On the Optimality of Velocity Averaging Lemmas
نویسندگان
چکیده
– Studying weak solutions of Burgers’ equation with finite entropy dissipation we show the sharpness of recent results of Jabin and Perthame on velocity averaging. Similar arguments give bounds on the regularity of asymptotic finite-energy states for some variational problems of Ginzburg–Landau type. 2003 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ. – Nous construisons des solutions faibles de l’équation de Burgers à dissipation finie de l’entropie et montrons que les exposants de régularité obtenus récemment par Jabin et Perthame pour les théorèmes de moyennes en vitesse sont optimal. Nous étudions aussi la régularité des états asymptotiques d’un problème variationnel de type Ginzburg–Landau. 2003 Éditions scientifiques et médicales Elsevier SAS MSC: 35L65; 35B65; 46E35
منابع مشابه
A Real Space Method for Averaging Lemmas
We introduce a new method to prove averaging lemmas, i.e. prove a regularizing effect on the average in velocity of a solution to a kinetic equation. The method does not require the use of Fourier transform and the whole procedure is performed in the ’real space’. We are consequently able to improve the known result when the integrability of the solution (or the right hand side of the equation)...
متن کاملThe Averaging Lemma
Averaging lemmas arise in the study of regularity of solutions to nonlinear transport equations. The present paper shows how techniques from Harmonic Analysis, such as wavelet decompositions, maximal functions, and interpolation, can be used to prove averaging lemmas and to establish their sharpness. Let f(x, v) be a real-valued function defined on R × Ω, where Ω is a bounded domain in R. In ap...
متن کاملA New Approach to Velocity Averaging Lemmas in Besov Spaces
We develop a new approach to velocity averaging lemmas based on the dispersive properties of the kinetic transport operator. This method yields unprecedented sharp results, which display, in some cases, a gain of one full derivative. Moreover, the study of dispersion allows to treat the case of LxL p v integrability with r ≤ p. We also establish results on the control of concentrations in the d...
متن کاملN ov 2 00 8 Averaging lemmas with a force term in the transport equation
We obtain several averaging lemmas for transport operator with a force term. These lemmas improve the regularity yet known by not considering the force term as part of an arbitrary right-hand side. We compare the obtained regularities according to the space and velocity variables. Our results are mainly in L 2 , and for constant force, in L p for 1 < p ≤ 2. Keywords: averaging lemma – force ter...
متن کاملRegularity in kinetic formulations via averaging lemmas
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 k...
متن کامل