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We adapt the vector field method of Klainerman to the study of relativistic transport equations. First, we prove robust decay estimates for velocity averages of solutions to the relativistic massive and massless transport equations, without any compact support requirements (in or ) for the distribution functions. In the second part of this article, we apply our method to the study of the massive and massless Vlasov–Nordström systems. In the massive case, we prove global existence and (almost) optimal decay estimates for solutions in dimensions under some smallness assumptions. In the massless case, the system decouples and we prove optimal decay estimates for the solutions in dimensions for arbitrarily large data, and in dimension under some smallness assumptions, exploiting a certain form of the null condition satisfied by the equations. The -dimensional massive case requires an extension of our method and will be treated in future work.
We are concerned with the problem of real analytic regularity of the solutions of sums of squares with real analytic coefficients. The Treves conjecture defines a stratification and states that an operator of this type is analytic hypoelliptic if and only if all the strata in the stratification are symplectic manifolds.
Albano, Bove, and Mughetti (2016) produced an example where the operator has a single symplectic stratum, according to the conjecture, but is not analytic hypoelliptic.
If the characteristic manifold has codimension 2 and if it consists of a single symplectic stratum, defined again according to the conjecture, it has been shown that the operator is analytic hypoelliptic.
We show here that the above assertion is true only if the stratum is single, by producing an example with two symplectic strata which is not analytic hypoelliptic.
The Whitney extension theorem is a classical result in analysis giving a necessary and sufficient condition for a function defined on a closed set to be extendable to the whole space with a given class of regularity. It has been adapted to several settings, including the one of Carnot groups. However, the target space has generally been assumed to be equal to for some .
We focus here on the extendability problem for general ordered pairs (with nonabelian). We analyse in particular the case and characterize the groups for which the Whitney extension property holds, in terms of a newly introduced notion that we call pliability. Pliability happens to be related to rigidity as defined by Bryant and Hsu. We exploit this relation in order to provide examples of nonpliable Carnot groups, that is, Carnot groups such that the Whitney extension property does not hold. We use geometric control theory results on the accessibility of control affine systems in order to test the pliability of a Carnot group. In particular, we recover some recent results by Le Donne, Speight and Zimmerman about Lusin approximation in Carnot groups of step 2 and Whitney extension in Heisenberg groups. We extend such results to all pliable Carnot groups, and we show that the latter may be of arbitrarily large step.
We investigate the rate of convergence to equilibrium for subcritical solutions to the Becker–Döring equations with physically relevant coagulation and fragmentation coefficients and mild assumptions on the given initial data. Using a discrete version of the log-Sobolev inequality with weights, we show that in the case where the coagulation coefficient grows linearly and the detailed balance coefficients are of typical form, one can obtain a linear functional inequality for the dissipation of the relative free energy. This results in showing Cercignani’s conjecture for the Becker–Döring equations and consequently in an exponential rate of convergence to equilibrium. We also show that for all other typical cases, one can obtain an “almost” Cercignani’s conjecture, which results in an algebraic rate of convergence to equilibrium.
Concerning and , we assume that is what we call an (unbounded) admissible -domain: satisfies a uniform Lipschitz condition, adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator , as well as an additional regularity condition formulated in terms of a Carleson measure. We prove that in admissible -domains the associated parabolic measure is absolutely continuous with respect to a surface measure and that the associated Radon–Nikodym derivative defines an weight with respect to this surface measure. Our result is sharp.
We compute the -Betti numbers of the free -tensor categories, which are the representation categories of the universal unitary quantum groups . We show that the -Betti numbers of the dual of a compact quantum group are equal to the -Betti numbers of the representation category and thus, in particular, invariant under monoidal equivalence. As an application, we obtain several new computations of -Betti numbers for discrete quantum groups, including the quantum permutation groups and the free wreath product groups. Finally, we obtain upper bounds for the first -Betti number in terms of a generating set of a -tensor category.