Fractional Burgers equation with singular initial condition. (July 2023) - Jakubowski, T.; Serafin, G.
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.