scholarly journals Generic global classical solutions of the Vlasov-Fokker-Planck-Poisson system in three dimensions

1992 ◽  
Vol 99 (1) ◽  
pp. 59-77 ◽  
Author(s):  
Gerhard Rein ◽  
Jürgen Weckler
Keyword(s):  
Three Dimensions ◽  
Classical Solutions ◽  
Poisson System ◽  
Global Classical Solutions ◽  
Fokker Planck

scholarly journals On the Existence of Global Smooth Solutions to the Parabolic–Elliptic Keller–Segel System with Irregular Initial Data

2021 ◽  
Author(s):  
Frederic Heihoff
Keyword(s):  
Initial Data ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Three Dimensions ◽  
Classical Solutions ◽  
List Type ◽  
Smooth Solutions ◽  
Smooth Bounded Domain ◽  
Global Classical Solutions ◽  
Global Smooth Solutions

AbstractWe consider the parabolic–elliptic Keller–Segel system $$\begin{aligned} \left\{ \begin{aligned} u_t&= \Delta u - \chi \nabla \cdot (u \nabla v), \\ 0&= \Delta v - v + u \end{aligned} \right. \end{aligned}$$ u t = Δ u - χ ∇ · ( u ∇ v ) , 0 = Δ v - v + u in a smooth bounded domain $$\Omega \subseteq {\mathbb {R}}^n$$ Ω ⊆ R n , $$n\in {\mathbb {N}}$$ n ∈ N , with Neumann boundary conditions. We look at both chemotactic attraction ($$\chi > 0$$ χ > 0 ) and repulsion ($$\chi < 0$$ χ < 0 ) scenarios in two and three dimensions. The key feature of interest for the purposes of this paper is under which conditions said system still admits global classical solutions due to the smoothing properties of the Laplacian even if the initial data is very irregular. Regarding this, we show for initial data $$\mu \in {\mathcal {M}}_+({\overline{\Omega }})$$ μ ∈ M + ( Ω ¯ ) that, if either $$n = 2$$ n = 2 , $$\chi < 0$$ χ < 0 or $$n = 2$$ n = 2 , $$\chi > 0$$ χ > 0 and the initial mass is small or $$n = 3$$ n = 3 , $$\chi < 0$$ χ < 0 and $$\mu = f \in L^p(\Omega )$$ μ = f ∈ L p ( Ω ) , $$p > 1$$ p > 1 holds, it is still possible to construct global classical solutions to ($$\star $$ ⋆ ), which are continuous in $$t = 0$$ t = 0 in the vague topology on $${\mathcal {M}}_+({\overline{\Omega }})$$ M + ( Ω ¯ ) .

scholarly journals Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck-Euler system

2011 ◽  
Vol 4 (1) ◽  
pp. 227-258 ◽  
Author(s):  
José A. Carrillo ◽  
◽  
Renjun Duan ◽  
Ayman Moussa ◽  
◽  
...  
Keyword(s):  
Euler System ◽  
Classical Solutions ◽  
Global Classical Solutions ◽  
Fokker Planck

Global Classical Solutions for the Vlasov--Nordström--Fokker--Planck System

2021 ◽  
Vol 53 (5) ◽  
pp. 6164-6190
Author(s):  
Renjun Duan ◽  
Shuangqian Liu ◽  
Tong Yang
Keyword(s):  
Classical Solutions ◽  
Global Classical Solutions ◽  
Fokker Planck
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