Harnack inequality for non-self-adjoint evolution equations
Harnack inequality for non-self-adjoint evolution equations
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DOI：
10.4310/mrl.1995.v2.n4.a2
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发表时间：
1995
影响因子：
1
通讯作者：
S. Yau
中科院分区：
文献类型：
--
作者：
S. Yau
多年前，S。Y. Cheng，P. Li和我[1]给出了拉普拉斯算子的热核估计。这一点后来由P.Li和我在[2]中加以改进。后来这篇论文的重点是一个抛物型Harnack不等式，它推广了我二十多年前提出的椭圆型Harnack不等式。由于应用数学中的许多发展方程不是自伴的，并且有一个对流项，我在这里建议推广李和我开发的抛物Harnack不等式，以覆盖非自伴的情况。Harnack不等式比热核的估计强得多。我希望它对控制理论和流体动力学的问题有用。这些申请将在稍后的场合进行研究。
Many years ago, S. Y. Cheng, P. Li and myself [1] gave an estimate of the heat kernel for the Laplacian. This was later improved by P. Li and myself in [2]. The key point of this later paper was a parabolic Harnack inequality that generalized an elliptic Harnack inequality that I developed more than twenty years ago. Since many evolution equations in applied mathematics are not selfadjoint and have a convection term, I propose here to generalize the parabolic Harnack inequality that Li and I developed to cover the non-selfadjoint case. The Harnack inequality is considerably stronger than the estimates of the heat kernel. I hope that it will be useful for questions in control theory and fluid dynamics. The applications will be studied on a later occasion.