2021/12/07 by Jin Jack Tan, Tan, Jin, François Vigneron +1
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2112.03545
openalex publication_date 2021/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is concerned with the study of a nonlinear non-local equation that has a commutator structure. The equation reads ∂t u-F(u) (-Δ)s/2 u+(-Δ)s/2 (uF(u))=0, x∈ \mathbbTd, with s ∈ (0, 1]. We are interested in solutions stemming from periodic positive bounded initial data. The given function F ∈ C^∞ (R+) must satisfy F' > 0 a.e. on (0, +∞). For instance, all the functions F (u) = un with n ∈ N * are admissible non-linearities. We construct global classical solutions starting from smooth positive data, and global weak solutions starting from positive data in L^∞. We show that any weak solution is instantaneously regularized into C^∞. We also describe the long-time asymptotics of all solutions. Our methods follow several recent advances in the regularity theory of parabolic integro-differential equations.