Fractional Burgers equation with singular initial condition

We consider the fractional Burgers equation ${\mathrm{\Delta }}^{\alpha \mathrm{/}2}u+b\cdot \mathrm{\nabla }(u\mathrm{\mid }u{\mathrm{\mid }}^{(\alpha -1)\mathrm{/}\beta })$ on ${\mathbf{R}}^d$, $d\ge 2$, with {$\alpha \in (1,2)$ and} $\beta >1$ and prove the existence of a solution for a large class of initial conditions, which contains functions that do not belong to any $L^p({\mathbf{R}}^d)$, $1\le p\le \mathrm{\infty }$. Next, we apply the general results to the initial condition $u_0(x)=M\mathrm{\mid }x{\mathrm{\mid }}^{-\beta }$, $1<\beta <d$, and show the existence of a selfsimilar solution and derive its properties such as smoothness, two-sided estimates, asymptotics and gradient estimates.