2019/03/24 by Zimo Hao, Hao, Zimo, Mingyan Wu +3
Mathematics · #35K65 #60H10 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Numerical methods in inverse problems #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1903.09967
openalex publication_date 2019/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we develop a new method based on Littlewood-Paley's decomposition and heat kernel estimates of integral form, to establish Schauder's estimate for the following degenerate nonlocal equation in \mathbb R2d with Hölder coefficients: ∂tu=\mathscr L(α)_κ;\rm v u+b⋅∇ u+f, u0=0, where u=u(t,x,\rm v) and \mathscr L(α)_κ;\rm v is a nonlocal α-stable-like operator with α∈(1,2) and kernel function κ, which acts on the variable \rm v. As an application, we show the strong well-posedness to the following degenerate stochastic differential equation with Hölder drift b: \rm dZt=b(t,Zt)\rm dt+(0,σ(t,Zt)\rm dL(α)t), Z0=(x,\rm v)∈\mathbb R2d, where L(α)t is a d-dimensional rotationally invariant and symmetric α-stable process with α∈(1,2), and b:\mathbb R+×\mathbb R2d→\mathbb R2d is a (γ,β)-Hölder continuous function in (x,\rm v) with γ∈((2+α)/(2(1+α)),1) and β∈(1-\fracα2,1), σ:\mathbb R+×\mathbb R2d→\mathbb Rd⊗\mathbb Rd is a Lipschitz function. Moreover, we also show that for almost all ω, the following random transport equation has a unique C1b-solution: ∂tu(t,x,ω)+(b(t,x)+L(α)t(ω))⋅∇x u(t,x,ω)=0, u(0,x)=φ(x), where φ∈ C1b(\mathbb Rd) and b:\mathbb R+×\mathbb Rd→\mathbb Rd is a bounded continuous function in (t,x) and γ-order Hölder continuous in x uniformly in t with γ∈((2+α)/(2(1+α)),1).