1. Non-uniqueness of forced active scalar equations with even drift operators
- Author
-
Dai, Mimi and Friedlander, Susan
- Subjects
Mathematics - Analysis of PDEs ,35Q35, 35Q86, 76D03 - Abstract
We consider forced active scalar equations with even and homogeneous degree 0 drift operator on $\mathbb T^d$. Inspired by the non-uniqueness construction for dyadic fluid models, by implementing a sum-difference convex integration scheme we obtain non-unique weak solutions for the active scalar equation in space $C_t^0C_x^\alpha$ with $\alpha<\frac{1}{2d+1}$. We note that in 1D, the regularity $\alpha<\frac13$ is sharp as the energy identity is satisfied for solutions in $C^\alpha$ with $\alpha>\frac13$. Without external forcing, Isett and Vicol constructed non-unique weak solutions for such active scalar equations with spatial regularity $C_x^\alpha$ for $\alpha<\frac{1}{4d+1}$., Comment: 30 pages
- Published
- 2023