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On wellposedness and limiting behavior of generalized SQG equations

  • Anuj Kumar
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Abstract

We consider the two-dimensional generalized surface quasi-geostrophic equations in ℝ2, given by ∂t θ+u· ∇ θ=0, u=−∇⊥Λβ−2θ, β∈ [1,2). When β=1, the equation defines the SQG equation and for β>1, it defines a family of more singular active scalar equations. We prove that if the interval of existence of the smooth solution to the generalized SQG equations for some β0∈[1,2) is [0,T], then with the same initial data, the interval of existence of the generalized SQG equations for β close to β0 also contains [0,T]. To prove these results, we develop estimates for a conservation law with flux modified around the generalized SQG equations. Furthermore, we also prove some new commutator estimates with bounds uniformly bounded as β→ 1 that may be of independent interest.

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Cite

BibTeXRIS
BibTeX
@misc{kumar2026wellposedness,
  title = {{On wellposedness and limiting behavior of generalized SQG equations}},
  author = {Anuj Kumar},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39580},
  eprint = {2609.39580},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39580},
}
RIS
TY  - UNPB
TI  - On wellposedness and limiting behavior of generalized SQG equations
AU  - Anuj Kumar
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39580
UR  - https://arxiv.org/abs/2609.39580
AB  - We consider the two-dimensional generalized surface quasi-geostrophic equations in ℝ^2, given by ∂_t θ+u· ∇ θ=0, u=−∇^⊥Λ^β−2θ, β∈ [1,2). When β=1, the equation defines the SQG equation and for β>1, it defines a family of more singular active scalar equations. We prove that if the interval of existence of the smooth solution to the generalized SQG equations for some β_0∈[1,2) is [0,T], then with the same initial data, the interval of existence of the generalized SQG equations for β close to β_0 also contains [0,T]. To prove these results, we develop estimates for a conservation law with flux modified around the generalized SQG equations. Furthermore, we also prove some new commutator estimates with bounds uniformly bounded as β→ 1 that may be of independent interest.
ER  -