In this paper, we give a systematic study of the boundary layer behavior for linear convection-di... more In this paper, we give a systematic study of the boundary layer behavior for linear convection-diffusion equation in the zero viscosity limit. We analyze the boundary layer structures in the viscous solution and derive the boundary condition satisfied by the viscosity limit as a solution of the inviscid equation. The results confirm that the Neumann type of far-field boundary condition is preferred in the outlet and characteristic boundary condition. Under some appropriate regularity and compatibility conditions on the initial and boundary data, we obtain optimal error estimates between the full viscous solution and the inviscid solution with suitable boundary layer corrections. These results hold in arbitrary space dimensions and similar statements also hold for the strip problem.
Mathematical Models and Methods in Applied Sciences, 2017
We investigate the long-time dynamics of an opinion formation model inspired by a work by Borghes... more We investigate the long-time dynamics of an opinion formation model inspired by a work by Borghesi, Bouchaud and Jensen. First, we derive a Fokker–Planck-type equation under the assumption that interactions between individuals produce little consensus of opinion (grazing collision approximation). Second, we study conditions under which the Fokker–Planck equation has non-trivial equilibria and derive the macroscopic limit (corresponding to the long-time dynamics and spatially localized interactions) for the evolution of the mean opinion. Finally, we compare two different types of interaction rates: the original one given in the work of Borghesi, Bouchaud and Jensen (symmetric binary interactions) and one inspired from works by Motsch and Tadmor (non-symmetric binary interactions). We show that the first case leads to a conservative model for the density of the mean opinion whereas the second case leads to a non-conservative equation. We also show that the speed at which consensus is ...
Transactions of the American Mathematical Society, 2016
The class of generating functions for completely monotone sequences (moments of finite positive m... more The class of generating functions for completely monotone sequences (moments of finite positive measures on [ 0 , 1 ] [0,1] ) has an elegant characterization as the class of Pick functions analytic and positive on ( − ∞ , 1 ) (-\infty ,1) . We establish this and another such characterization and develop a variety of consequences. In particular, we characterize generating functions for moments of convex and concave probability distribution functions on [ 0 , 1 ] [0,1] . Also we provide a simple analytic proof that for any real p p and r r with p > 0 p>0 , the Fuss-Catalan or Raney numbers r p n + r ( p n + r n ) \frac {r}{pn+r}\binom {pn+r}{n} , n = 0 , 1 , … n=0,1,\ldots , are the moments of a probability distribution on some interval [ 0 , τ ] [0,\tau ] if and only if p ≥ 1 p\ge 1 and p ≥ r ≥ 0 p\ge r\ge 0 . The same statement holds for the binomial coefficients ( p n + r − 1 n ) \binom {pn+r-1}n , n = 0 , 1 , … n=0,1,\ldots \, .
In this paper, we give a systematic study of the boundary layer behavior for linear convection-di... more In this paper, we give a systematic study of the boundary layer behavior for linear convection-diffusion equation in the zero viscosity limit. We analyze the boundary layer structures in the viscous solution and derive the boundary condition satisfied by the viscosity limit as a solution of the inviscid equation. The results confirm that the Neumann type of far-field boundary condition is preferred in the outlet and characteristic boundary condition. Under some appropriate regularity and compatibility conditions on the initial and boundary data, we obtain optimal error estimates between the full viscous solution and the inviscid solution with suitable boundary layer corrections. These results hold in arbitrary space dimensions and similar statements also hold for the strip problem.
Mathematical Models and Methods in Applied Sciences, 2017
We investigate the long-time dynamics of an opinion formation model inspired by a work by Borghes... more We investigate the long-time dynamics of an opinion formation model inspired by a work by Borghesi, Bouchaud and Jensen. First, we derive a Fokker–Planck-type equation under the assumption that interactions between individuals produce little consensus of opinion (grazing collision approximation). Second, we study conditions under which the Fokker–Planck equation has non-trivial equilibria and derive the macroscopic limit (corresponding to the long-time dynamics and spatially localized interactions) for the evolution of the mean opinion. Finally, we compare two different types of interaction rates: the original one given in the work of Borghesi, Bouchaud and Jensen (symmetric binary interactions) and one inspired from works by Motsch and Tadmor (non-symmetric binary interactions). We show that the first case leads to a conservative model for the density of the mean opinion whereas the second case leads to a non-conservative equation. We also show that the speed at which consensus is ...
Transactions of the American Mathematical Society, 2016
The class of generating functions for completely monotone sequences (moments of finite positive m... more The class of generating functions for completely monotone sequences (moments of finite positive measures on [ 0 , 1 ] [0,1] ) has an elegant characterization as the class of Pick functions analytic and positive on ( − ∞ , 1 ) (-\infty ,1) . We establish this and another such characterization and develop a variety of consequences. In particular, we characterize generating functions for moments of convex and concave probability distribution functions on [ 0 , 1 ] [0,1] . Also we provide a simple analytic proof that for any real p p and r r with p > 0 p>0 , the Fuss-Catalan or Raney numbers r p n + r ( p n + r n ) \frac {r}{pn+r}\binom {pn+r}{n} , n = 0 , 1 , … n=0,1,\ldots , are the moments of a probability distribution on some interval [ 0 , τ ] [0,\tau ] if and only if p ≥ 1 p\ge 1 and p ≥ r ≥ 0 p\ge r\ge 0 . The same statement holds for the binomial coefficients ( p n + r − 1 n ) \binom {pn+r-1}n , n = 0 , 1 , … n=0,1,\ldots \, .
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