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

TL;DR

This paper investigates self-similar solutions to hypoviscous Burgers and SQG equations at criticality, developing approximation schemes, numerically constructing solutions, and exploring implications for linearization and fluid dynamics.

Contribution

It introduces a governing equation for self-similar solutions, constructs first-order approximations, and numerically solves the hypoviscous Burgers equation, extending analysis to SQG equations at criticality.

Findings

First-order approximation matches numerical solutions well.

Self-similar solutions suggest potential for linearization.

Implications for 2D incompressible fluid equations and SQG dynamics.

Abstract

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.