A note on blowup criterion of the 3d magnetic Bxc3xa9nard equations Abstract   This note establishes a Serrintype blowup criterion for local smooth solution to the 3D magnetic Benard equations in term of the corresponding pressure. More precisely, it is proved that for     T  u003e  0    , if     P  xe2x88x88    L      2    2  xe2x88x92  r       (  0  ,  T  ;    L      3    r       (    R    3    )   )      with     0    r  xe2x89xa4  1    , then the corresponding solution     (  u  ,  b  ,  xcexb8  )     to the 3D magnetic Benard equations can be extended beyond time     t    T    .

Existence of solution for nonlinear integral equations of Hxc3xa9non type in critical case Abstract   In this paper, we will show that there exists a       xcexb4    xe2x88x97    u003e  0     such that when     xcexb1  u003e  0  ,  xcexb2  u003e  0     and     xcexb1    xcexb2      xcexb4    xe2x88x97      , the following problem    (0.1)       f      q    xcexb3    xe2x88x92  1     (  x  )       xe2x88xab    xcexa9           x       xcexb1         y       xcexb2    f   (  y  )          x  xe2x88x92  y       n  xe2x88x92  xcexb3        d  y    xcexbb    xe2x88xab    xcexa9      f   (  y  )          x  xe2x88x92  y       n  xe2x88x92  xcexb3  xe2x88x92  1        d  y  ,    x  xe2x88x88    xcexa9   xc2xaf       has at least one positive solution     f  xe2x88x88  C   (    xcexa9        )     , where    xcexa9    is the unit ball in       R    n     (  n  xe2x89xa5  3  )      centered at the origin,     1    xcexb3    n    ,     xcexbb  u003e  0     and       q    xcexb3    xe2x89x94    2  n    n    xcexb3      .