may be supersonic, and the stronger one is transonic. Since any bistable shock does not actually occur, then the study on stability the shocks becomes extremely important. It is known that, supersonic shocks are stable, but only a little is known on stability of transonic shocks. This dissertation is concerned with the stability of transonic shocks in supersonic flow past a wedge. Taking the potential equation and the full Euler system as the model to describe the steady compressible fluids, we prove that transonic shocks are conditionally stable with respect to perturbation of the upstream flow or the wedge boundary, that is, if the upstream flow or the wedge boundary is slightly perturbed, and the perturbation satisfies some control conditions at infinity, then there exists a new position of shock front and downstream flow, which is also a perturbation of the original state.The whole dissertation is organized as follows.Chapter One is Preface. This chapter is devoted to the physical and mathematical background. The main results of this Ph.D. dissertation are also illustrated.
当超音速流遇到障碍物时,会在障碍物之前产生激波,在[19]中,Courant和Friedrichs证明了当直楔的顶角小于某一临界值时,在直楔之前产生的激波与其顶点相连。所有可能出现的连体激波可以分为两类:超音速激波(激波后为超音速流)及跨音速激波(激波后为亚音速流)。当直楔的顶角小于一个临界值时,有两个可能出现的连体激波,它们都满足Rankine-Hugoniot条件与熵条件,其中强度较弱的一个可能是超音速激波,而较强的一个是跨音速激波。由于不稳定的激波实际并不出现,因此关于激波稳定性的研究是十分重要的。关于超音速激波的稳定性已较清楚,但对跨音速激波稳定性的了解尚不多。本文对跨音速激波的稳定性进行研究,分别对描述定常可压缩流体的位势流方程与Euler方程组进行了分析和讨论,证明了跨音速激波关于上游流场及直楔边界的扰动是条件稳定的,也就是说,如果上游流场或直楔边界有一个小的扰动,且在无穷远处满足一定的控制条件,那么存在新的激波位置与下游流场,它们也是原始状态的一个扰动。
下面对全文的结构安排作一简单介绍。
第一章是绪论。在这一章中,我们简单介绍了超音速绕流问题的物理、数学背景,并说明了论文中的主要结果与证明方法。
第二章是准备工作。我们引入了加权H(¨|o)lder空间及加权Sobolev空间,并在这类函数空间中求解无界扇形区域上的二阶线性椭圆型方程边值问题与一阶常系数椭圆型方程组边值问题,给出了解的存在性、唯一性及关于解的先验估计。
第三章以位势流方程为模型研究跨音速激波的稳定性。它可以归结为一个无界区域上二阶非线性椭圆-双曲复合型方程的自由边界问题。我们利用第二章中的结果通过部分速度图变换及非线性迭代得到自由边界问题的解,从而证明了激波的条件稳定性。
The present Ph.D. dissertation is concerned with stability of transonic shocks in supersonic flow past a wedge.It is well known that when a supersonic flow attacks a given body, a shock will be generated ahead of it. R.Courant and K.O.Friedrichs indicated in their famous book [19] that the generated shock will attach its edge if the given body is a straight wedge with its vertex angle less than the criticle value. The attached shocks can be supersonic( with supersonic flow behind the shock fronts) or transonic( with subsonic flow behind the shock fronts). For a given straight wedge with a vertex angle less than a critical value, these two kinds of shocks are both possible and satisfy the Rankine-Hugoniot conditions and the entropy condition. The weaker one may be supersonic, and the stronger one is transonic. Since any bistable shock does not actually occur, then the study on stability the shocks becomes extremely important. It is known that, supersonic shocks are stable, but only a little is known on stability of transonic shocks. This dissertation is concerned with the stability of transonic shocks in supersonic flow past a wedge. Taking the potential equation and the full Euler system as the model to describe the steady compressible fluids, we prove that transonic shocks are conditionally stable with respect to perturbation of the upstream flow or the wedge boundary, that is, if the upstream flow or the wedge boundary is slightly perturbed, and the perturbation satisfies some control conditions at infinity, then there exists a new position of shock front and downstream flow, which is also a perturbation of the original state.The whole dissertation is organized as follows.Chapter One is Preface. This chapter is devoted to the physical and mathematical background. The main results of this Ph.D. dissertation are also illustrated.
