2017/04/07 by Zhongmin Qian, Qian, Zhongmin, Guangyu Xi +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1704.02173
openalex publication_date 2017/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study the fundamental solution \varGamma(t,x;τ,ξ) of the parabolic operator Lt=∂t-Δ+b(t,x)⋅∇, where for every t, b(t,⋅) is a divergence-free vector field, and we consider the case that b belongs to the Lebesgue space Ll(0,T;Lq(ℝn)). The regularity of weak solutions to the parabolic equation Ltu=0 depends critically on the value of the parabolic exponent γ=(2)/(l)+(n)/(q). Without the divergence-free condition on b, the regularity of weak solutions has been established when γ≤1, and the heat kernel estimate has been obtained as well, except for the case that l=∞,q=n. The regularity of weak solutions was deemed not true for the critical case L∞(0,T;Ln(ℝn)) for a general b, while it is true for the divergence-free case, and a written proof can be deduced from the results in [Semenov, 2006]. One of the results obtained in the present paper establishes the Aronson type estimate for critical and supercritical cases and for vector fields b which are divergence-free. We will prove the best possible lower and upper bounds for the fundamental solution one can derive under the current approach. The significance of the divergence-free condition enters the study of parabolic equations rather recently, mainly due to the discovery of the compensated compactness. The interest for the study of such parabolic equations comes from its connections with Leray's weak solutions of the Navier-Stokes equations and the Taylor diffusion associated with a vector field where the heat operator Lt appears naturally.