Nontrivial integrable weak stationary solutions to active scalar equations with non-odd drift
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917214682087424 |
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| author | Gismondi, Nicholas |
| author_facet | Gismondi, Nicholas |
| contents | In this paper we construct nontrivial weak solutions to a class of stationary active scalar equations with a non-odd nonlocal operator in the drift term using a convex integration scheme. We show our solutions lie in $$ \bigcap_{0 < ε< 1} \dot{B}^{-ε}_{\infty,\infty}(\mathbb{T}^d) \cap L^{2-ε}(\mathbb{T}^d) $$ for $d \geq 2$. The key ingredient of the construction is the use of highly oscillatory corrections with a variable degree of intermittency, which is arranged to decrease to zero at higher stages of the iteration procedure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_14592 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nontrivial integrable weak stationary solutions to active scalar equations with non-odd drift Gismondi, Nicholas Analysis of PDEs In this paper we construct nontrivial weak solutions to a class of stationary active scalar equations with a non-odd nonlocal operator in the drift term using a convex integration scheme. We show our solutions lie in $$ \bigcap_{0 < ε< 1} \dot{B}^{-ε}_{\infty,\infty}(\mathbb{T}^d) \cap L^{2-ε}(\mathbb{T}^d) $$ for $d \geq 2$. The key ingredient of the construction is the use of highly oscillatory corrections with a variable degree of intermittency, which is arranged to decrease to zero at higher stages of the iteration procedure. |
| title | Nontrivial integrable weak stationary solutions to active scalar equations with non-odd drift |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.14592 |