引文
[1] Bressan, A. The semigroup approach to systems of conservation laws, Math. Comtemp. 1996, 10: 21-74.
[2] Bressan, A. & Crasta, G. & Piccoli, B. Well-posedness of the Cauchy problem for n×n system of conservation laws, Memoirs A. M. S.
[3] Canic, S.; Keyfitz, B. L.; Lieberman, G. M. A proof of existence of perturbed steady transonic shocks via a free boundary problem. Comm. Pure Appl. Math. 2000, 53: 484-511.
[4] Canic, S.; Keyfitz, B. L.; Kim, E. H. A free boundary problems for unsteady transonic small disturbance equation: transonic regular reflection. Methods and Appl. Math. 2000, 7: 313-336.
[5] Canic, S.; Keyfitz, B. L.; Kim. E. H. A free boundary problems for a quasilinear degenerate elliptic equation: transonic regular reflection. Comm. Pure Appl. Math. 2002, 55: 71-92.
[6] Chen, G. Q. Convergence of the Lax-Friedrichs schemes for isentropic gas dynamics(Ⅲ). Acta Math. Scientia 1986, 6: 75-120.
[7] Chen, G. Q.; Feldman, M. Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type. J. Amer. Math. Soc. 2003, 16: 461-494.
[8] Chen, G. Q.; Feldman, M. Steady transonic shocks and free goundary problems in infinite cylinders for the Euler equations. Comm. Pure Appl. Math. 2004, 57.
[9] Chen, G. Q.; Feldman, M. Free boundary problems and transonic shocks for the Euler equations in unbounded domains, in press.
[10] Chen, S. X. Existence of local solution to supersonic flow past a three-dimensional wing, Adv. in Appl. Math. 1992, 13(3): 273-304.
[11] Chen, S. X. A free boundary problem of elliptic equation arising in supersonic flow past a conical body. Zeitschrift fur Angewandte Mathematik und Physik 2003, 54: 1-23.
[12] Chen, S. X. Existence of stationary supersonic flow past a pointed body, Arch. Rational Mech. Anal. 2001, 156: 141-181.
[13] Chen, S. X. A free boundary value problem of Euler system arising in supersonic flow past a curved cone, Tohoku Math. J. 2002, 54: 105-120.
[14] Chen, S. X. New development of the theory on multidimensional quasilinear hyperbolic systems, New Studies in Advanced Mathematics, International Press, v.4(2003): 47-56.
[15] Chen, S. X. Stability of stationary shock front in M-D Euler system. Trans. Amer. Math. Soc. 2005, 357: 287-308.
[16] Chen, S. X. Stability of Mach configuration, in press, 2003.
[17] Chen, S. X. Stability of oblique shock fronts, Science in China, 2002, 45(8): 1012-1019.
[18] Chen, S. X. & Xin, Z. P. & Yin, H. C. Unsteady supersonic flow past a wedge, IMS of CUHK, Preprint(2002).
[19] Courant, R. & Priedrichs, K. O. Supersonic Flow and Shock Waves, New York: Interscience Publishers Inc., 1948.
[20] Dafermos, C. M. Hyperbolic Conservation Laws in. Continuum Physics, New York: Springer, 2000.
[21] Diperna, R. J. Compensated compactness and general systems of conservation laws, Trans. A. M. S. 1985, 292: 283-420.
[22] Gilbarg, D. & Hormander, L. Intermediate Schauder estimates. Arch. Rational Mech. Anal., 1980, 74: 297-314.
[23] Gilbarg, D.; Trudinger, N. S. Elliptic Partial Differential Equations of Sencond Order, 2nd Edition; Berlin: Springer-Verlag, 1983.
[24] Glimm, J. Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 1965, 18: 697-715.
[25] Glimm, J. & Majda, A. Multidimensional Hyperbolic Problems and Computations; New York: Springer-Verlag, 1991.
[26] Grisvard, P. Elliptic problems in nonsmooth domains; Boston: Monographs and Studies in Mathematics, 24, Pitman, 1985.
