Strichartz estimates for parabolic equations with higher order differential operators

The present paper first obtains Strichartz estimates for parabolic equations with nonnegative elliptic operators of order 2m by using both the abstract Strichartz estimates of Keel-Tao and the Hardy-Littlewood-Sobolev inequality. Some conclusions can be viewed as the improvements of the previously known ones. Furthermore, an endpoint homogeneous Strichartz estimates on BMO (x) (a"e (n) ) and a parabolic homogeneous Strichartz estimate are proved. Meanwhile, the Strichartz estimates to the Sobolev spaces and Besov spaces are generalized. Secondly, the local well-posedness and small global well-posedness of the Cauchy problem for the semilinear parabolic equations with elliptic operators of order 2m, which has a potential V (t, x) satisfying appropriate integrable conditions, are established. Finally, the local and global existence and uniqueness of regular solutions in spatial variables for the higher order elliptic Navier-Stokes system with initial data in L (r) (a"e (n) ) is proved.