ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

10. Partial Differential Equations

1991 MS Classification: 35Q30, 47H17 Bogovskii, Mikhail E., Friendship of Nations University of Russia The problem of global smooth solutions for multidimensional Navier-Stokes equations solved Existence of global solutions of the initial boundary value problem for the non-linear Navier-Stokes equations is established for the number of space dimensions n ≥ 3 with velocities vector field in the anisotropic Sobolev space W_{p,x,t}^{2,1} in case (n + 2)/3 ≤ p < ∞. No restrictions are imposed either on the time interval or on the values of appropriate norms of prescribed data. In case n = 3 the existence theorem covers the functional class with finite Dirichlet integral, implying existence of classically smooth solutions should the prescribed data be sufficiently smooth. The indirect proof is based on a new method of solving non-linear problems.