For a compact Riemannian manifold NRK without boundary, we establish the existence of strong solutions to the heat flow for harmonic maps from Rn to N, and the regularizing rate estimate of the strong solutions. Moreover, we obtain the analyticity in spatial variables of the solutions. The uniqueness of the mild solutions in C([0,T]; W1,n) is also considered in this paper.
Let T2 be a flat two-dimensional torus with fundamental cell domain [-21,21] × [-21,12],h(x) a positive smooth function satisfying the symmetric property(8) on T2. In this paper we give some suffcient condition under which the mean field equation u = 16π-16πheu,has a smooth solution.
In this paper, the Lp(Rn)-boundedness of the commutators generalized by BMO(Rn) function and the singular integral operator T with rough kernel Ω∈ Llog+ L(Sn-1) is proved by using the Bony's formula for the paraproduct of two functions.
