The Cauchy problem for systems through the normal form of systems and theory of weighted determinant

Matsumoto, Waichiro

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1998-1999), Exposé no. 18, 29 p.

The author propose what is the principal part of linear systems of partial differential equations in the Cauchy problem through the normal form of systems in the meromorphic formal symbol class and the theory of weighted determinant. As applications, he choose the necessary and sufficient conditions for the analytic well-posedness ( Cauchy-Kowalevskaya theorem ) and $C^{\infty }$ well-posedness (Levi condition).

Mots clés : normal form of systems, p-determinant of matrix of pseudo-differential operators, p-evolution, the Cauchy-Kowalevskaya theorem for systems, $C^\infty $ well-posedness for systems

[1] K.Adjamagbo; Problème de Cauchy non caractéristique pour le système général d${}^{\prime }$équations différentielles linéaires, Compt. Rendus Acad. Sciences Paris, 294, Série I, (1982), 159-162. | MR | Zbl

[2] —; Panorama de la théorie des déterminants sur un anneau non commutatif, Bull.Sc.Math., 2e Série, 117, (1993), 401-420. | MR | Zbl

[3] —; Les fondements de la théorie des déterminants sur un domaine de Ore, Thèse Doc. État, Univ. Paris VI (1991).

[4] V.I.Arnold; Matrices depending on parameters, Uspehi Mat Nauk, 26-2 (158) (1971), 101-114, (English translation) Russ. Math. Surveys, 26-2(1971), 29-43. | MR | Zbl

[5] E.Artin; Geometric Algebra, Chap. IV, 1, Interscience Publishers (1957). | MR | Zbl

[6] L.Boutet de Monvel; Opérateurs pseudo-différentiels analytiques et opérateurs d${}^{\prime }$ordre infini , Ann.Inst.Fourier, Grenoble, 22 (1972), 229-268. | Numdam | MR | Zbl

[7] L.Boutet de Monvel and P.Krée; Pseudo-differential operators and Gevrey classes, Ann.Inst.Fourier, Grenoble, 17 (1967), 295-323. | Numdam | MR | Zbl

[8] J.Chazarin; Opérateurs hyperboliques à caractéristiques de multiplicité constante, Ann.Inst.Fourier Grenoble 24 (1974), 173-202. | Numdam | MR | Zbl

[9] A.D’Agnolo and G.Taglialatela; Sato-Kashiwara determinant and Levi conditions for systems, ( Preliminary version).

[10] A.D’Agnolo and F.Tonin; Cauchy problem for hyperbolic D-modules with regular singularities, Pacific Jour. Math., 184 (1998), 1-22. | Zbl

[11] Y.Demay; Paramétrix pour des systèmes hyperboliques du premier ordre à multiplicité constante, Jour.Math.Pure Appl., 56 (1977), 393-422. | MR | Zbl

[12] J.Dieudonné; Les déterminants sur un corps non commutatif, Bull.Soc.Math.France, 71 (1943), 27-45. | Numdam | MR | Zbl

[13] H.Flaschka and G.Strang; The correctness of the Cauchy problem, Adv.Math. 6 (1971), 347-379. | MR | Zbl

[14] G.Hufford; On the characteristic matrix of a matrix of differential operators, Jour. Diff. Eq. 1 (1965), 27-38. | MR | Zbl

[15] M.Kashiwara; Algebraic study of systems of partial differential equations, Mém.Soc.Math.France, 63 (1995), ( Translation of M.Kashiwara’s thesis in 1971 written in Japanese to English by A.D${}^{\prime }$Agnolo and J.-P.Schneiders ). | Numdam | Zbl

[16] S.von Kowalevsky; Zur Theorie der partiellen Differentialgleichungen, Jour.reine angew.Math., 80 (1875), 1-32.

[17] H.Kumano-go; A calculus of Fourier integral operators on ${ℝ}^n$ and the fundamental solution for an operator of hyperbolic type, Comm.Part.Diff.Eq. 1 (1976), 1-44. | MR | Zbl

[18] —; Pseudo-Differential Operators , Chap.10, MIT Press (1981).

[19] ; J.Leray; Uniformisation de la solution du problème linéaire analytique de Cauchy près de la variété qui porte les données de Cauchy, ( Problème de Cauchy I ), Bull.Soc.Math.France, 85 (1957), 389-429. | Numdam | MR | Zbl

[20] E.E.Levi; Caracteristiche multiple e problema di Cauchy, Ann.Math.Pura Appl., Ser.III, 16 (1909), 161-201.

[21] W.Matsumoto; On the conditions for the hyperbolicity of systems with double characteristics, I, Jour.Math.Kyoto Univ. 21 (1981), 47-84. | MR | Zbl

[22] —; On the conditions for the hyperbolicity of systems with double characteristics, II, Jour.Math.Kyoto Univ. 21 (1981), 251-271. | Zbl

[23] —; Sur l’espace de données admissibles dans le problème de Cauchy, C.R.Acad.Sc.Paris, Série I 292 (1981), 621-623. | Zbl

[24] —; Theory of pseudo-differential operators of ultradifferentiable class, Jour.Math.Kyoto Univ., 27 (1987), 453-500. | Zbl

[25] —; Normal form of systems of partial differential and pseudo-differential operators in formal symbol classes , Jour.Math.Kyoto Univ. 34 (1994), 15-40. | MR | Zbl

[26] —; Levi condition for general systems, Physics on Manifolds, Proc.Intern.Colloq., Honour Y.Choquet-Bruhat, 1992, Ed. M.Flato et al. Kluwer Academic Publishers (1994), 303-307. | MR | Zbl

[27] —; Direct proof of the perfect block diagonalization of systems of pseudo-differential operators in the ultradifferentiable classes, ( presented to Jour.Math.Kyoto Univ. ). | Zbl

[28] —; On the Cauchy Kowalevskaya theorem of Nagumo type for systems , ( to appear ).

