The Cauchy problem for differential operators with double characteristics
HTML articles powered by AMS MathViewer
- by
Tatsuo Nishitani
Translated by: Tatsuo Nishitani - Sugaku Expositions 31 (2018), 169-197
- DOI: https://doi.org/10.1090/suga/433
- Published electronically: September 19, 2018
- PDF | Request permission
Abstract:
In this monograph we provide a general picture of the Cauchy problem for differential operators with double characteristics from the viewpoint that the Hamilton map and the geometry of orbits of the Hamilton flow completely characterizes the well/ill-posedness of the Cauchy problem.References
- Enrico Bernardi and Antonio Bove, Geometric results for a class of hyperbolic operators with double characteristics, Comm. Partial Differential Equations 13 (1988), no. 1, 61–86. MR 914814, DOI 10.1080/03605308808820538
- Enrico Bernardi, Antonio Bove, and Cesare Parenti, Geometric results for a class of hyperbolic operators with double characteristics. II, J. Funct. Anal. 116 (1993), no. 1, 62–82. MR 1237986, DOI 10.1006/jfan.1993.1104
- Enrico Bernardi, Cesare Parenti, and Alberto Parmeggiani, The Cauchy problem for hyperbolic operators with double characteristics in presence of transition, Comm. Partial Differential Equations 37 (2012), no. 7, 1315–1356. MR 2942985, DOI 10.1080/03605302.2012.668258
- Enrico Bernardi and Tatsuo Nishitani, On the Cauchy problem for non-effectively hyperbolic operators, the Gevrey 5 well-posedness, J. Anal. Math. 105 (2008), 197–240. MR 2438425, DOI 10.1007/s11854-008-0035-3
- Enrico Bernardi and Tatsuo Nishitani, On the Cauchy problem for noneffectively hyperbolic operators: the Gevrey 4 well-posedness, Kyoto J. Math. 51 (2011), no. 4, 767–810. MR 2854152, DOI 10.1215/21562261-1424857
- Enrico Bernardi and Tatsuo Nishitani, On the Cauchy problem for non-effectively hyperbolic operators. The Gevrey 3 well-posedness, J. Hyperbolic Differ. Equ. 8 (2011), no. 4, 615–650. MR 2864542, DOI 10.1142/S0219891611002512
- M. D. Bronšteĭn, The Cauchy problem for hyperbolic operators with characteristics of variable multiplicity, Trudy Moskov. Mat. Obshch. 41 (1980), 83–99 (Russian). MR 611140
- Jacques Chazarain, Opérateurs hyperboliques a caractéristiques de multiplicité constante, Ann. Inst. Fourier (Grenoble) 24 (1974), no. 1, 173–202 (French). MR 390512
- Ferruccio Colombini and Sergio Spagnolo, An example of a weakly hyperbolic Cauchy problem not well posed in $C^{\infty }$, Acta Math. 148 (1982), 243–253. MR 666112, DOI 10.1007/BF02392730
- J. J. Duistermaat and L. Hörmander, Fourier integral operators. II, Acta Math. 128 (1972), no. 3-4, 183–269. MR 388464, DOI 10.1007/BF02392165
- Hermann Flaschka and Gilbert Strang, The correctness of the Cauchy problem, Advances in Math. 6 (1971), 347–379 (1971). MR 279425, DOI 10.1016/0001-8708(71)90021-1
- Lars Gȧrding, Solution directe du problème de Cauchy pour les équations hyperboliques, La théorie des équations aux dérivées partielles. Nancy, 9-15 avril 1956, Colloques Internationaux du Centre National de la Recherche Scientifique, LXXI, Centre National de la Recherche Scientifique, Paris, 1956, pp. 71–90 (French). MR 0116142
- Lars Gårding, Hyperbolic differential operators, Perspectives in mathematics, Birkhäuser, Basel, 1984, pp. 215–247. MR 779678
- Lars Gårding, Hyperbolic equations in the twentieth century, Matériaux pour l’histoire des mathématiques au XX$^\textrm {e}$ siècle (Nice, 1996) Sémin. Congr., vol. 3, Soc. Math. France, Paris, 1998, pp. 37–68 (English, with English and French summaries). MR 1640255
- Lars Hörmander, The Cauchy problem for differential equations with double characteristics, J. Analyse Math. 32 (1977), 118–196. MR 492751, DOI 10.1007/BF02803578
- Lars Hörmander, The analysis of linear partial differential operators. III, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274, Springer-Verlag, Berlin, 1985. Pseudodifferential operators. MR 781536
- Lars Hörmander, Quadratic hyperbolic operators, Microlocal analysis and applications (Montecatini Terme, 1989) Lecture Notes in Math., vol. 1495, Springer, Berlin, 1991, pp. 118–160. MR 1178557, DOI 10.1007/BFb0085123
- V. Ja. Ivriĭ and V. M. Petkov, Necessary conditions for the correctness of the Cauchy problem for non-strictly hyperbolic equations, Uspehi Mat. Nauk 29 (1974), no. 5(179), 3–70 (Russian). Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), III. MR 0427843
- V. Ja. Ivriĭ, Sufficient conditions for regular and completely regular hyperbolicity, Trudy Moskov. Mat. Obšč. 33 (1975), 3–65 (1976) (Russian). MR 0492904
- V. Ja. Ivrii, The well-posed Cauchy problem for non-strictly hyperbolic operators, III. The energy integral, English transl., Trans. Moscow Math. Soc. 34 (1978), 149–168.
