Regularity in kinetic formulations via averaging lemmas

Pierre-Emmanuel Jabin; Benoît Perthame

ESAIM: Control, Optimisation and Calculus of Variations (2002)

  • Volume: 8, page 761-774
  • ISSN: 1292-8119

Abstract

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We present a new class of averaging lemmas directly motivated by the question of regularity for different nonlinear equations or variational problems which admit a kinetic formulation. In particular they improve the known regularity for systems like γ = 3 in isentropic gas dynamics or in some variational problems arising in thin micromagnetic films. They also allow to obtain directly the best known regularizing effect in multidimensional scalar conservation laws. The new ingredient here is to use velocity regularity for the solution to the transport equation under consideration. The method of proof is based on a decomposition of the density in Fourier space, combined with the K -method of real interpolation.

How to cite

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Jabin, Pierre-Emmanuel, and Perthame, Benoît. "Regularity in kinetic formulations via averaging lemmas." ESAIM: Control, Optimisation and Calculus of Variations 8 (2002): 761-774. <http://eudml.org/doc/245168>.

@article{Jabin2002,
abstract = {We present a new class of averaging lemmas directly motivated by the question of regularity for different nonlinear equations or variational problems which admit a kinetic formulation. In particular they improve the known regularity for systems like $ \gamma =3$ in isentropic gas dynamics or in some variational problems arising in thin micromagnetic films. They also allow to obtain directly the best known regularizing effect in multidimensional scalar conservation laws. The new ingredient here is to use velocity regularity for the solution to the transport equation under consideration. The method of proof is based on a decomposition of the density in Fourier space, combined with the $K$-method of real interpolation.},
author = {Jabin, Pierre-Emmanuel, Perthame, Benoît},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {regularizing effects; kinetic formulation; averaging lemmas; hyperbolic equations; line-energy Ginzburg–Landau; isentropic gas dynamics; transport equation},
language = {eng},
pages = {761-774},
publisher = {EDP-Sciences},
title = {Regularity in kinetic formulations via averaging lemmas},
url = {http://eudml.org/doc/245168},
volume = {8},
year = {2002},
}

TY - JOUR
AU - Jabin, Pierre-Emmanuel
AU - Perthame, Benoît
TI - Regularity in kinetic formulations via averaging lemmas
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2002
PB - EDP-Sciences
VL - 8
SP - 761
EP - 774
AB - We present a new class of averaging lemmas directly motivated by the question of regularity for different nonlinear equations or variational problems which admit a kinetic formulation. In particular they improve the known regularity for systems like $ \gamma =3$ in isentropic gas dynamics or in some variational problems arising in thin micromagnetic films. They also allow to obtain directly the best known regularizing effect in multidimensional scalar conservation laws. The new ingredient here is to use velocity regularity for the solution to the transport equation under consideration. The method of proof is based on a decomposition of the density in Fourier space, combined with the $K$-method of real interpolation.
LA - eng
KW - regularizing effects; kinetic formulation; averaging lemmas; hyperbolic equations; line-energy Ginzburg–Landau; isentropic gas dynamics; transport equation
UR - http://eudml.org/doc/245168
ER -

References

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