Weighted a priori estimates for the swirl component of the vorticity of the axisymmetric Navierxe2x80x93Stokes system Abstract   We study the axisymmetric Navierxe2x80x93Stokes equations. By invoking a twodimensional Gagliardoxe2x80x93Nirenberg inequality, we could show         r      7    2    xe2x88x92    4    p        xcfx89    xcexb8    xe2x88x88    L    xe2x88x9e     (  0  ,  T  ;    L    p     (    R    3    )   )   ,      for any       3    2    xe2x89xa4  p  xe2x89xa4    8    5      .

Axially symmetric solutions of the Schrxc3xb6dingerxe2x80x93Poisson system with zero mass potential in R2 Abstract   In this paper, we consider the following planar Schrodingerxe2x80x93Poisson system           xe2x88x92  xcex94  u    xcfx95  u    K   (  x  )   f   (  u  )   ,          x  xe2x88x88    R    2    ,      xcex94  xcfx95      u    2    ,          x  xe2x88x88    R    2    ,          where     K  xe2x88x88  C   (    R    2    ,   (  0  ,  xe2x88x9e  )   )      and     f  xe2x88x88  C   (  R  ,  R  )     . By constructing a new energy estimate inequality and applying some new tricks, we prove that the above system has an axially symmetric solution which has a special minimax characterisation. Our result extends previous results in the literature.