PreprintCC BY 4.0arXiv
Self-similar solutions for the gSQG patch equations
Abstract
The inviscid generalized surface quasi-geostrophic (gSQG) equations are a family of active scalar equations parametrized by a scalar α that includes the two-dimensional Euler equations (α= 0) and the SQG equation (α= 1). We prove that, for every α∈ (7/4, 2) and every T > 0, there exists a self-similar patch solution whose boundary is a smooth, unbounded graph for 0 ≤ t < T and develops a corner singularity at time T.
Related
- Self-similar Imploding Solutions of the Relativistic Euler Equations
- Self-similar imploding solutions of the 1D compressible Euler equations with a far field cutoff
- Self-similar solutions to the steady Navier-Stokes equations in arbitrarily high dimensions
- Self-Similar Solutions of the 2D Incompressible Euler Equations via a Lagrangian Construction
- Homothetic Self-Similar Solutions to the Incompressible Navier-Stokes Equations
BibTeX
@misc{gomezserrano2026selfsimilar,
title = {{Self-similar solutions for the gSQG patch equations}},
author = {Javier Gomez-Serrano and Marcus Pasquariello},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32857},
eprint = {2610.32857},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32857},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Self-similar solutions for the gSQG patch equations
AU - Javier Gomez-Serrano
AU - Marcus Pasquariello
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32857
UR - https://arxiv.org/abs/2610.32857
AB - The inviscid generalized surface quasi-geostrophic (gSQG) equations are a family of active scalar equations parametrized by a scalar α that includes the two-dimensional Euler equations (α= 0) and the SQG equation (α= 1). We prove that, for every α∈ (7/4, 2) and every T > 0, there exists a self-similar patch solution whose boundary is a smooth, unbounded graph for 0 ≤ t < T and develops a corner singularity at time T.
ER -