Large self-similar solutions of a convection--diffusion equation

Abstract

We construct self-similar solutions to a nonlinear convection--diffusion equation posed on the whole Euclidean space, for a class of homogeneous initial data. The corresponding self-similar profiles solve a nonlinear elliptic equation on ${\mathbb{R}}^N$. We obtain these profiles as limits of solutions to Dirichlet problems on bounded domains with truncated nonlinearities, using a priori estimates that are uniform with respect to both the domain size and the truncation level. Our approach is inspired by recent constructions of large self-similar solutions for the Navier--Stokes system. The convection--diffusion equation considered here shares some structural features with the Navier--Stokes equations, but its scalar parabolic nature allows us to exploit tools such as the maximum principle and thereby obtain a more direct construction of self-similar solutions.