查看“︁热核”︁的源代码

{{not|热核聚变}}

[[File:Fundamental solution to the heat equation.gif|right|thumb|upright=2|一维热方程的基本解（红线）]]
'''热核'''（{{lang-en|heat kernel}}）在数学中是指[[热方程]]的基本解。其也是[[拉普拉斯算子]]谱分析中的重要工具之一。对于固定边界的区域，当边界温度给定、并于''t'' = 0时在其中某一点放置一单位热能时，热核表示此后区域内温度的变化过程。

最常见的热核为''d''维[[欧几里得空间]]'''R'''<sup>''d''</sup>上的热核。该热核为随时间变化的[[高斯函数]]，其表达式为
:<math>K(t,x,y) = \frac{1}{(4\pi t)^{d/2}} e^{-|x-y|^2/4t}\,</math>

该热核是热方程
:<math>\frac{\partial K}{\partial t}(t,x,y) = \Delta_x K(t,x,y)\,</math>

的解，其中''t''&nbsp;>&nbsp;0，''x'',''y''&nbsp;∈&nbsp;'''R'''<sup>''d''</sup>，Δ则表示拉普拉斯算子。方程的初始条件为
:<math>\lim_{t \to 0} K(t,x,y) = \delta(x-y)=\delta_x(y)</math>

其中δ表示[[狄拉克δ函数]]。对任一[[紧支撑]]的光滑函数φ，有
:<math>\lim_{t \to 0}\int_{\mathbf{R}^d} K(t,x,y)\phi(y)\,dy = \phi(x).</math>

对于'''R'''<sup>''d''</sup>上的一般区域，热核并没有显式的表达式。当区域为圆盘或方形时，热核则分别为[[贝塞尔函数]]与[[Θ函數|雅可比Θ函数]]。可以证明，对任意[[黎曼流形]]，当边界条件充分正则时，热核存在且在''t''>0时光滑。

== 参考文献 ==
* {{Citation | last1=Berline | first1=Nicole | last2=Getzler | first2=E. | last3=Vergne | first3=Michèle | title=Heat Kernels and Dirac Operators | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=2004}}
* {{Citation | last1=Chavel | first1=Isaac | title=Eigenvalues in Riemannian geometry | publisher=[[Academic Press]] | location=Boston, MA | series=Pure and Applied Mathematics | isbn=978-0-12-170640-1 | mr=768584 | year=1984 | volume=115}}.
* {{Citation | last1=Evans | first1=Lawrence C. | title=Partial differential equations | publisher=[[American Mathematical Society]] | location=Providence, R.I. | isbn=978-0-8218-0772-9 | year=1998}}
* {{Citation | last1=Gilkey | first1=Peter B. | title=Invariance Theory, the Heat Equation, and the Atiyah–Singer Theorem | url=http://www.emis.de/monographs/gilkey/ | isbn=978-0-8493-7874-4 | year=1994 | accessdate=2018-04-05 | archive-date=2019-12-29 | archive-url=https://web.archive.org/web/20191229083135/http://www.emis.de/monographs/gilkey/ | dead-url=no }}
*{{Citation | last1=Grigor'yan | first1=Alexander | title=Heat kernel and analysis on manifolds | url=https://books.google.com/books?id=X7QQcVa2EWsC | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=AMS/IP Studies in Advanced Mathematics | isbn=978-0-8218-4935-4 | mr=2569498 | year=2009 | volume=47 | accessdate=2018-04-05 | archive-date=2019-06-09 | archive-url=https://web.archive.org/web/20190609050423/https://books.google.com/books?id=X7QQcVa2EWsC | dead-url=no }}
*{{citation | first1=Motoko | last1=Kotani | first2=Toshikazu |last2=[[Toshikazu Sunada|Sunada]]|  title=Albanese maps and an off diagonal long time asymptotic for the heat kernel | journal=Comm. Math. Phys. | volume=209 | year=2000 | pages=633–670 | doi=10.1007/s002200050033|bibcode = 2000CMaPh.209..633K }}

[[Category:热传导]]
[[Category:拋物型偏微分方程]]
[[Category:谱理论]]