Non-divergence Parabolic Equations of Second Order with Critical Drift in Lebesgue Spaces

2015/11/04 by Gong Chen, Chen, Gong · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1511.01215

openalex publication_date 2015/11/04 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We consider uniformly parabolic equations and inequalities of second order in the non-divergence form with drift -u_t+Lu=-u_t+∑_ija_ijD_iju+∑ b_iD_iu=0 (≥0, ≤0) in some domain Q⊂ ℝⁿ⁺¹. We prove growth theorems and the interior Harnack inequality as the main results. In this paper, we will only assume the drift b is in certain Lebesgue spaces which are critical under the parabolic scaling but not necessarily to be bounded. In the last section, some applications of the interior Harnack inequality are presented. In particular, we show there is a "universal" spectral gap for the associated elliptic operator. The counterpart for uniformly elliptic equations of second order in non-divergence form is shown in \citeS10.

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