- V. Ja. Ivrii, Wave fronts of solutions of some hyperbolic pseudodifferential equations, English transl., Trans. Moscow Math. Soc. 39 (1981), 87–119
- V. Ya. Ivriĭ, Linear hyperbolic equations, Partial differential equations, 4 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988, pp. 157–247, 254 (Russian, with Russian summary). MR 1175404
- Nobuhisa Iwasaki, The Cauchy problem for effectively hyperbolic equations (a special case), J. Math. Kyoto Univ. 23 (1983), no. 3, 503–562. MR 721384, DOI 10.1215/kjm/1250521480
- Nobuhisa Iwasaki, The Cauchy problem for effectively hyperbolic equations. A standard type, Publ. Res. Inst. Math. Sci. 20 (1984), no. 3, 543–584. MR 759681, DOI 10.2977/prims/1195181410
- N. Iwasaki, The Cauchy problem for effectively hyperbolic equations, Sugaku Expositions 36 (1984), 227–238.
- Nobuhisa Iwasaki, The Cauchy problem for effectively hyperbolic equations (general cases), J. Math. Kyoto Univ. 25 (1985), no. 4, 727–743. MR 810976, DOI 10.1215/kjm/1250521020
- Kunihiko Kajitani and Tatsuo Nishitani, The hyperbolic Cauchy problem, Lecture Notes in Mathematics, vol. 1505, Springer-Verlag, Berlin, 1991. MR 1166190, DOI 10.1007/BFb0090882
- Gen Komatsu and Tatsuo Nishitani, Continuation of bicharacteristics for effectively hyperbolic operators, Publ. Res. Inst. Math. Sci. 28 (1992), no. 6, 885–911. MR 1203754, DOI 10.2977/prims/1195167731
- Peter D. Lax, Asymptotic solutions of oscillatory initial value problems, Duke Math. J. 24 (1957), 627–646. MR 97628
- Anneli Lax, On Cauchy’s problem for partial differential equations with multiple characteristics, Comm. Pure Appl. Math. 9 (1956), 135–169. MR 81406, DOI 10.1002/cpa.3160090203
- Jean Leray, Hyperbolic differential equations, Institute for Advanced Study (IAS), Princeton, N.J., 1953. MR 0063548
- E. E. Levi, Carateristiche multiple e problema di Cauchy, Ann. Mat. Pura Appl. 16 (1909), 161–201.
- Anders Melin, Lower bounds for pseudo-differential operators, Ark. Mat. 9 (1971), 117–140. MR 328393, DOI 10.1007/BF02383640
- R. Melrose, The Cauchy problem and propagation of singularities, In: Seminar on Nonlinear Partial Differential Equations, Papers from the seminar, Berkeley, Calif., 1983 (ed S. S.Chern), Math. Sci. Res. Inst. Publ. 2, Springer, 1984, pp. 185–201.
