Asymptotic behavior for the vorticity equations in dimensions two and three
Asymptotic behavior for the vorticity equations in dimensions two and three
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DOI：
10.1080/03605309408821037
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发表时间：
1994
影响因子：
1.9
通讯作者：
A. Carpio
中科院分区：
文献类型：
--
作者：
A. Carpio
对于某些类型的初值，我们建立了二维和三维涡度方程解的自相似行为。更确切地说，如果|M|足够小，则以有限Radon测度V0为初值的二维涡度方程的任何解v都与质量为M=∫R2 V0的热方程的基本解渐近等价：t1−1 p‖v(T)−MG(T)‖Lp(R2)→0当t对所有1∞p≤趋于≤∞时。这推广了Giga和Kambe的结果，其中假设V0的总变化很小。设v是Morrey空间(M32(R))3中初值为V0的三维涡度方程的零散度小范数解，使得λV0(λx)→μ在测度意义下为λ→∞和‖V0‖M32(|x|>；R)→0 When R→∞Then Lim t→∞T1−3 2p‖v(T)−ν(T)‖p=0，对所有32个≤p≤∞，其中ν是具有初始数据μ的涡度方程的唯一解，已被Giga和Miyakawa证明是自相似的。AMS分类：35B40、35K05、35K55、76Rxx、35Q35、76D05。
We establish the selfsimilar behavior of the solutions of the two and three dimensional vorticity equations for some classes of initial data. More precisely, any solution v of the two dimensional vorticity equation taking as initial data a finite Radon measure v0 is shown to be asymptotically equivalent to the fundamental solution of the heat equation with mass M = ∫ R2 v0 provided that |M | is small enough, in the following sense: t1− 1 p ‖v(t)−MG(t)‖Lp(R2) → 0 when t tends to ∞ for all 1 ≤ p ≤ ∞. This extends a result due to Giga and Kambe where the total variation of v0 was assumed to be small. If v is a solution of the three dimensional vorticity equation with initial data v0 in the Morrey space (M 3 2 (R))3 with zero divergence and small norm, such that λv0(λx) → μ in the sense of measures as λ →∞ and ‖v0‖ M 3 2 (|x|>R) → 0 when R →∞ then lim t→∞ t1− 3 2p ‖v(t)− ν(t)‖p = 0 for all 32 ≤ p ≤ ∞, where ν is the unique solution of the vorticity equation with initial data μ, which has been proved to be selfsimilar by Giga and Miyakawa. AMS Classification: 35B40, 35K05, 35K55, 76Rxx, 35Q35, 76D05.