We show the existence of dissipative Hhlder continuous solutions of the Boussi- nesq equations. More precise, for anyβ∈ （0, 1/5）, a time interval [0, T] and any given smooth energy profile e ： [0, T] → （0, ∞...We show the existence of dissipative Hhlder continuous solutions of the Boussi- nesq equations. More precise, for anyβ∈ （0, 1/5）, a time interval [0, T] and any given smooth energy profile e ： [0, T] → （0, ∞）, there exist a weak solution （v, θ） of the 3d Boussinesq equations such that （v, 8） ∈ Cβ（T3 × [0, T]） with e（t） = ∫T3 ｜v（x, t）｜2dx for all t ∈ [0, T]. This extend the result of [2] about Onsager＇s conjecture into Boussinesq equation and improve our previous result in [30].展开更多
We show the existence of Holder continuous periodic weak solutions of the 2D Boussinesq equation with thermal diffusion which satisfy the prescribed kinetic energy.More precisely,for any smooth e(t):[0,1]→R+andε∈(0...We show the existence of Holder continuous periodic weak solutions of the 2D Boussinesq equation with thermal diffusion which satisfy the prescribed kinetic energy.More precisely,for any smooth e(t):[0,1]→R+andε∈(0,110),there exist v∈C 110−ε([0,1]×T2)andθ∈C 1,120−εt 2 C 2,1 x 10−ε([0,1]×T2),which satisfy(1.1)in the sense of distribution and e(t)=ˆT2|v(t,x)|2 dx,∀t∈[0,1].展开更多
基金partially supported by the NSFC(11471320 and 11631008)
文摘We show the existence of dissipative Hhlder continuous solutions of the Boussi- nesq equations. More precise, for anyβ∈ （0, 1/5）, a time interval [0, T] and any given smooth energy profile e ： [0, T] → （0, ∞）, there exist a weak solution （v, θ） of the 3d Boussinesq equations such that （v, 8） ∈ Cβ（T3 × [0, T]） with e（t） = ∫T3 ｜v（x, t）｜2dx for all t ∈ [0, T]. This extend the result of [2] about Onsager＇s conjecture into Boussinesq equation and improve our previous result in [30].
基金supported by National Natural Science Foundation of China(Grant No.11971464)supported by National Natural Science Foundation of China(Grant No.11901349)supported by National Natural Science Foundation of China(Grant Nos.11471320 and 11631008)。
文摘We show the existence of Holder continuous periodic weak solutions of the 2D Boussinesq equation with thermal diffusion which satisfy the prescribed kinetic energy.More precisely,for any smooth e(t):[0,1]→R+andε∈(0,110),there exist v∈C 110−ε([0,1]×T2)andθ∈C 1,120−εt 2 C 2,1 x 10−ε([0,1]×T2),which satisfy(1.1)in the sense of distribution and e(t)=ˆT2|v(t,x)|2 dx,∀t∈[0,1].