- Richard Melrose, The Cauchy problem for effectively hyperbolic operators, Hokkaido Math. J. 12 (1983), no. 3, 371–391. MR 725587, DOI 10.14492/hokmj/1525852964
- Sigeru Mizohata, The theory of partial differential equations, Cambridge University Press, New York, 1973. Translated from the Japanese by Katsumi Miyahara. MR 0599580
- Sigeru Mizohata, Systèmes hyperboliques, J. Math. Soc. Japan 11 (1959), 205–233 (French). MR 123835, DOI 10.2969/jmsj/01130205
- Sigeru Mizohata, Note sur le traitement par les opérateurs d’intégrale singulière du problème de Cauchy, J. Math. Soc. Japan 11 (1959), 234–240 (French). MR 123836, DOI 10.2969/jmsj/01130234
- Sigeru Mizohata, Some remarks on the Cauchy problem, J. Math. Kyoto Univ. 1 (1961/62), 109–127. MR 170112, DOI 10.1215/kjm/1250525109
- Sigeru Mizohata and Yujiro Ohya, Sur la condition de E. E. Levi concernant des équations hyperboliques, Publ. Res. Inst. Math. Sci. Ser. A 4 (1968/1969), 511–526 (French). MR 0276606, DOI 10.2977/prims/1195194888
- Sigeru Mizohata and Yujiro Ohya, Sur la condition d’hyperbolicité pour les équations à caractéristiques multiples. II, Jpn. J. Math. 40 (1971), 63–104 (French). MR 303100, DOI 10.4099/jjm1924.40.0_{6}3
- Tatsuo Nishitani, The Cauchy problem for weakly hyperbolic equations of second order, Comm. Partial Differential Equations 5 (1980), no. 12, 1273–1296. MR 593968, DOI 10.1080/03605308008820169
- Tatsuo Nishitani, On wave front sets of solutions for effectively hyperbolic operators, Sci. Rep. College Gen. Ed. Osaka Univ. 32 (1983), no. 2, 1–7. MR 751377
- Tatsuo Nishitani, Note on some noneffectively hyperbolic operators, Sci. Rep. College Gen. Ed. Osaka Univ. 32 (1983), no. 2, 9–17. MR 751378
- Tatsuo Nishitani, Local energy integrals for effectively hyperbolic operators. I, II, J. Math. Kyoto Univ. 24 (1984), no. 4, 623–658, 659–666. MR 775976, DOI 10.1215/kjm/1250521222
- Tatsuo Nishitani, Microlocal energy estimates for hyperbolic operators with double characteristics, Hyperbolic equations and related topics (Katata/Kyoto, 1984) Academic Press, Boston, MA, 1986, pp. 235–255. MR 925251, DOI 10.1016/B978-0-12-501658-2.50017-8
- Tatsuo Nishitani, Note on Ivriĭ-Petkov-Hörmander condition of hyperbolicity, Sci. Rep. College Gen. Ed. Osaka Univ. 39 (1990), no. 1-2, 7–9. MR 1116094
- Tatsuo Nishitani, Non effectively hyperbolic operators, Hamilton map and bicharacteristics, J. Math. Kyoto Univ. 44 (2004), no. 1, 55–98. MR 2062708, DOI 10.1215/kjm/1250283583
- T. Nishitani, Effectively hyperbolic Cauchy problem, In: Phase Space Analysis of Partial Differential Equations, vol II (eds. F. Colombini and L. Pernazza), Publ. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm., 2004, pp. 363–449.
- T. Nishitani, On Gevrey well-posedness of the Cauchy problem for some noneffectively hyperbolic operators, In: Advances in Phase Space Analysis of PDEs, In honor of F. Colombini’s 60th birthday (eds. A. Bove et al.), Progr. Nonlinear Differential Equations Appl. 78, Birkhäuser, 2009, pp.217–233.
- Tatsuo Nishitani, A note on zero free regions of the Stokes multipliers for second order ordinary differential equations with cubic polynomial coefficients, Funkcial. Ekvac. 54 (2011), no. 3, 473–483. MR 2918148, DOI 10.1619/fesi.54.473
- Tatsuo Nishitani, Local and microlocal Cauchy problem for non-effectively hyperbolic operators, J. Hyperbolic Differ. Equ. 11 (2014), no. 1, 185–213. MR 3190116, DOI 10.1142/S0219891614500052
- Cesare Parenti and Alberto Parmeggiani, On the Cauchy problem for hyperbolic operators with double characteristics, Comm. Partial Differential Equations 34 (2009), no. 7-9, 837–888. MR 2560303, DOI 10.1080/03605300902892360
- I. G. Petrovsky, Über das Cauchysche Problem für Systeme von partiellen Differentialgleichungen, Mat. Sb. N.S., 2 (44) (1937), 815–870.
- Yasutaka Sibuya, Global theory of a second order linear ordinary differential equation with a polynomial coefficient, North-Holland Mathematics Studies, Vol. 18, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0486867
- Trinh Duc Tai, On the simpleness of zeros of Stokes multipliers, J. Differential Equations 223 (2006), no. 2, 351–366. MR 2214939, DOI 10.1016/j.jde.2005.07.020
Bibliographic Information
- Tatsuo Nishitani
- Affiliation: Department of Mathematics, Osaka University, Toyonaka 560-0043, Osaka, Japan
- Email: nishitani@math.sci.osaka-u.ac.jp
- Published electronically: September 19, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Sugaku Expositions 31 (2018), 169-197
- MSC (2010): Primary 35L15, 35L30; Secondary 58J47, 35A21
- DOI: https://doi.org/10.1090/suga/433
- MathSciNet review: 3863902