Fractional Burgers equation with singular initial condition. (July 2023)

• Record Type: Journal Article
• Authors: Jakubowski, T.; Serafin, G.
• Is Part Of: Nonlinear analysis. Volume 232(2023)
• Publication Date: 2023-07
• Subjects: 35A01 -- 35B40 -- 35K55 -- 35S10; Fractional Burgers equation -- Singular initial condition -- Existence of solutions -- Estimates of solutions; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2023.113269
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider the fractional Burgers equation u t = Δ α / 2 u + b ⋅ ∇ ( u | u | ( α − 1 ) / β ) on R d, d ≥ 2, with α ∈ ( 1, 2 ) and β > 1 and prove the existence of a solution for a large class of initial conditions, which contains functions that do not belong to any L p ( R d ), 1 ≤ p ≤ ∞ . Next, we apply the general results to the initial condition u 0 ( x ) = M | x | − β, 1 < β < d, and show the existence of a selfsimilar solution and derive its properties such as smoothness, two-sided estimates, asymptotics and gradient estimates.