scholarly journals Ergodicity effects on transport-diffusion equations with localized damping

2021 ◽  
Vol 18 (1) ◽  
pp. 1-10
Author(s):  
Kaïs Ammari ◽  
Taoufik Hmidi
Keyword(s):  
Diffusion Equations ◽  
Transport Diffusion ◽  
Localized Damping

UNIFORM ESTIMATES FOR TRANSPORT-DIFFUSION EQUATIONS

2007 ◽  
Vol 04 (01) ◽  
pp. 1-17 ◽  
Author(s):  
R. DANCHIN
Keyword(s):  
Reynolds Number ◽  
Vector Field ◽  
Fluid Mechanics ◽  
Incompressible Fluid ◽  
High Reynolds Number ◽  
Stokes Equations ◽  
Diffusion Equations ◽  
Uniform Estimates ◽  
Transport Diffusion ◽  
High Reynolds

We prove new estimates in Besov spaces [Formula: see text] or [Formula: see text] for transport-diffusion equations. Comparing with previous works, there is no restriction on the index p (hence our estimates apply in Hölder spaces) and the transport vector-field need not be divergence-free. Besides, similar estimates can be proved for the non-stationary Stokes equations with convection. We expect our results to be useful for studying compressible or incompressible fluid mechanics with high Reynolds number.

scholarly journals ASTROPARTICLE THEORY: SOME NEW INSIGHTS INTO HIGH ENERGY COSMIC RAYS

2004 ◽  
Vol 19 (07) ◽  
pp. 1133-1141 ◽  
Author(s):  
E. ROULET
Keyword(s):  
Cosmic Rays ◽  
High Energy ◽  
Diffusion Equations ◽  
New Developments ◽  
Diffusive Behavior ◽  
High Energy Cosmic Rays ◽  
Transport Diffusion ◽  
Neutrino Fluxes ◽  
Observed Behavior

Some new developments obtained in the last few years concerning the propagation of high energy cosmic rays are discussed. In particular, it is shown how the inclusion of drift effects in the transport diffusion equations leads naturally to an explanation for the knee, for the second knee and for the observed behavior of the composition and anisotropies between the knee and the ankle. It is shown that the trend towards a heavier composition above the knee has significant impact on the predicted neutrino fluxes above 1014 eV. The effects of magnetic lensing on the cosmic rays with energies above the ankle are also discussed, analyzing the main features of the different regimes that appear between the diffusive behavior that takes place at lower energies and the regime of small deflections present at the highest ones.

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