[27] Gu, C. H. A method for solving the supersonic flow past a curved wedge. Fudan J.(Natural Science) 1962, 7: 11-14.
[28] Keyfitz, B. L.; Warnecke, G. G. The existence of viscous profiles and admissibility for transonic shocks. Comm. P. D. E. 1991, 16(6&7): 1197-1221.
[29] Kozlov, V. A.; Maz'ya, V. G.; Rossmann, J. Elliptic boundary value problems in domains with point singularities; Mathematical survey and Monographs, Volume 52, AMS, 1997.
[30] Kuz'min, A. G., Boundary- Value Problems for Transonic Flow; John Wiley & Sons, Ltd, 2002.
[31] Lax, P. D., Hyperbolic System of Conservation Laws Ⅱ. Comm. Pure Appl. Math., 1957, 10: 537-566.
[32] Lax, P. D., Hyperbolic System of Conservation Laws and The Mathematical Theory of Shock Waves. CBMS Regional Conference Series in Mathematics 11. Philadelphia: SIAM, 1973.
[33] Li, Ta-tsien. Global Classical Solutions for Quasilinear Hyperbolic Systems; John Wiley & Sons: Masson, 1994.
[34] 李大潜,秦铁虎,物理学与偏微分方程.北京:高等教育出版社,1997.
[35] Lieberman, G. M. The Perron process applied to oblique derivative problems. Adv. in Math. 1985, 55: 161-172.
[36] Lieberman, G. M. Mixed boundary value problems for elliptic and parabolic differential equa- tions of second order. J. Math. Anal. Appl. 1986,113: 422-440.
[37] Lieberman, G.M. Oblique derivative problems in Lipschitz domains,I. Continuous boundary data. Boll. Un. Mat. Ital. 1987, 7(1-B): 1185-1210.
[38] Lieberman, G.M. Oblique derivative problems in Lipschitz domains,II. Discontinuous bound ary data. J. reine angew. Math. 1988,389: 1-21. .
[39] Lions, P.L. & Perthame, B. & Souganidis, P.E. Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure Appl. Math. 1996,49: 599-638.
[40] Liu, T.P. Hyperbolic and Viscous Conservation Laws, Philadelphia: SIAM, 1999.
[41] Liu, T.P. & Yang, T. Well-posedness theory for hyperbolic conservation laws, Comm. Pure Appl. Math. 1999,52: 1553-1586.
[42] Majda, A. Compressible Fluid Flow and Systems of Conservation Laws; Applied Mathematical Sciences, 53, New York: Springer-Verlag, 1984.
[43] Majda, A. One perspective on open problems in rnulti-dimensional conservation laws; IMA vol. Math. Appl. 1991,29: 217-237, MR1087085(91m:35142).
[44] Majda, A. & Thomann, E. Multi-dimensional shock fronts for second order wave equations, Comm.P.D.E., 1987, 12: 777-728.
[45] Metivier, G. Stability of multi-dimensional weak shocks. Comm. P.D.E. 1990, 15: 983-1028.
[46] Miller, K. Barriers on cones for uniformly elliptic operators. Ann. Mat. Pura. Appl. 1967, 76 (4): 93-105.
[47] Morawetz, C.S. On the non-existence of continuous transonic flows past profiles I-III. Comm. Pure Appl. Math. 1956,9: 45-68; 1957,10: 107-131; 1958,11: 129-144.
[48] Morawetz, C.S. Mathematical problems in transonic flow. Canad. Math. Bull. 1986,29(2): 129-139.
[49] Morawetz, C.S. Potential theory for regular and Mach reflection of a shock at a wedge. Comm. Pure Appl. Math. 1994,47: 593-624.
[50] Schaeffer, D.G. Supersonic flow past a nearly straight wedge, Duke Math. J., 1976, 43: 637- 670.
[51] Smoller, J. Shock Waves and Reaction- Diffusion Equations (second edition). New York: Springer, 1994.
[52] Xin, Z.P. Some current topics in nonlinear conservation laws. AMS/IP Studies in Advanced Mathematics, 2000, 15: xiii-xxxi
[53] Xin, Z.P.; Yin, H.C. Transonic shock in a nozzle I, 2-D case. Research Report, CUHK, IMS- 2003-12.
