Precise finite speed and uniqueness in the Cauchy problem for symmetrizable hyperbolic systems

Rauch, Jeffrey

doi:10.1090/S0002-9947-2010-05047-0

Precise finite speed and uniqueness in the Cauchy problem for symmetrizable hyperbolic systems
HTML articles powered by AMS MathViewer

by Jeffrey Rauch PDF

Trans. Amer. Math. Soc. 363 (2011), 1161-1182 Request permission

Abstract:

Precise finite speed, in the sense of the domain of influence being a subset of the union of influence curves through the support of the initial data, is proved for hyperbolic systems symmetrized by pseudodifferential operators in the spatial variables. From this, uniqueness in the Cauchy problem at spacelike hypersurfaces is derived by a Hölmgren style duality argument. Sharp finite speed is derived from an estimate for propagation in each direction. Propagation in a fixed direction is proved by regularizing the problem in the orthogonal directions. Uniform estimates for the regularized equations are proved using pseudodifferential techniques of Beals-Fefferman type.

References

Richard Beals and Charles Fefferman, Spatially inhomogeneous pseudodifferential operators. I, Comm. Pure Appl. Math. 27 (1974), 1–24. MR 352747, DOI 10.1002/cpa.3160270102
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
Alberto-P. Calderón, Integrales singulares y sus aplicaciones a ecuaciones diferenciales hiperbolicas, Universidad de Buenos Aires, Buenos Aires, 1960 (Spanish). Cursos y Seminarios de Matemática, Fasc. 3. MR 0123834
R. Courant, Methods of Mathematical Physics vol II, Interscience Publ., 1962.
K. O. Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math. 7 (1954), 345–392. MR 62932, DOI 10.1002/cpa.3160070206
K. O. Friedrichs and P. D. Lax, Boundary value problems for first order operators, Comm. Pure Appl. Math. 18 (1965), 355–388. MR 174999, DOI 10.1002/cpa.3160180127
K. O. Friedrichs and P. D. Lax, On symmetrizable differential operators, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) Amer. Math. Soc., Providence, R.I., 1967, pp. 128–137. MR 0239256
Lars Gårding, Linear hyperbolic partial differential equations with constant coefficients, Acta Math. 85 (1951), 1–62. MR 41336, DOI 10.1007/BF02395740
L. Hörmander, The Analysis of Linear Partial Differential Operators vol. I (§9.6), Grundlehren der mathematischen Wissenschaften #256, Springer-Verlag, 1983.
L. Hörmander, The Analysis of Linear Partial Differential Operators vol. III, Grundlehren der mathematischen Wissenschaften #274, Springer-Verlag, 1985.
Heinz-Otto Kreiss, Über Matrizen die beschränkte Halbgruppen erzeugen, Math. Scand. 7 (1959), 71–80 (German). MR 110952, DOI 10.7146/math.scand.a-10563
Jean-Luc Joly, Guy Métivier, and Jeffrey Rauch, Hyperbolic domains of determinacy and Hamilton-Jacobi equations, J. Hyperbolic Differ. Equ. 2 (2005), no. 3, 713–744. MR 2172702, DOI 10.1142/S0219891605000609
P. D. Lax, Lectures on Hyperbolic Partial Differential Equations, Stanford Lecture Notes, 1963.
P. D. Lax and L. Nirenberg, On stability for difference schemes: A sharp form of Gȧrding’s inequality, Comm. Pure Appl. Math. 19 (1966), 473–492. MR 206534, DOI 10.1002/cpa.3160190409
Jean Leray, Hyperbolic differential equations, Institute for Advanced Study (IAS), Princeton, N.J., 1953. MR 0063548
Jeffrey Rauch, Precise finite speed with bare hands, Methods Appl. Anal. 12 (2005), no. 3, 267–277. MR 2254010, DOI 10.4310/MAA.2005.v12.n3.a4
J. Rauch, Finite speed for symmetrizable hyperbolic systems, J. Hyp. Diff. Eq., to appear.
S. Spagnolo and G. Taglialatela, Quasisymmetrizers and semilinear hyperbolic systems, preprint.
Rémi Vaillancourt, A simple proof of Lax-Nirenberg theorems, Comm. Pure Appl. Math. 23 (1970), 151–163. MR 257824, DOI 10.1002/cpa.3160230203
Rémi Vaillancourt, On the stability of Friedrichs’ scheme and the modified Lax-Wendroff scheme, Math. Comp. 24 (1970), 767–770. MR 277125, DOI 10.1090/S0025-5718-1970-0277125-6
Masaya Yamaguti, Sur l’inégalité d’énergie pour le système hyperbolique, Proc. Japan Acad. 35 (1959), 37–41 (French). MR 105565
Masaya Yamaguti and Tatsuo Nogi, An algebra of pseudo difference schemes and its application, Publ. Res. Inst. Math. Sci. Ser. A 3 (1967/1968), 151–166. MR 0226172, DOI 10.2977/prims/1195195561