On the analyticity of solutions of first-order nonlinear PDE. (English) Zbl 0758.35018

Hanges, Nicholas; Treves, François

Trans. Am. Math. Soc. 331, No. 2, 627-638 (1992).

(From the author’s abstract:) Let \(x\in R^ m\), \(t\in R^ 1\) and \(u\in C^ 2\). We discuss local and microlocal analyticity for solutions \(u\) to the nonlinear equation \(u_ t=f(x,t,u,u_ x)\). Here \(f(x,t,\zeta_ 0,\zeta)\) is complex valued and analytic in all arguments. We also assume \(f\) to be holomorphic in \((\zeta_ 0,\zeta)\in C\times C^ m\). In particular we show that \(WF_ Au\subseteq\text{Char}(L^ u)\), where \(WF_ A\) denotes the analytic wave front set and \(\text{Char}(L^ u)\) is the characteristic of the linearized operator \(L^ u=\partial/\partial t-\sum\partial f/\partial\zeta_ j(x,t,u,u_ x)\partial/\partial x_ j\). If we assume that \(u\in C^ 3(R^ m\times R)\), then we show the analyticity of \(u\) propagates along the elliptic submanifolds of \(L^ u\).

Reviewer: E.Barron (Chicago)

MSC:

35F20	Nonlinear first-order PDEs
35A20	Analyticity in context of PDEs
35A27	Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35S30	Fourier integral operators applied to PDEs

References:

[1]	M. S. Baouendi, C. H. Chang, and F. Trèves, Microlocal hypo-analyticity and extension of CR functions, J. Differential Geom. 18 (1983), no. 3, 331 – 391. · Zbl 0575.32019
[2]	M. S. Baouendi and F. Trèves, A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. of Math. (2) 113 (1981), no. 2, 387 – 421. · Zbl 0491.35036 · doi:10.2307/2006990
[3]	J.-Y. Chemin, Calcul paradifférentiel précisé et applications à des équations aux dérivées partielles non semilinéaires, Duke Math. J. 56 (1988), no. 3, 431 – 469 (French). · Zbl 0676.35009 · doi:10.1215/S0012-7094-88-05619-0
[4]	Nicholas Hanges and François Trèves, Propagation of holomorphic extendability of CR functions, Math. Ann. 263 (1983), no. 2, 157 – 177. · Zbl 0494.32004 · doi:10.1007/BF01456878
[5]	J.-M. Trépreau, Sur la propagation des singularités dans les variétés CR, Bull. Soc. Math. France 118 (1990), no. 4, 403 – 450 (French, with English summary). · Zbl 0742.58053
[6]	François Trèves, Hypo-analytic structures, Princeton Mathematical Series, vol. 40, Princeton University Press, Princeton, NJ, 1992. Local theory. · Zbl 0565.35079