[54] Yin, H.C. Globel existence of a shock for the supersonic flow past a curved wedge, in press.
[2] Bressan, A. & Crasta, G. & Piccoli, B. Well-posedness of the Cauchy problem for n×n system of conservation laws, Memoirs A. M. S.
[3] Canic, S.; Keyfitz, B. L.; Lieberman, G. M. A proof of existence of perturbed steady transonic shocks via a free boundary problem. Comm. Pure Appl. Math. 2000, 53: 484-511.
[4] Canic, S.; Keyfitz, B. L.; Kim, E. H. A free boundary problems for unsteady transonic small disturbance equation: transonic regular reflection. Methods and Appl. Math. 2000, 7: 313-336.
[5] Canic, S.; Keyfitz, B. L.; Kim. E. H. A free boundary problems for a quasilinear degenerate elliptic equation: transonic regular reflection. Comm. Pure Appl. Math. 2002, 55: 71-92.
[6] Chen, G. Q. Convergence of the Lax-Friedrichs schemes for isentropic gas dynamics(Ⅲ). Acta Math. Scientia 1986, 6: 75-120.
[7] Chen, G. Q.; Feldman, M. Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type. J. Amer. Math. Soc. 2003, 16: 461-494.
[8] Chen, G. Q.; Feldman, M. Steady transonic shocks and free goundary problems in infinite cylinders for the Euler equations. Comm. Pure Appl. Math. 2004, 57.
[9] Chen, G. Q.; Feldman, M. Free boundary problems and transonic shocks for the Euler equations in unbounded domains, in press.
[10] Chen, S. X. Existence of local solution to supersonic flow past a three-dimensional wing, Adv. in Appl. Math. 1992, 13(3): 273-304.
[11] Chen, S. X. A free boundary problem of elliptic equation arising in supersonic flow past a conical body. Zeitschrift fur Angewandte Mathematik und Physik 2003, 54: 1-23.
[12] Chen, S. X. Existence of stationary supersonic flow past a pointed body, Arch. Rational Mech. Anal. 2001, 156: 141-181.
[13] Chen, S. X. A free boundary value problem of Euler system arising in supersonic flow past a curved cone, Tohoku Math. J. 2002, 54: 105-120.
[14] Chen, S. X. New development of the theory on multidimensional quasilinear hyperbolic systems, New Studies in Advanced Mathematics, International Press, v.4(2003): 47-56.
[15] Chen, S. X. Stability of stationary shock front in M-D Euler system. Trans. Amer. Math. Soc. 2005, 357: 287-308.
[16] Chen, S. X. Stability of Mach configuration, in press, 2003.
[17] Chen, S. X. Stability of oblique shock fronts, Science in China, 2002, 45(8): 1012-1019.
[18] Chen, S. X. & Xin, Z. P. & Yin, H. C. Unsteady supersonic flow past a wedge, IMS of CUHK, Preprint(2002).
[19] Courant, R. & Priedrichs, K. O. Supersonic Flow and Shock Waves, New York: Interscience Publishers Inc., 1948.
[20] Dafermos, C. M. Hyperbolic Conservation Laws in. Continuum Physics, New York: Springer, 2000.
[21] Diperna, R. J. Compensated compactness and general systems of conservation laws, Trans. A. M. S. 1985, 292: 283-420.
[22] Gilbarg, D. & Hormander, L. Intermediate Schauder estimates. Arch. Rational Mech. Anal., 1980, 74: 297-314.
[23] Gilbarg, D.; Trudinger, N. S. Elliptic Partial Differential Equations of Sencond Order, 2nd Edition; Berlin: Springer-Verlag, 1983.
[24] Glimm, J. Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 1965, 18: 697-715.
[25] Glimm, J. & Majda, A. Multidimensional Hyperbolic Problems and Computations; New York: Springer-Verlag, 1991.
[26] Grisvard, P. Elliptic problems in nonsmooth domains; Boston: Monographs and Studies in Mathematics, 24, Pitman, 1985.
[27] Gu, C. H. A method for solving the supersonic flow past a curved wedge. Fudan J.(Natural Science) 1962, 7: 11-14.