[29] —; Levi condition for systems with characteristics of constant multiplicity, ( to appear ).

[30] W.Matsumoto and M.Murai; On the necessary and sufficient condition for the Cauchy-Kowalevskaya theorem of Nagumo type - one of the simplest cases -, ( to appear ).

[31] W.Matsumoto and H.Yamahara; On the Cauchy-Kowalevskaya theorem for systems, Proc.Japan Acad., 67, Ser.A (1991), 181-185. | MR | Zbl

[32] —; The Cauchy-Kowalevskaya theorem for systems, ( to appear ).

[33] M.Miyake; On Cauchy-Kowalevski’s theorem for general systems, Publ.RIMS, Kyoto Univ. 15 (1979), 315-337. | Zbl

[34] —; Reduction to the first order systems of the Kowalevskian systems in the sense of Volevič, Publ.RIMS, Kyoto Univ. 15 (1979), 339-355. | MR | Zbl

[35] S.Mizohata; Some remarks on the Cauchy problem, Jour.Math.Kyoto Univ. 1 (1961), 109-127. | MR | Zbl

[36] —; On Kowalevskian systems, Uspehi Mat.Nauk. 29 (1974), translated in English: Russ.Math.Survey 29 (1974), 223-235.

[37] —; On Cauchy-kowalevski’s theorem; A necessary condition, Publ.RIMS, Kyoto Univ. 10 (1975), 509-519. | Zbl

[38] —; On the Cauchy- Kowalevski theorem, Math.Anal.Appl., Part B, Adv.Math.Supplem.Studies 7B (1981), 615-652. | Zbl

[39] —; On the hyperbolicity in the domain of real analytic functions and Gevrey classes, Hokkaido Math.Jour. 12 (1983), 298-310. | MR | Zbl

[40] S.Mizohata and Y.Ohya; Sur la condition de E.E.levi concernant des équations hyperboliques, Pub.RIMS Kyoto Univ. Ser.A 4 (1968), 511-526. | MR | Zbl

[41] —; Sur la condition d’hyperbolicité pour les équations à caractéristiques multiples, II Japan Jour.Math. 40 (1971), 63-104. | Zbl

[42] M.Nagumo; Über des Anfangswertproblem partieller Differentialgleichungen Japan Jour.Math. 18 (1941-43), 41-47. | MR | Zbl

[43] T.Nishitani; On the Lax-Mizohata theorem in the analytic and Gevrey classes, Jour.Math.Kyoto Univ. 18 (1978), 509-521. | MR | Zbl

[44] O.Ore; Linear equations in non-commutative fields, Ann.Math. 32 (1931), 463-477. | MR

[45] V.M.Petkov; On the Cauchy problem for first-order hyperbolic systems with multiple characteristics Dokl.Acad.Nauk.SSSR, 209 (1973), 795-797, ( English translation ) Soviet Math.Dokl. 14 (1973), 1-13. | MR | Zbl

[46] —; Le problème de Cauchy et la propagation des singularités pour une classe des systèmes hyperboliques non symmétrizables, École Polyt. Centre Math. Sémin. G-L-S, (1974-1975) éxp. V. | Numdam | MR | Zbl

[47] —; Microlocal forms for hyperbolic systems, Math.Nachr. 93 (1979), 117-131. | MR | Zbl

[48] —; The Parametrix of the Cauchy problem for nonsymmetrizable hyperbolic systems with characteristics of constant multiplicity, Trans.Moscow Math.Soc. 37 (1980), 1-47. | Zbl

[49] I.G.Petrowsky; Über des Cauchysche Probleme für Systeme von partiellen Differentialgleichungen, Mat.Sbornik 2:5 (1937), 815-870, ( English transl. ) I.G.Petrowsky Selected Works, Part 1, Systems PDE and Alg.Geom., ed. O.A.Oleinik, (1996), Gordon Breach Publishers, 42-101.

[50] M.Sato and M.Kashiwara; The determinant of matrices of pseudo-differential operators, Proc.Japan Acad. 51, Ser.A, (1975), 17-19. | MR | Zbl

[51] G.Taglialatela and J.Vaillant; Conditions invariantes d’hyperbolicité des systèmes et réduction des systèmes, Bull.Sci.Math., 120 (1996), 19-97. | Zbl

[52] J.Vaillant; Caractéristiques multiples et bicaractéristiques des systèmes d’équations aux dérivées partielles linéaires et à coefficients constantes, Ann.Inst.Fourier, Grenoble, 51 (1965), 225-311. | Numdam | Zbl

[53] —; Conditions d’hyperbolicité des systèmes, C.R.Acad.Sci.Paris, 313 (1991), 227-230. | Zbl

[54] —; Conditions invariantes pour un système du type conditions de Levi, Physics on Manifolds, Proc.Intern.Colloq., Honour Y.Choquet-Bruhat, 1992, Ed. M.Flato et al. Kluwer Academic Publishers (1994), | MR | Zbl