Vector fields associated with the nonlinear interaction of progressing waves. (English) Zbl 0644.35069

Beals, Michael

Indiana Univ. Math. J. 37, No. 3, 637-666 (1988).

Let u be a sufficiently smooth solution to the semilinear equation \(p(t,x_ 1,x_ 2;\partial _ t,\partial _{x_ 1},\partial _{x_ 2})u=f(t,x_ 1,x_ 2,u)\) on an open set in \({\mathbb{R}}\times {\mathbb{R}}^ 2\) containing the origin. Here f is assumed to be a smooth function of its arguments, and \(p(t,x_ 1,x_ 2;\partial _ t,\partial _{x_ 1},\partial _{x_ 2})\) is a second order differential operator with strictly hyperbolic partial part. If \(\Sigma _ 1\), \(\Sigma _ 2\), and \(\Sigma _ 3\) are smooth characteristic hypersurfaces for p intersecting transversally at the origin, suppose that u is conormal in the past with respect to the family \((\Sigma _ 1,\Sigma _ 2,\Sigma _ 3)\), which means u does not lose regularity in \((t<t_ 0)\) when differentiated by any smooth vector fields simultaneously tangent to \(\Sigma _ 1\), \(\Sigma _ 2\), and \(\Sigma _ 3\). J. Rauch and M. Reed [Commun. Partial Differ. Equations 7, 1117-1133 (1982; Zbl 0502.35060)] demonstrated that there are choices of nonlinear f for which u has additional singularities on C, the surface of the light cone over the origin. R. Melrose and N. Ritter [Ann. Math., II. Ser. 121, 187-213 (1985; Zbl 0575.35063)] and J.-M. Bony [Semin. Goulaouic- Meyer-Schwartz, Equations Deriv. Partielles 1983-1984, Exp. No.10, 27 p. (1984; Zbl 0555.35118)] proved that u remains conormal with respect to the family \((\Sigma _ 1,\Sigma _ 2,\Sigma _ 3,C)\). In this paper a new proof of this property is provided, involving no microlocalization, and only the usual energy estimate for the linear inhomogeneous equation. Commutator arguments for a special collection of smooth vector fields, each of which is tangent to at least three of the four hypersurfaces \(\Sigma _ 1\), \(\Sigma _ 2\), \(\Sigma _ 3\), and C, are used on cones over the origin. The equation itself is used to handle the remaining vector fields needed to establish conormal smoothness.

Reviewer: M.Beals

MSC:

35L70	Second-order nonlinear hyperbolic equations
35L67	Shocks and singularities for hyperbolic equations
35B45	A priori estimates in context of PDEs

Keywords:

nonlinear interaction; semilinear; strictly hyperbolic; smooth characteristic hypersurfaces; conormal; singularities; surface of the light cone; energy estimate; commutator arguments

Citations:

Zbl 0502.35060; Zbl 0575.35063; Zbl 0555.35118