TY - JOUR
T1 - The Dissipative Quasi-geostrophic Equation in Spaces Admitting Singular Solutions
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 - <p>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.</p>