Removable singularities and unbounded asymptotic profiles of multi-dimensional Burgers equations

This paper is concerned with the unbounded radially symmetric profiles and their asymptotic stability of the multi-dimensional Burgers equations on the whole space \(\mathbb R^n\) ${\mathbb{R}}^n$ . The possible singularities near the origin of the unbounded stationary solutions are investigated and we show that both the convection \(({\varvec{u}}\cdot \nabla ){\varvec{u}}\) $(\mathit{u}·\mathrm{\nabla })\mathit{u}$ and the diffusion \(\mu \Delta {\varvec{u}}\) $\mu \mathrm{\Delta }\mathit{u}$ have singularities in the sense of distributions for three dimensional Burgers equations, such that their singularities cancel with each other in the Burgers equations. We present the decay estimates of perturbations for unbounded solutions around the singular profile for \(n=3\) $n=3$ and the asymptotic convergence to the unbounded profile for \(n\ge 4\) $n\ge 4$ , both with large perturbations. We also find a better asymptotic profile for three dimensional case, a superposition of the singular stationary solution and a self-similar diffusion wave, such that the perturbation decays faster.