TY - JOUR T1 - The Dissipative Quasi-geostrophic Equation in Spaces Admitting Singular Solutions AU - Baoquan Yuan & Jia Yuan JO - Journal of Partial Differential Equations VL - 3 SP - 203 EP - 219 PY - 2007 DA - 2007/08 SN - 20 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5303.html KW - Dissipative quasi-geostrophic equation KW - singular solutions KW - pseudomeasure spaces KW - Lorentz space KW - global well-posedness AB -
This paper studies the Cauchy problem of the dissipative quasi-geostrophic equation in pseudomeasure space PM^{n+1-2α}(\mathbb{R}^n) or Lorentz space L\frac{n}{2α-1, ∞}(\mathbb{R}^n), which admit the singular solutions. The global well-posedness is established provided initial data θ_0(x) are small enough in these spaces. Moreover, the asymptotic stability of solutions in pseudomeasure space is proved. In particular, if the initial data are homo-geneous functions of degree 1 - 2α, the self-similar solutions are also obtained.