Bernt Wennberg

Overproduction of VLDL1 Driven by Hyperglycemia Is a Dominant Feature of Diabetic Dyslipidemia

Objective—We sought to compare the synthesis and metabolism of VLDL1 and VLDL2 in patients with type 2 diabetes mellitus (DM2) and nondiabetic subjects. Methods and Results—We used a novel multicompartmental model to simultaneously determine the kinetics of apolipoprotein (apo) B and triglyceride (TG) in VLDL1 and VLDL2 after a bolus injection of [2H3]leucine and [2H5]glycerol and to follow the catabolism and transfer of the lipoprotein particles. Our results show that the overproduction of VLDL particles in DM2 is explained by enhanced secretion of VLDL1 apoB and TG. Direct production of VLDL2 apoB and TG was not influenced by diabetes per se. The production rates of VLDL1 apoB and TG were closely related, as were the corresponding pool sizes. VLDL1 and VLDL2 compositions did not differ in subjects with DM2 and controls, and the TG to apoB ratio of newly synthesized particles was very similar in the 2 groups. Plasma glucose, insulin, and free fatty acids together explained 55% of the variation in VLDL1 TG production rate. Conclusion—Insulin resistance and DM2 are associated with excess hepatic production of VLDL1 particles similar in size and composition to those in nondiabetic subjects. We propose that hyperglycemia is the driving force that aggravates overproduction of VLDL1 in DM2.

Metrics for Probability Distributions and the Trend to Equilibrium for Solutions of the Boltzmann Equation

This paper deals with the trend to equilibrium of solutions to the spacehomogeneous Boltzmann equation for Maxwellian molecules with angular cutoff as well as with infinite-range forces. The solutions are considered as densities of probability distributions. The Tanaka functional is a metric for the space of probability distributions, which has previously been used in connection with the Boltzmann equation. Our main result is that, if the initial distribution possesses moments of order 2+ε, then the convergence to equilibrium in his metric is exponential in time. In the proof, we study the relation between several metrics for spaces of probability distributions, and relate this to the Boltzmann equation, by proving that the Fourier-transformed solutions are at least as regular as the Fourier transform of the initial data. This is also used to prove that even if the initial data only possess a second moment, then ∫∣v∣>Rf(v, t) ∣v∣2dv→0 asR→∞, and this convergence is uniform in time.

Smoothness of the Solution of the Spatially Homogeneous Boltzmann Equation without Cutoff

Abstract For regularized hard potentials cross sections, the solution of the spatially homogeneous Boltzmann equation without angular cutoff lies in Schwartzs space 𝒮(ℝ N ) for all (strictly positive) time. The proof is presented in full detail for the two-dimensional case, and for a moderate singularity of the cross section. Then we present those parts of the proof for the general case, where the dimension, or the strength of the singularity play an essential role.

ENTROPY DISSIPATION AND MOMENT PRODUCTION FOR THE BOLTZMANN EQUATION

LetH(f/M)=∫flog(f/M)dv be the relative entropy off and the Maxwellian with the same mass, momentum, and energy, and denote the corresponding entropy dissipation term in the Boltzmann equation byD(f)=∫Q(f,f) logf dv. An example is presented which shows that |D(f)/H(f/M)| can be arbitrarily small. This example is a sequence of isotropic functions, and the estimates are very explicitly given by a simple formula forD which holds for such functions. The paper also gives a simplified proof of the so-called Povzner inequality, which is a geometric inequality for the magnitudes of the velocities before and after an elastic collision. That inequality is then used to prove that ∫f(v) |v|sdt<C(t), wheref is the solution of the spatially homogeneous Boltzmann equation. HereC(t) is an explicitly given function dependings and the mass, energy, and entropy of the initial data.

A Maxwellian lower bound for solutions to the Boltzmann equation

We prove that the solution of the spatially homogeneous Boltzmann equation is bounded pointwise from below by a Maxwellian, i.e. a function of the formc1 exp(-c2v2). This holds for any initial data with bounded mass, energy and entropy, and for any positive timet≧t0. The constantsc1, andc2, depend on the mass, energy and entropy of the initial data, and ont0>0 only.A similar result is obtained for the Kac caricature of the Boltzmann equation, where the proof is easier.

Brief paper: State elimination and identifiability of the delay parameter for nonlinear time-delay systems

The identifiability of the delay parameter for nonlinear systems with a single constant time delay is analyzed. We show the existence of input-output equations and relate the identifiability of the delay parameter to their form. Explicit criteria based on rank calculations are formulated. The identifiability of the delay parameter is shown not to be directly related to the well-characterized identifiability/observability of the other system parameters/states.

Stability and exponential convergence for the Boltzmann equation

We prove existence, uniqueness and stability for solutions of the nonlinear Boltzmann equation in a periodic box in the case when the initial data are sufficiently close to a spatially homogeneous function. The results are given for a range of spaces, including L1, and extend previous results in L∞ for the non-homogeneous equation, as well as the more developed Lp-theory for the spatially homogeneous Boltzmann equation.We also give new L∞-estimates for the spatially homogeneous equation in the case of Maxwellian interactions.

On moments and uniqueness for solutions to the space homogeneous Boltzmann equation

Abstract Recently Desvillettes proved that the solutions to the space homogeneous Boltzmann equation possess all moments, provided that the initial data have 2 + ∊ moments in L 1. Here an alternative proof of this statement is presented, together with a proof that the result does not hold for pseudo-Maxwellian molecules. Two implications of the result are discussed: an analogous L p-result and an improved uniqueness result.

Lattice points on circles and discrete velocity models for the Boltzmann equation

The construction of discrete velocity models or numerical methods for the Boltzmann equation, may lead to the necessity of computing the collision operator as a sum over lattice points. The collision operator involves an integral over a sphere, which corresponds to the conservation of energy and momentum. In dimension two there are difficulties even in proving the convergence of such an approximation since many circles contain very few lattice points, and some circles contain many badly distributed lattice points. However, by showing that lattice points on most circles are equidistributed we find that the collision operator can indeed be approximated as a sum over lattice points in the two-dimensional case. The proof uses a weak form of the Halberstam-Richert inequality for multiplicative functions (a proof is given in the paper), and estimates for the angular distribution of Gaussian primes. For higher dimensions, this result has already been obtained by Palczewski, Schneider, and Bobylev [SIAM J. Numer. Anal., 34 (1997), pp. 1865-1883].

A new approach to quantitative propagation of chaos for drift, diffusion and jump processes

This paper is devoted the study of the mean field limit for many-particle systems undergoing jump, drift or diffusion processes, as well as combinations of them. The main results are quantitative estimates on the decay of fluctuations around the deterministic limit and of correlations between particles, as the number of particles goes to infinity. To this end we introduce a general functional framework which reduces this question to the one of proving a purely functional estimate on some abstract generator operators (consistency estimate) together with fine stability estimates on the flow of the limiting nonlinear equation (stability estimates). Then we apply this method to a Boltzmann collision jump process (for Maxwell molecules), to a McKean–Vlasov drift-diffusion process and to an inelastic Boltzmann collision jump process with (stochastic) thermal bath. To our knowledge, our approach yields the first such quantitative results for a combination of jump and diffusion processes.