$L^2$ well-posed Cauchy problems and symmetrizability of first order systems

Métivier, Guy

L^{2}

well-posed Cauchy problems and symmetrizability of first order systems
[Problèmes de Cauchy bien posés dans

L^{2}

et symétrisabilité pour les systèmes du premier ordre]

Métivier, Guy

Journal de l'École polytechnique - Mathématiques, Tome 1 (2014), p. 39-70 / Harvested from Numdam

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Résumé
Abstract

Le problème de Cauchy est bien posé dans $L^{2}$ pour les systèmes du premier ordre $L (t, x, \partial_{t}, \partial_{x})$ qui admettent un symétriseur microlocal $S (t, x, ξ)$ $C^{\infty}$ en $ξ \neq 0$ et lipschitzien en $(t, x)$ . Cet article contient trois principaux résultats. D’abord, il est montré qu’une régularité lipschitzienne globale en $(t, x, ξ)$ pour le symétriseur est suffisante. Ensuite, il est établi que l’existence de symétriseurs microlocaux est équivalente à l’existence de symétriseurs complets $Σ (t, x, τ, ξ)$ de même régularité, notion introduite dans [FL67]. Cette étape est le point clé dans la démonstration du troisième résultat qui affirme que l’existence de symétriseurs microlocaux est préservée par changement de variable de temps. Un corollaire en est l’unicité locale et la vitesse finie de propagation.

The Cauchy problem for first order system $L (t, x, \partial_{t}, \partial_{x})$ is known to be well-posed in $L^{2}$ when it admits a microlocal symmetrizer $S (t, x, ξ)$ which is smooth in $ξ$ and Lipschitz continuous in $(t, x)$ . This paper contains three main results. First we show that a Lipschitz smoothness globally in $(t, x, ξ)$ is sufficient. Second, we show that the existence of symmetrizers with a given smoothness is equivalent to the existence of full symmetrizers having the same smoothness. This notion was first introduced in [FL67]. This is the key point to prove the third result saying that the existence of microlocal symmetrizer is preserved if one changes the direction of time, implying local uniqueness and finite speed of propagation.

Publié le : 2014-01-01
DOI : https://doi.org/10.5802/jep.3
Classification: 35L
Mots clés: Systèmes d’équations aux dérivées partielles, problème de Cauchy, hyperbolicité, hyperbolicité forte, symétriseurs, estimation d’énergie, unicité locale, propagation à vitesse finie

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     author = {M\'etivier, Guy},
     title = {$L^2$ well-posed Cauchy problems and symmetrizability of first order systems},
     journal = {Journal de l'\'Ecole polytechnique - Math\'ematiques},
     volume = {1},
     year = {2014},
     pages = {39-70},
     doi = {10.5802/jep.3},
     language = {en},
     url = {http://dml.mathdoc.fr/item/JEP_2014__1__39_0}
}

Métivier, Guy. $L^2$ well-posed Cauchy problems and symmetrizability of first order systems. Journal de l'École polytechnique - Mathématiques, Tome 1 (2014) pp. 39-70. doi : 10.5802/jep.3. http://gdmltest.u-ga.fr/item/JEP_2014__1__39_0/

Bibliographie

[Bro79] Bronshtejn, M. D. Smoothness of roots of polynomials depending on parameters, Sibirsk. Mat. Zh., Tome 20 (1979) no. 3, pp. 493-501 (English transl. Siberian Math. J. 20 (1980), p. 347–352) | Article | MR 537355 | Zbl 0415.30003

[Cal60] Calderón, A.-P. Integrales singulares y sus aplicaciones a ecuaciones diferenciales hiperbolicas, Universidad de Buenos Aires, Buenos Aires, Cursos y Seminarios de Matemática, Fasc. 3 (1960) | MR 123834 | Zbl 0096.06702

[FL67] Friedrichs, K. O.; Lax, P. D. On symmetrizable differential operators, Singular Integrals (Chicago, 1966), Amer. Math. Soc., Providence, R.I. (Proc. Sympos. Pure Math.) (1967), pp. 128-137 | MR 239256 | Zbl 0184.36603

