Stability of a supersonic flow past a wedge with adjoint weak neutrally stable shock wave

A. M. Blokhin ; D. L. Tkachev

Matematičeskie trudy, Tome 19 (2016) no. 2, pp. 3-41.

Résumé

We study the classical problem of a supersonic stationary flow of a nonviscous nonheat-conducting gas in local thermodynamic equilibrium past an infinite plane wedge. Under the Lopatinskiĭ­ condition on the shock wave (neutral stability), we prove the well-posedness of the linearized mixed problem (the main solution is a weak shock wave), obtain a representation of the classical solution, where, in this case (in contrast to the case of the uniform Lopatinskiĭ­ condition — an absolutely stable shock wave), plane waves additionally appear in the representation. If the initial data have compact support, the solution reaches the given regime in infinite time.

Bibliographie

[1] Blokhin A. M., Integraly energii i ikh prilozheniya k zadacham gazovoi dinamiki, Nauka, Novosibirsk, 1986 | MR

[2] Blokhin A. M., Bushmanov R. S., Tkachev D. L., O vypolnenii usloviya Lopatinskogo v zadache ob obtekanii klina normalnym gazom, Preprint No 271, Izd-vo In-ta matematiki SO RAN, Novosibirsk, 2011

[3] Blokhin A. M., Bychkov A. S., Myakishev V. O., “Chislennyi analiz realizuemosti uslovii neitralnoi ustoichivosti udarnykh voln v zadache ob obtekanii klina gazom Van-der-Vaalsa”, Sib. zhurn. ind. matem., 15:4(52) (2012), 51–63 | Zbl

[4] Blokhin A. M., Tkachev D. L., “Ustoichivost sverkhzvukovogo obtekaniya klina so slaboi udarnoi volnoi”, Matem. sb., 200:2 (2009), 3–30 | DOI | MR

[5] Blokhin A. M., Tkachev D. L., Obosnovanie gipotezy Kuranta — Fridrikhsa v zadache ob obtekanii klina, Belaya seriya v matematike i fizike, 10, «Tamara Rozhkovskaya», Novosibirsk, 2011

[6] Bulakh B. M., Nelineinye konicheskie techeniya gaza, Nauka, M., 1970

[7] Vladimirov V. S., Obobschennye funktsii v matematicheskoi fizike, Nauka, M., 1979 | MR

[8] Vladimirov V. S., Uravneniya matematicheskoi fiziki, 5-e izd., Nauka, M., 1988 | MR

[9] Godunov S. K., Uravneniya matematicheskoi fiziki, Nauka, M., 1979 | MR

[10] Dobrushkin V. A., Kraevye zadachi dinamicheskoi teorii uprugosti dlya klinovidnykh oblastei, Nauka i tekhnika, Minsk, 1988

[11] Dyakov S. P., “Ob ustoichivosti udarnykh voln”, ZhETF, 27:3 (1954), 288–296

[12] Iordanskii S. V., “Ob ustoichivosti statsionarnoi udarnoi volny”, PMM, 21:4 (1957), 465–472 | MR

[13] Kontorovich V. M., “K voprosu ob ustoichivosti udarnykh voln”, ZhETF, 33:6 (1957), 1525–1526

[14] Konyukhov A. V., Likhachev A. P., Fortov V. E., Oparin A. M., Anisimov S. I., “Neitralno ustoichivye udarnye volny v termodinamicheski neitralnykh sredakh: teoriya i vychislitelnyi eksperiment”, Tr. mezhdunar. konf. po fizike vysokikh plotnostei energii-IX (Snezhinsk, 2007)

[15] Nikolskii A. A., “O ploskikh vikhrevykh techeniyakh gaza // Teoreticheskie issledovaniya po mekhanike zhidkosti i gaza”, Trudy TsAGI, 2122, 1984, 74–85

[16] Ovsyannikov L. V., Lektsii po osnovam gazovoi dinamiki, In-t kompyuternykh issledovanii, M.–Izhevsk, 2003

[17] Rozhdestvenskii B. L., “Utochnenie teorii obtekaniya klina sverkhzvukovym potokom nevyazkogo gaza”, Matem. modelir., 1:8 (1989), 99–102 | MR

[18] Rylov A. I., “O vozmozhnykh rezhimakh obtekaniya zaostrennykh tel konechnoi tolschiny pri proizvolnykh sverkhzvukovykh skorostyakh nabegayuschego potoka”, PMM, 55:1 (1991), 95–99 | MR | Zbl

