Parabolic equations with singular divergence‐free drift vector fields
Parabolic equations with singular divergence‐free drift vector fields
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具有奇异无散度漂移向量场的抛物线方程
DOI：
10.1112/jlms.12202
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
Guangyu Xi
中科院分区：
文献类型：
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作者：
Z. Qian;Guangyu Xi
本文研究了一个散度型但不一定对称的椭圆算子。特别地，我们的结果可以应用于椭圆算子L=νΔ+u（x，t）·μ，其中u（·，t）是Rn中的依赖于时间的向量场，它在分布意义下是无发散的，即μ·u=0。设u∈L∞（0，∞; BMO −1（Rn））。我们证明了抛物型算子L− <$t的基本解Γ（x，t; τ，τ）的存在性，并证明了Γ满足Aronson估计，其常数仅依赖于维数n、椭圆常数λ和范数<$u <$Lt∞（BMO x−1）.因此，在L2空间中，对于初始数据，建立了抛物方程（L− ω t）v=0解的存在唯一性，并得到了其正则性理论。事实上，我们建立了一个一般的非对称椭圆算子的发散形式的这些结果。
In this paper, we study an elliptic operator in divergence form but not necessarily symmetric. In particular, our results can be applied to elliptic operator L=νΔ+u(x,t)·∇ , where u(·,t) is a time‐dependent vector field in Rn , which is divergence‐free in the distributional sense, that is ∇·u=0 . Suppose u∈L∞(0,∞; BMO −1(Rn)) . We show the existence of the fundamental solution Γ(x,t;ξ,τ) of the parabolic operator L−∂t and show that Γ satisfies the Aronson estimate with a constant depending only on the dimension n , the elliptic constant λ and the norm ∥u∥Lt∞( BMO x−1) . Therefore, the existence and uniqueness of solutions to the parabolic equation (L−∂t)v=0 are established for initial data in L2 ‐space, and their regularity theory is obtained too. In fact, we establish these results for a general non‐symmetric elliptic operator in divergence form.