scholarly journals Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data

1996 ◽  
Vol 176 (2) ◽  
pp. 475-478 ◽  
Author(s):  
G. Rein ◽  
A. D. Rendall
2021 ◽  
Author(s):  
Salah Badraoui

We prove in this work the existence of a unique global nonnegative classical solution to the class of reaction–diffusion systems uttx=aΔutx−guvm,vttx=dΔvtx+λtxguvm, where a>0, d>0, t>0,x∈Rn, n,m∈N∗, λ is a nonnegative bounded function with λt.∈BUCRn for all t∈R+, the initial data u0, v0∈BUCRn, g:BUCRn→BUCRn is a of class C1,dgudu∈L∞R, g0=0 and gu≥0 for all u≥0. The ideas of the proof is inspired from the work of Collet and Xin who proved the same result in the particular case d>a=1, λ=1,gu=u. Moreover, they showed that the L∞-norm of v can not grow faster than Olnlnt for any space dimension.


2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Yongqiang Xu

This paper is concerned with the fractional quasigeostrophic equation with modified dissipativity. We prove the local existence of solutions in Sobolev spaces for the general initial data and the global existence for the small initial data when1/2≤α<1.


1997 ◽  
Vol 07 (04) ◽  
pp. 457-476 ◽  
Author(s):  
T. Goudon

We are interested in the initial value problem for the Boltzmann equation, when the initial data u0 belongs to a set B0 = {δ0m1 (0,x,v) ≤ u0(x,v) ≤ C0m2 (0,x,v)} where m1, m2 are traveling Maxwellians. We consider soft or Maxwell's interactions with cutoff (7/3 < s ≤ 5) and C0 smaller than a bound depending on the coefficients of m2. We obtain global existence of solutions remaining in a "generalized invariant set" Bt ⊂ B∞, characterized by these particular states.


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