305,161 research outputs found
Remarks on the H theorem for a non involutive Boltzmann like kinetic model
No full textIn this paper, we consider a one-dimensional kinetic equation of Boltzmann type in which the binary collision process is described by the linear transformation v* = pv + qw, w* = qv + pw, where (v, w) are the pre-collisional velocities and (v*, w*) the post-collisional ones and p ≥ q > 0 are two positive parameters. This kind of model has been extensively studied by Pareschi and Toscani (in J. Stat. Phys., 124(2–4):747–779, 2006) with respect to the asymptotic behavior of the solutions in a Fourier metric. In the conservative case p2 + q2 = 1, even if the transformation has Jacobian J ≠ 1 and so it is not involutive, we remark that the H Theorem holds true. As a consequence we prove exponential convergence in L1 of the solution to the stationary state, which is the Maxwellian
Sharp cooling rates in nonlinear friction equations
No full textWe study some extremal properties of the self-similar solutions of certain onedimensional kinetic models of granular flows, usually known with the name of nonlinear friction equations. This analysis, inspired by some recent results on nonlinear diffusion equations [6], allows to obtain various sharp inequalities, which can be fruitfully used to better clarify the large-time behavior of the solution density
Besov spaces and unconditional well-posedness for the nonlinear Schrodinger equation in H-s (R-n)
No full textWe extend some results about uniqueness, without any extra condition, of solutions for the nonlinear Schrödinger equation with polynomial nonlinearities in low dimensions. The proof lies on paraproduct techniques and Besov spaces
Temi d'esame di Analisi Matematica I. Vol. 2, Limiti e studio di funzione
No full textI due volumi di esercizi di cui questo è il secondo, raccolgono temi d’esame svolti dei corsi di Analisi Matematica 1 (precedentemente denominati Matematica 1) tenuti pres- so la facoltà di Ingegneria dell’Università di Bergamo negli anni 2004-2012. Tutti gli esercizi presentati sono completamente e dettagliatamente svolti. Gli argomenti trattati nel Volume 2 sono limiti e studio di funzione reale di una variabile reale. In particolare, nel capitolo sullo studio di funzione, oltre ai classici studi del grafico di una funzione tramite le sue proprietà essenziali, sono presentati anche esercizi specifici sullo studio di continuità e derivabilità, sulla determinazione della retta tangente e sui polinomi di Taylor, oltre che sulla composizione e inversione di funzioni
Wright–Fisher–type equations for opinion formation, large time behavior and weighted logarithmic-Sobolev inequalities
No full textWe study the rate of convergence to equilibrium of the solution of a Fokker–Planck type equation introduced in [19] to describe opinion formation in a multi-agent system. The main feature of this Fokker–Planck equation is the presence of a variable diffusion coefficient and boundaries, which introduce new challenging mathematical problems in the study of its long-time behavior
Temi d’esame di Analisi Matematica I. Vol. 1, Numeri complessi, numeri reali, successioni, serie ed integrali
No full textI due volumi di esercizi di cui questo è il primo, raccolgono temi d’esame svolti dei corsi di Analisi Matematica 1 (precedentemente denominati Matematica 1) tenuti presso la facoltà di Ingegneria dell’Università di Bergamo negli anni 2004-2012. Tutti gli esercizi presentati sono completamente e dettagliatamente svolti. Gli argomenti trattati nel Volume 1 sono numeri complessi, numeri reali (in particolare ricerca di massimo, minimo, estremo superiore ed estremo inferiore di insiemi numerici), successioni, serie ed integrali, definiti, indefiniti e generalizzati
Strong convergence towards self-similarity for one-dimensional dissipative Maxwell models
No full textAbstractWe prove the propagation of regularity, uniformly in time, for the scaled solutions of the one-dimensional dissipative Maxwell models introduced in [D. Ben-Avraham, E. Ben-Naim, K. Lindenberg, A. Rosas, Self-similarity in random collision processes, Phys. Rev. E 68 (2003) R050103]. This result together with the weak convergence towards the stationary state proven in [L. Pareschi, G. Toscani, Self-similarity and power-like tails in nonconservative kinetic models, J. Stat. Phys. 124 (2–4) (2006) 747–779] implies the strong convergence in Sobolev norms and in the L1 norm towards it depending on the regularity of the initial data. As a consequence, the original nonscaled solutions are also proved to be convergent in L1 towards the corresponding self-similar homogeneous cooling state. The proof is based on the (uniform in time) control of the tails of the Fourier transform of the solution, and it holds for a large range of values of the mixing parameters. In particular, in the case of the one-dimensional inelastic Boltzmann equation, the result does not depend of the degree of inelasticity
Heat equation with an exponential nonlinear boundary condition in the half space
No full textWe consider the initial-boundary value problem for the heat equation in the half space with an
exponential nonlinear boundary condition. We prove the existence of global-in-time solutions
under the smallness condition on the initial data in the Orlicz space expL2(RN
+ ). Furthermore,
we derive decay estimates and the asymptotic behavior for small global-in-time solutions
Non-Maxwellian kinetic equations modeling the dynamics of wealth distribution
No full textWe introduce a class of new one-dimensional linear Fokker-Planck-type equations describing the dynamics of the distribution of wealth in a multi-agent society. The equations are obtained, via a standard limiting procedure, by introducing an economically relevant variant to the kinetic model introduced in 2005 by Cordier, Pareschi and Toscani according to previous studies by Bouchaud and Mézard. The steady state of wealth predicted by these new Fokker-Planck equations remains unchanged with respect to the steady state of the original Fokker-Planck equation. However, unlike the original equation, it is proven by a new logarithmic Sobolev inequality with weight and classical entropy methods that the solution converges exponentially fast to equilibrium
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