[19] Tikhonov A. N., Samarskii A. A., Uravneniya matematicheskoi fiziki, 4-e izd., Nauka, M., 1972 | MR

[20] Chernyi G. G., Gazovaya dinamika, Nauka, M., 1988

[21] Abramowitz M., Stegun I. A. (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, John Wiley and Sons, New York, 1972 ; M. Abramovits, I. Stigan (red.), Spravochnik po spetsialnym funktsiyam s formulami, grafikami i matematicheskimi tablitsami, Nauka, M., 1979 | MR | Zbl | MR

[22] Blokhin A. M., Tkachev D. L., Mixed Problems for the Wave Equation in Coordinate Domains, Nova Science Publishers, Inc., Commack, New York, 1998 | MR | Zbl

[23] Blokhin A. M., Tkachev D. L., Baldan L. O., “Study of the stability in the problem on flowing around a wedge. The case of strong wave”, J. Math. Anal. Appl., 319:1 (2006), 248–277 | DOI | MR | Zbl

[24] Blokhin A. M., Tkachev D. L., Pashinin Yu. Yu., “Stability condition for strong shock waves in the problem of flow around an infinite plane wedge”, Nonlinear Anal., Hybrid Syst., 2:1 (2008), 1–17 | DOI | MR | Zbl

[25] Blokhin A. M., Tkachev D. L., Pashinin Yu. Yu., “The strong shock wave in the problem on flow around infinite plane wedge”, Hyperbolic Problems. Theory, Numerics and Applications, Proc. of the 11th Int. Conf. on Hyperbolic Problems (École Norm. Supérieure, Lyon, France, 2006), Springer-Verlag, Berlin, 2008, 1037–1044 | MR | Zbl

[26] Cagniard L., Reflection and Refraction of Progressive Seismic Waves, McGraw-Hill Book Co., New York, 1962 | Zbl

[27] Courant R., Friedrichs K. O., Supersonic Flow and Shock Waves, Interscience Publishers, New York, 1948 ; Kurant P., Fridrikhs K. O., Sverkhzvukovoe techenie i udarnye volny, Izd-vo inostr. lit., M., 1950 | MR | Zbl

[28] De Hoop A. T., “A modification of Cagniard's method for solving seismic pulse problems”, Appl. Sci. Res. B, 8:4 (1960), 349–356 | DOI | Zbl

[29] Elling V., Liu T.-P., “Physicality of weak Prandtl–Meyer reflection”, RIMS Kokyuroku, 1495:5 (2006), 112–117

[30] Elling V., Liu T.-P., “Exact solutions to supersonic flow onto a solid wedge”, Hyperbolic problems. Theory, Numerics and Applications, Proc. of the 11th Int. Conf. on Hyperbolic Problems (École Norm. Supérieure, Lyon, France, 2006), Springer-Verlag, Berlin, 2008, 101–112 | DOI | MR | Zbl

[31] Hadamard J., Le Probléme de Cauchy et les Équations aux Dérivées Partielles Linéaires Hyperboliques, Hermann, Paris, 1932; Adamar Zh., Zadacha Koshi dlya lineinykh uravnenii s chastnymi proizvodnymi giperbolicheskogo tipa, Nauka, M., 1978 | MR

[32] Hörmander L., Linear Partial Differential Operators, Springer-Verlag, New York, 1963 ; Khërmander L., Lineinye differentsialnye operatory s chastnymi proizvodnymi, Mir, M., 1965 | Zbl | MR

[33] Ikawa M., “Mixed problem for the wave equation with an oblique derivative boundary condition”, Osaka J. Math., 7:2 (1970), 495–527 | MR

[34] Sakamoto R., Hyperbolic Boundary Value Problems, Iwanami Shoten, Tokyo, 1978 | MR | Zbl

[35] Salas M. D., Morgan B. D., “Stability of shock waves attached to wedges and cones”, AIAA J., 21:12 (1983), 1611–1617 | DOI | Zbl

[36] Tkachev D. L., “Mixed problem for the wave equation in a quadrant”, Sib. J. Diff. Eq., 1:3 (1998), 269–283 | MR

[37] Tkachev D. L., Blokhin A. M., “Courant–Friedrich's hypothesis and stability of the weak shock”, Hyperbolic Problems. Theory, Numerics and Applications, Proc. of the 12th Int. Conf. on Hyperbolic Problems (June 9–13, 2008), v. 2, Proceedings of Symposia in Applied Mathematics, 67, Amer. Math. Soc., Providence, RI, 2009, 959–966 | DOI | MR | Zbl