Pointwise estimates for solutions of fractal Burgers equation

In this paper, we provide two-sided estimates and uniform asymptotics for the solution of d-dimensional critical fractal Burgers equation ut−Δα/2u+b⋅∇(u|u|q)=0, α∈(1,2), b∈Rd for q=(α−1)/d and u0∈L1(Rd). We consider also q>(α−1)/d under additional condition u0∈L∞(Rd). In both cases we assume u0≥0, which implies that the solution is non-negative. The estimates are given in the terms of the function Ptu0, where Pt denotes the semigroup for the operator ∂t−Δα/2.