Layering methods for nonlinear partial differential equations of first order

Douglis, Avron

Annales de l'Institut Fourier, Tome 22 (1972) no. 3, pp. 141-227.

Cette étude s’intéresse aux solutions discontinues généralisées de problèmes aux données initiales d’équations de premier ordre aux dérivées partielles. “Par couche” est la méthode d’approximation d’une solution arbitraire généralisée, en divisant son domaine, disons un demi-espace $t\ge 0$, en couches minces $(i-1)h\le t\le ih$, $i=2,...,(h>0)$, et employant une solution précise $u_i$ dans la $i$-ième couche. Sur le plan $t=(i-1)h$, $u_i$ est requis de se réduire à une fonction égale approximativement aux valeurs de $u_{i-1}$ sur ce plan. Les configurations stratifiées finales de solutions précises de l’équation sont nommées solution “en couche”. Sous des conditions appropriées, chaque solution généralisée peut être réalisée comme la limite d’une séquence de solutions “en couche” pour lesquelles l’aplanissement est de plus en plus fin et $h\to 0$ ; l’estimation nécessaire pour prouver ceci appartient uniquement aux solutions précises de l’équation en question. “Par couche”, fut employé premièrement par N.N. Kuznetsov en connexion avec les lois de conservation et avec les données initiales de variations bornées (dans un sens multi-dimensionnel). Ces sujets sont discutés ici également, les méthodes sont étendues aux données initiales bornées et mesurables, et une large catégorie d’opérations possibles d’aplanissement est discutée. En plus, la méthode est adaptée aux équations du genre Hamilton-Jacobi.

This paper is concerned with generalized, discontinuous solutions of initial value problems for nonlinear first order partial differential equations. “Layering” is a method of approximating an arbitrary generalized solution by dividing its domain, say a half-space $t\ge 0$, into thin layers $(i-\ell )h\le t\le ih$, $i=1,2,...(h>0)$, and using a strict solution $u_i$ in the $i$-th layer. On the interface $t=(i-\ell )h$, $u_t$ is required to reduce to a smooth function approximating the values on that plane of $u_{i-\ell }$. The resulting stratified configuration of strict solutions of the equation is called a “layered solution” . Under appropriate conditions, any generalized solution can be realized as the limit of a sequence of layered solutions for which smoothing is made finer and finer and $h\to 0$; the estimates needed to prove this pertain solely to strict solutions of the equation concerned. Layering was first used by N.N. Kuznetsov in connection with conservation laws and with initial data of bounded variation (in a multi-dimensional sense). These matters are also discussed here, the method extended to the case of bounded, measurable initial data, and a large class of possible smoothing operations discussed. In addition, the method is adapted to equations of Hamilton-Jacobi type.

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