Global existence and optimal decay rates of solutions to conservation laws with diffusion-type terms

2016 ◽  
Vol 40 (8) ◽  
pp. 3040-3054
Author(s):  
Lijuan Wang
Keyword(s):  
Global Existence ◽  
Conservation Laws ◽  
Decay Rates ◽  
Diffusion Type ◽  
Optimal Decay Rates ◽  
Optimal Decay

scholarly journals OPTIMAL DECAY RATES TO CONSERVATION LAWS WITH DIFFUSION-TYPE TERMS OF REGULARITY-GAIN AND REGULARITY-LOSS

2012 ◽  
Vol 22 (07) ◽  
pp. 1250012 ◽  
Author(s):  
RENJUN DUAN ◽  
LIZHI RUAN ◽  
CHANGJIANG ZHU
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
Source Term ◽  
Decay Rates ◽  
Heat Diffusion ◽  
Time Behavior ◽  
Time Decay ◽  
Diffusion Type ◽  
Linear Solution ◽  
Optimal Decay Rates

We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term related to an index s ∈ ℝ over the whole space ℝn for any spatial dimension n ≥ 1. Here, the diffusion-type source term behaves as the usual diffusion term over the low frequency domain while it admits on the high frequency part a feature of regularity-gain and regularity-loss for s < 1 and s > 1, respectively. For all s ∈ ℝ, we not only obtain the Lp–Lq time-decay estimates on the linear solution semigroup but also establish the global existence and optimal time-decay rates of small-amplitude classical solutions to the nonlinear Cauchy problem. In the case of regularity-loss, the time-weighted energy method is introduced to overcome the weakly dissipative property of the equation. Moreover, the large-time behavior of solutions asymptotically tending to the heat diffusion waves is also studied. The current results have general applications to several concrete models arising from physics.

scholarly journals Global Existence and Temporal Decay of Large Solutions for the Poisson--Nernst--Planck Equations in Low Regularity Spaces

2021 ◽  
Author(s):  
Jihong Zhao ◽  
Xilan Liu
Keyword(s):  
Global Existence ◽  
Decay Rates ◽  
Large Solution ◽  
Low Regularity ◽  
Charged Species ◽  
Temporal Decay ◽  
Large Solutions ◽  
Optimal Decay Rates ◽  
Optimal Decay ◽  
Positively Charged

We are concerned with the global existence and decay rates of large solutions for the Poisson–Nernst–Planck equations. Based on careful observation of algebraic structure of the equations and using the weighted Chemin–Lerner type norm, we obtain the global existence and optimal decay rates of large solutions without requiring the summation of initial densities of a negatively and positively charged species is small enough. Moreover, the large solution is obtained for initial data belonging to the low regularity Besov spaces with different regularity and integral indices for the different charged species, which indicates more specific coupling relations between the negatively and positively charged species.

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