Parabolic equations with rough coefficients and singular forcing.
Parabolic equations with rough coefficients and singular forcing.
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具有粗略系数和奇异强迫的抛物线方程。
DOI：
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发表时间：
2018
期刊：
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通讯作者：
H. Weber
中科院分区：
文献类型：
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作者：
F. Otto;J. Sauer;Scott A. Smith;H. Weber
本文主要研究一类具有粗糙扩散系数的抛物型方程，由于奇异强迫的存在，该方程在经典分布意义下是不适定的。受哲学的启发，粗糙的路径和规律性结构，我们引入了一个概念的模型化分布，这是适合在这种情况下。我们证明了两个用于重建和积分的通用工具，以及为粗糙扩散算子的重建量身定制的积引理。这产生了一个部分自动的确定性理论，我们应用它获得了一个存在性和唯一性理论的抛物方程的粗糙扩散系数和奇异强迫的负抛物H”{o}lder空间的订单大于$-frac{3}{2}$。
This article focuses on parabolic equations with rough diffusion coefficients which are ill-posed in the classical sense of distributions due to the presence of a singular forcing. Inspired by the philosophy of rough paths and regularity structures, we introduce a notion of modelled distribution which is suitable in this context. We prove two general tools for reconstruction and integration, as well as a product lemma which is tailor made for the reconstruction of the rough diffusion operator. This yields a partially automated deterministic theory, which we apply to obtain an existence and uniqueness theory for parabolic equations with rough diffusion coefficients and a singular forcing in the negative parabolic H"{o}lder space of order larger than $-frac{3}{2}$.