Layering methods for nonlinear partial differential equations of first order

Avron Douglis

Annales de l'institut Fourier (1972)

  • Volume: 22, Issue: 3, page 141-227
  • ISSN: 0373-0956

Abstract

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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 ≥ 0 , into thin layers ( i - ℓ ) h ≤ t ≤ i h , i = 1 , 2 , ... ( h > 0 ) , and using a strict solution u i in the i -th layer. On the interface t = ( i - ℓ ) h , u t is required to reduce to a smooth function approximating the values on that plane of u i - ℓ . 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 → 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.

How to cite

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Douglis, Avron. "Layering methods for nonlinear partial differential equations of first order." Annales de l'institut Fourier 22.3 (1972): 141-227. <http://eudml.org/doc/74087>.

@article{Douglis1972,
abstract = {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 $(i-\ell ) h \le t\le ih$, $i=1,2,\ldots (h&gt;0)$, 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 $h\rightarrow 0$; 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.},
author = {Douglis, Avron},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {3},
pages = {141-227},
publisher = {Association des Annales de l'Institut Fourier},
title = {Layering methods for nonlinear partial differential equations of first order},
url = {http://eudml.org/doc/74087},
volume = {22},
year = {1972},
}

TY - JOUR
AU - Douglis, Avron
TI - Layering methods for nonlinear partial differential equations of first order
JO - Annales de l'institut Fourier
PY - 1972
PB - Association des Annales de l'Institut Fourier
VL - 22
IS - 3
SP - 141
EP - 227
AB - 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 $(i-\ell ) h \le t\le ih$, $i=1,2,\ldots (h&gt;0)$, 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 $h\rightarrow 0$; 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.
LA - eng
UR - http://eudml.org/doc/74087
ER -

References

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