Bounded, stratified and striated solutions of hyperbolic systems. (English) Zbl 0695.35124

Nonlinear partial differential equations and their applications, Lect. Coll. de France Semin., Vol. IX, Paris/Fr. 1985-86, Pitman Res. Notes Math. Ser. 181, 334-351 (1988).
Summary: [For the entire collection see Zbl 0653.00012.]
Stratified solutions are those whose derivatives tangent to a foliation by regular characteristic hypersurfaces all lie in \(L^ 2\). Striated solutions of two speed systems are differentiable tangent to a codimension two foliation, the transverse intersection of two characteristic foliations. Our main results are local existence and continuous dependence theorems for bounded, stratified and striated solutions. If \(L^ 2\) is replaced by \(H^ 2\) with \(s>N/2\) these results are known or follows easily from known results using standard techniques. Bounded, stratified solutions are important for the study of the semilinear analogue of the oscillating solutions of P. D. Lax [Duke Math. J. 24, 627-646 (1957; Zbl 0083.318)]. For pairwise interactions of oscillations in two speed systems, the striated category is appropriate.


35L60 First-order nonlinear hyperbolic equations
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs