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Abstract. We develop a new approach to study the well-posedness theory of the Prandtl equation in Sobolev spaces by using a direct energy method under a monotonicity condition on the tangential velocity field instead of using the Crocco transformation. Precisely, we firstly investigate the linearized Prandtl equation in some weighted Sobolev spaces when the tangential velocity of the background state is monotonic in the normal variable. Then to cope with the loss of regularity of the perturbation with respect to the background state due to the degeneracy of the equation, we apply the Nash-Moser-Hörmander iteration to obtain a well-posedness theory of classical solutions to the nonlinear Prandtl equation when the initial data is a small perturbation of a monotonic shear flow.

Propagating waves occur in many excitable media and were recently found in neural systems from retina to neocortex. While propagating waves are clearly present under anaesthesia, whether they also appear during awake and conscious states remains unclear. One possibility is that these waves are systematically missed in trial-averaged data, due to variability. Here we present a method for detecting propagating waves in noisy multichannel recordings. Applying this method to single-trial voltage-sensitive dye imaging data, we show that the stimulus-evoked population response in primary visual cortex of the awake monkey propagates as a travelling wave, with consistent dynamics across trials. A network model suggests that this reliability is the hallmark of the horizontal fibre network of superficial cortical layers. Propagating waves with similar properties occur independently in secondary visual cortex, but maintain precise phase relations with the waves in primary visual cortex. These results show that, in response to a visual stimulus, propagating waves are systematically evoked in several visual areas, generating a consistent spatiotemporal frame for further neuronal interactions.

We study the Boltzmann equation without Grad's angular cutoff assumption. We introduce a suitable renormalized formulation that allows the cross section to be singular in both the angular and the relative velocity variables. Angular singularities occur as soon as one is interested in long-range interactions, while singularities in the relative velocity variable occur in the study of soft potentials, in particular, Coulomb interaction. Together with several new estimates, this new formulation enables us to prove existence of weak solutions and to give a proof of a conjecture by Lions (appearance of strong compactness) under general, fully realistic assumptions.

It is known that the singularity in the non-cutoff cross-section of the Boltzmann equation leads to the gain of regularity and gain of weight in the velocity variable. By defining and analyzing a non-isotropy norm which precisely captures the dissipation in the linearized collision operator, we first give a new and precise coercivity estimate for the non-cutoff Boltzmann equation for general physical cross sections. Then the Cauchy problem for the Boltzmann equation is considered in the framework of small perturbation of an equilibrium state. In this part, for the soft potential case in the sense that there is no positive power gain of weight in the coercivity estimate on the linearized operator, we derive some new functional estimates on the nonlinear collision operator. Together with the coercivity estimates, we prove the global existence of classical solutions for the Boltzmann equation in weighted Sobolev spaces. Contents2000 Mathematics Subject Classification. 35A05, 35B65, 35D10, 35H20, 76P05, 84C40.

The Boltzmann equation without Grad's angular cutoff assumption is believed to have regularizing effect on the solution because of the non-integrable angular singularity of the cross-section. However, even though so far this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially inhomogeneous Boltzmann equation. In this paper, by sharpening the coercivity and upper bound estimates for the collision operator, establishing the hypo-ellipticity of the Boltzmann operator based on a generalized version of the uncertainty principle, and analyzing the commutators between the collision operator and some weighted pseudo-differential operators, we prove the regularizing effect in all (time, space and velocity) variables on solutions when some mild regularity is imposed on these solutions. For completeness, we also show that when the initial data has this mild regularity and Maxwellian type decay in velocity variable, there exists a unique local solution with the same regularity, so that this solution acquires the C ∞ regularity for positive time.2000 Mathematics Subject Classification. 35A05, 35B65, 35D10, 35H20, 76P05, 84C40.

Abstract. We prove the global existence and uniqueness of classical solutions around an equilibrium to the Boltzmann equation without angular cutoff in some Sobolev spaces. In addition, the solutions thus obtained are shown to be non-negative and C ∞ in all variables for any positive time. In this paper, we study the Maxwellian molecule type collision operator with mild singularity. One of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator. This norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated H-theorem.

This paper studies the approximation of the Boltzmann equation by the Landau equation in a regime when grazing collisions prevail. While all previous results in the subject were limited to the spatially homogeneous case, here we manage to cover the general, space-dependent situation, assuming only basic physical estimates of finite mass, energy, entropy and entropy production. The proofs are based on the recent results and methods introduced previously in [R. Alexandre, C. Villani, Comm. Pure Appl. Math. 55 (1) (2002) 30-70] by both authors, and the entropy production smoothing effects established in [R. Alexandre et al., Arch. Rational Mech. Anal. 152 (4) (2000) 327-355]. We are able to treat realistic singularities of Coulomb type, and approximations of the Debye cut. However, our method only works for finite-time intervals, while the Landau equation is supposed to describe long-time corrections to the Vlasov-Poisson equation. If the mean-field interaction is neglected, then our results apply to physically relevant situations after a time rescaling.  2003 Elsevier SAS. All rights reserved. RésuméNous étudions l'approximation de l'équation de Boltzmann par l'équation de Landau quand les collisions rasantes sont dominantes. Alors que tous les résultats connus auparavant en la matière concernaient le cas spatialement homogène, ici nous parvenons à couvrir le cas général, spatialement inhomogène, supposant seulement des estimations a priori physiquement réalistes portant sur la masse, l'énergie, l'entropie et la production d'entropie. Les preuves reposent sur les résultats et méthodes mis au point récemment par les auteurs dans [R. Alexandre, C. Villani, Comm. Pure Appl. Math. 55 (1) (2002) 30-70], et sur les effets de régularisation par production d'entropie établis dans [R. Alexandre et al., Arch. Rational Mech. Anal. 152 (4) (2000) 327-355]. Nos résultats couvrent certaines singularités physiquement réalistes de type coulombien, et des approximations de la coupure de Debye. Cependant, nos résultats s'appliquent sur un intervalle de temps fini, alors que l'équation de Landau est censée décrire les corrections en temps grand de l'équation de Vlasov-Poisson. Si le terme d'interaction de champ moyen est négligé, nos résultats s'appliquent à des situations physiquement réalistes après un changement d'échelle de temps.