By Solonnikov
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Extra resources for Estimates for solutions of nonstationary Navier-Stokes equations
Example text
2) is an arbitrary bounded subdomain of ~ such that S N(~\fl')--0. The constants 6, and C, do not depend on T . Proof~ It is s~ficient to estimate L~:,~' since . ( _sSol~Lec~,t)-~,gc~,bl)- 9) for ~ >~o . 6. 2) in the s p a c e s of functions which a r e HSlder continuous together with their d e r i v a t i v e s . We give their definitions. :,. 6). Basically, we shall have to do with the spaces rives of the functions ~ e ~ Finally, for any ~ (~) and ~"((~T) . It is known that the d e r i v a - [Q~ have a finite s e m i n o r m ~,,g~ (0, 1)we define the s e m i n o r m I "~ . ~,,D ~10. -wc~,%)+ u(~,~. 2) :=,~,t,~ It is easy to see that Ittl~ ~ ~ P It-~l In particular, if St =V~, E ~ l~(m,t)-~(m,~)l * ~ , ~ t ( ~ , t ) > n. 121 (1963). 12. A. Solonnikov, "Estimates for the solutions of a nonstationary linearized system of N a v i e r Stokes equations," Tro MIAN SSSR, 70, 213-317 (1964)~ 13. V~ A. Solonnikov, "On the differential properties of the solutions of the first boundary-value problem for a nonstationary system of Navier-Stokes equations," Tr. MIAN SSSR, 73, 221-291 (1964). O~ A. Ladyzhenskaya, "On the classicality of the generalized solutions of general nonlinear nonstationary Navier-Stokes equations," Tr. Estimates for solutions of nonstationary Navier-Stokes equations by Solonnikov
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