scholarly journals Local and global solvability and blow up for the drift-diffusion equation with the fractional dissipation in the critical space

2015 ◽  
Vol 258 (9) ◽  
pp. 2983-3010 ◽  
Author(s):  
Yuusuke Sugiyama ◽  
Masakazu Yamamoto ◽  
Keiichi Kato
Keyword(s):  
Diffusion Equation ◽  
Blow Up ◽  
Global Solvability ◽  
Critical Space ◽  
Drift Diffusion

Non-uniform bound and finite time blow up for solutions to a drift–diffusion equation in higher dimensions

2016 ◽  
Vol 14 (01) ◽  
pp. 145-183 ◽  
Author(s):  
Takayoshi Ogawa ◽  
Hiroshi Wakui
Keyword(s):  
Diffusion Equation ◽  
Finite Time ◽  
Blow Up ◽  
Elliptic Type ◽  
Sharp Constant ◽  
Uniform Bound ◽  
Critical Space ◽  
Drift Diffusion ◽  
Entropy Functional ◽  
Blowing Up

We show the non-uniform bound for a solution to the Cauchy problem of a drift–diffusion equation of a parabolic–elliptic type in higher space dimensions. If an initial data satisfies a certain condition involving the entropy functional, then the corresponding solution to the equation does not remain uniformly bounded in a scaling critical space. In other words, the solution grows up at [Formula: see text] in the critical space or blows up in a finite time. Our presenting results correspond to the finite time blowing up result for the two-dimensional case. The proof relies on the logarithmic entropy functional and a generalized version of the Shannon inequality. We also give the sharp constant of the Shannon inequality.

scholarly journals Formulation of the Boltzmann Equation as a Multi-Mode Drift-Diffusion Equation

VLSI Design ◽  
1998 ◽  
Vol 8 (1-4) ◽  
pp. 539-544
Author(s):  
K. Banoo ◽  
F. Assad ◽  
M. S. Lundstrom
Keyword(s):  
Diffusion Coefficient ◽  
Boltzmann Equation ◽  
Diffusion Equation ◽  
Momentum Space ◽  
Bulk Silicon ◽  
Rigorous Solution ◽  
Drift Diffusion ◽  
The Boltzmann Equation ◽  
And Diffusion ◽  
Multi Mode

We present a multi-mode drift-diffusion equation as reformulation of the Boltzmann equation in the discrete momentum space. This is shown to be similar to the conventional drift-diffusion equation except that it is a more rigorous solution to the Boltzmann equation because the current and carrier densities are resolved into M×1 vectors, where M is the number of modes in the discrete momentum space. The mobility and diffusion coefficient become M×M matrices which connect the M momentum space modes. This approach is demonstrated by simulating electron transport in bulk silicon.

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