[28] Keyfitz, B. L.; Warnecke, G. G. The existence of viscous profiles and admissibility for transonic shocks. Comm. P. D. E. 1991, 16(6&7): 1197-1221.
[29] Kozlov, V. A.; Maz'ya, V. G.; Rossmann, J. Elliptic boundary value problems in domains with point singularities; Mathematical survey and Monographs, Volume 52, AMS, 1997.
[30] Kuz'min, A. G., Boundary- Value Problems for Transonic Flow; John Wiley & Sons, Ltd, 2002.
[31] Lax, P. D., Hyperbolic System of Conservation Laws Ⅱ. Comm. Pure Appl. Math., 1957, 10: 537-566.
[32] Lax, P. D., Hyperbolic System of Conservation Laws and The Mathematical Theory of Shock Waves. CBMS Regional Conference Series in Mathematics 11. Philadelphia: SIAM, 1973.
[33] Li, Ta-tsien. Global Classical Solutions for Quasilinear Hyperbolic Systems; John Wiley & Sons: Masson, 1994.
[34] 李大潜,秦铁虎,物理学与偏微分方程.北京:高等教育出版社,1997.
[35] Lieberman, G. M. The Perron process applied to oblique derivative problems. Adv. in Math. 1985, 55: 161-172.
[36] Lieberman, G. M. Mixed boundary value problems for elliptic and parabolic differential equa- tions of second order. J. Math. Anal. Appl. 1986,113: 422-440.
[37] Lieberman, G.M. Oblique derivative problems in Lipschitz domains,I. Continuous boundary data. Boll. Un. Mat. Ital. 1987, 7(1-B): 1185-1210.
[38] Lieberman, G.M. Oblique derivative problems in Lipschitz domains,II. Discontinuous bound ary data. J. reine angew. Math. 1988,389: 1-21. .
[39] Lions, P.L. & Perthame, B. & Souganidis, P.E. Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure Appl. Math. 1996,49: 599-638.
[40] Liu, T.P. Hyperbolic and Viscous Conservation Laws, Philadelphia: SIAM, 1999.
[41] Liu, T.P. & Yang, T. Well-posedness theory for hyperbolic conservation laws, Comm. Pure Appl. Math. 1999,52: 1553-1586.
[42] Majda, A. Compressible Fluid Flow and Systems of Conservation Laws; Applied Mathematical Sciences, 53, New York: Springer-Verlag, 1984.
[43] Majda, A. One perspective on open problems in rnulti-dimensional conservation laws; IMA vol. Math. Appl. 1991,29: 217-237, MR1087085(91m:35142).
[44] Majda, A. & Thomann, E. Multi-dimensional shock fronts for second order wave equations, Comm.P.D.E., 1987, 12: 777-728.
[45] Metivier, G. Stability of multi-dimensional weak shocks. Comm. P.D.E. 1990, 15: 983-1028.
[46] Miller, K. Barriers on cones for uniformly elliptic operators. Ann. Mat. Pura. Appl. 1967, 76 (4): 93-105.
[47] Morawetz, C.S. On the non-existence of continuous transonic flows past profiles I-III. Comm. Pure Appl. Math. 1956,9: 45-68; 1957,10: 107-131; 1958,11: 129-144.
[48] Morawetz, C.S. Mathematical problems in transonic flow. Canad. Math. Bull. 1986,29(2): 129-139.
[49] Morawetz, C.S. Potential theory for regular and Mach reflection of a shock at a wedge. Comm. Pure Appl. Math. 1994,47: 593-624.
[50] Schaeffer, D.G. Supersonic flow past a nearly straight wedge, Duke Math. J., 1976, 43: 637- 670.
[51] Smoller, J. Shock Waves and Reaction- Diffusion Equations (second edition). New York: Springer, 1994.
[52] Xin, Z.P. Some current topics in nonlinear conservation laws. AMS/IP Studies in Advanced Mathematics, 2000, 15: xiii-xxxi
[53] Xin, Z.P.; Yin, H.C. Transonic shock in a nozzle I, 2-D case. Research Report, CUHK, IMS- 2003-12.
[54] Yin, H.C. Globel existence of a shock for the supersonic flow past a curved wedge, in press.