Generalized singular integral with rough kernel and approximation of surface quasi-geostrophic equation. (English) Zbl 1513.76039
Summary: This paper is concerned with the generalized singular integral operator with rough kernel and the approximation problem for the generalized surface quasi-geostrophic equation. For the generalized singular integral operator, we obtain uniform \(L^p - L^q\) estimates with respect to a parameter \(\beta \). From this one can cover the \(L^p\)-boundedness of the Calderón-Zygmund operator with rough kernel by letting \(\beta \to 0\). We applied this estimate to study the Cauchy problem of the generalized surface quasi-geostrophic (SQG) equation. Local well-posedness in the Besov space \(B_{p, q}^s\) and some limit behaviour of the solutions are obtained. Our results improve the previous ones by H. Yu et al. [Arch. Ration. Mech. Anal. 232, No. 1, 265–301 (2019; Zbl 1410.35148)] in 2019 and by H. Yu et al. [J. Funct. Anal. 280, No. 5, Article ID 108887, 23 p. (2021; Zbl 1454.42012)] in 2021.
MSC:
| 76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 35Q35 | PDEs in connection with fluid mechanics |
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