[Fri54] Friedrichs, K. O. Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math., Tome 7 (1954), pp. 345-392 | MR 62932 | Zbl 0059.08902

[Fri58] Friedrichs, K. O. Symmetric positive linear differential equations, Comm. Pure Appl. Math., Tome 11 (1958), pp. 333-418 | MR 100718 | Zbl 0083.31802

[Går51] Gårding, L. Linear hyperbolic partial differential equations with constant coefficients, Acta Math., Tome 85 (1951), pp. 1-62 | MR 41336 | Zbl 0045.20202

[Går63] Gårding, L. Problèmes de Cauchy pour des systèmes quasi-linéaires d’ordre un, strictement hyperboliques, Les EDP, CNRS, Paris (Colloques Internationaux du CNRS) Tome 117 (1963), pp. 33-40 | Zbl 0239.35013

[Går98] Gårding, L. Hyperbolic equations in the twentieth century, Matériaux pour l’histoire des mathématiques au XX $^{e}$ siècle (Nice, 1996), Soc. Math. France, Paris (Séminaires & Congrès) Tome 3 (1998), pp. 37-68 | MR 1640255 | Zbl 1044.01525

[Hör83] Hörmander, L. The analysis of linear partial differential operators. II, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften, Tome 257 (1983), pp. viii+391 | Article | MR 705278 | Zbl 0521.35002

[IP74] Ivriĭ, V. Ja.; Petkov, V. M. Necessary conditions for the Cauchy problem for non-strictly hyperbolic equations to be well-posed, Uspehi Mat. Nauk, Tome 29 (1974) no. 5(179), pp. 3-70 (English Transl. Russian Math. Surveys 29 (1974), p. 1–70) | Article | MR 427843 | Zbl 0312.35049

[JMR05] Joly, J.-L.; Métivier, G.; Rauch, J. Hyperbolic domains of determinacy and Hamilton-Jacobi equations, J. Hyperbolic Differ. Equ., Tome 2 (2005) no. 3, pp. 713-744 | Article | MR 2172702 | Zbl 1090.35114

[Lax57] Lax, P. D. Asymptotic solutions of oscillatory initial value problems, Duke Math. J., Tome 24 (1957), pp. 627-646 | MR 97628 | Zbl 0083.31801

[Lax63] Lax, P. D. Lectures on hyperbolic partial differential equations, Stanford University, Stanford Lecture Notes (1963)

[Ler53] Leray, J. Hyperbolic differential equations, The Institute for Advanced Study, Princeton, N.J. (1953), pp. 238 pp. Reprinted November, 1955 | MR 63548

[LN66] Lax, P. D.; Nirenberg, L. On stability for difference schemes: A sharp form of Gårding’s inequality, Comm. Pure Appl. Math., Tome 19 (1966), pp. 473-492 | MR 206534 | Zbl 0185.22801

[Mét08] Métivier, G. Para-differential calculus and applications to the Cauchy problem for nonlinear systems, Edizioni della Normale, Pisa, Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series, Tome 5 (2008), pp. xii+140 | MR 170112 | Zbl 0104.31903

[Miz62] Mizohata, S. Some remarks on the Cauchy problem, J. Math. Kyoto Univ., Tome 1 (1961/1962), pp. 109-127 | MR 2418072 | Zbl 1156.35002

[Rau05] Rauch, J. Precise finite speed with bare hands, Methods Appl. Anal., Tome 12 (2005) no. 3, pp. 267-277 | Article | MR 2254010 | Zbl 1123.35334

[Str66] Strang, G. Necessary and insufficient conditions for well-posed Cauchy problems, J. Differential Equations, Tome 2 (1966), pp. 107-114 | MR 194722 | Zbl 0131.09102

[Vai70] Vaillancourt, R. On the stability of Friedrichs’ scheme and the modified Lax-Wendroff scheme, Math. Comput., Tome 24 (1970), pp. 767-770 | MR 277125 | Zbl 0228.65070

[Yam59] Yamaguti, M. Sur l’inégalité d’énergie pour le système hyperbolique, Proc. Japan Acad., Tome 35 (1959), pp. 37-41 | MR 105565 | Zbl 0197.07603