Global regularity results for the fractional heat equation and application to a class of non-linear KPZ problems

2025/06/07 by Boumediene Abdellaoui, Abdellaoui, Boumediene, Somia Atmani +5
Mathematics · #35B05 #35B40 #35K15 #35K55 #35K65 #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2506.06875

openalex publication_date 2025/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the first part of this paper, we prove the global regularity, in an adequate parabolic Bessel-Potential space and then in the corresponding parabolic fractional Sobolev space, of the unique solution to following fractional heat equation w_t+(-Δ)^sw= h ; w(x,t)=0 in (ℝ^N∖Ω)×(0,T) ; w(x,0)=w₀(x) in Ω, where Ω is an open bounded subset of ℝ^N. The proof is based on a new pointwise estimate on the fractional gradient of the corresponding kernel. Moreover, we establish the compactness of (w₀,h)↦ w. As a majeur application, in the second part , we establish existence and regularity of solutions to a class of Kardar--Parisi--Zhang equations with fractional diffusion and a nonlocal gradient term. Additionally, several auxiliary results of independent interest are obtained.

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Related

• Regularity of solutions for fully fractional parabolic equations
• On the KPZ equation with fractional diffusion: global regularity and existence results
• Regularity in L_p Sobolev spaces of solutions to fractional heat equations
• Local regularity for fractional heat equations
• Regularity theory and extension problem for fractional nonlocal parabolic equations and the master equation