论文标题
用于重新连接涡流线与II型超导体中边界的梯度轮廓
Gradient profile for the reconnection of vortex lines with the boundary in type-II superconductors
论文作者
论文摘要
在最近的一项工作中，Duong，Ghoul和Zaag确定了标准半线性热方程的爆炸解决方案的梯度曲线，并在（应该是）通用情况下具有功率非线性。他们的方法完善了Bricmont和Kupiainen引入的建设性技术，并由Merle和Zaag进一步开发。在本文中，我们将它们扩展到有关在平面近似下与边界重新连接与边界重新连接的问题，这是由Chapman，Hunton和Ockendon得出的物理模型，该模型在非线性热热方程式$ frac $ frac \ frac \ frac frac {\ partial h} { h} {\ partial x^2}+e^{ - h} - \ frac {1} {h^β}，\quadβ> 0 $$符合初始边界价值条件$$ H（\ cdot，0）= h_0> 0> h_0> 0，\ quad h（\ quad h（\ quad h（\ quad h（\ pm1，t）= 1。时间$ t $和灭绝点$ 0 $，梯度配置文件作为$ x \ rightarrow0 $喜欢$ \ lim_ {t \ rightarrow t} \，（\ nabla） h）（x，t）\ quad \ sim \ quad \ frac {1} {\ sqrt {2β}}} \ frac {x} {x} {| x |} \ frac {1} {\ sqrt {\ sqrt {左[\ frac {（β+1）^2} {8β} \ frac {| x |^2} {| \ log | x || x ||} \ right]^{\ frac {1} {β+1} - \ frac12}} - \ frac12}}，$$同意Merle和Zaag先前得出的灭绝曲线的梯度。我们的结果与一般的边界条件和更高的维度相处。
In a recent work, Duong, Ghoul and Zaag determined the gradient profile for blowup solutions of standard semilinear heat equation with power nonlinearities in the (supposed to be) generic case. Their method refines the constructive techniques introduced by Bricmont and Kupiainen and further developed by Merle and Zaag. In this paper, we extend their refinement to the problem about the reconnection of vortex lines with the boundary in a type-II superconductor under planar approximation, a physical model derived by Chapman, Hunton and Ockendon featuring the finite time quenching for the nonlinear heat equation $$\frac{\partial h}{\partial t}=\frac{\partial^2 h}{\partial x^2}+e^{-h}-\frac{1}{h^β},\quadβ>0$$ subject to initial boundary value conditions $$h(\cdot,0)=h_0>0,\quad h(\pm1,t)=1.$$ We derive the intermediate extinction profile with refined asymptotics, and with extinction time $T$ and extinction point $0$, the gradient profile behaves as $x\rightarrow0$ like $$\lim_{t\rightarrow T}\,(\nabla h)(x,t)\quad\sim\quad\frac{1}{\sqrt{2β}}\frac{x}{|x|}\frac{1}{\sqrt{|\log|x||}}\left[\frac{(β+1)^2}{8β}\frac{|x|^2}{|\log|x||}\right]^{\frac{1}{β+1}-\frac12},$$ agreeing with the gradient of the extinction profile previously derived by Merle and Zaag. Our result holds with general boundary conditions and in higher dimensions.
