Self-similar solutions to the hypoviscous Burgers and SQG equations at criticality

After reviewing the source-type solution of the Burgers equation with standard dissipativity, we study the hypoviscous counterpart of the Burgers equation. (1) We determine an equation that governs the near-identity transformation underlying its self-similar solution. (2) We develop its approximation scheme and construct the first-order approximation. (3) We obtain the source-type solution numerically by the Newton-Raphson iteration scheme and find it to agree well with the first-order approximation. Implications of the source-type solution are given, regarding the possibility of linearisation of the hypoviscous Burgers equation. Finally we address the problems of the incompressible fluid equations in two dimensions, centering on the surface quasi-geostrophic equation with standard and hypoviscous dissipativity.