# Estimates of solutions of an initial- and boundary-value by Solonnikov By Solonnikov

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Extra resources for Estimates of solutions of an initial- and boundary-value problem for the linear nonstationary Navier-Stokes system

Sample text

L (-~-'-' < ~ then (~+~',tP ~" " U +'~" ~. Ipl~? , - ~" L~'Jt,ll~, The constants t, fl~ G L do not depend on T . 3. 5. 2) in Holder norms. We preface the proof of this result by the following auxiliary proposition. 1. _.. ~. Proof. 3) is the solution of Eq. 1) and dit~on We P~l, - P. 2). The lemma is proved. 2). is established under the condition that the boundary 5 of the domain ~ Ot+~. , in the neighborhood of any point the surface ~6S It belongs to the class ~ can be given by the equation 2+~.

J Z where II:'IK~" ,= H, IPlcxK~)-~Mo and the constants 376 M, ~ , , d . 1. ~(~) ~ Let ~ and with the ~ axis pointing along the Therefore S E C z''&. +. m§ lU,,os 4- IE,=I~ ,t~+ ~ 1 '~ + [B] , ~ I "Q-t ' I~'IQt + [ -~ C~ depend only on ~. , u " , t q . , - J'Qt "({')t + L~J%,Q qe =E-~(~) and on the domain (they and depend as a power function is the tangential component of the vector ~ the tangential part of the gradient. 27), understood in the generalized sense as the identities t. 6). Proof. -~14x~,in we shall distinguish two cases: where IEI~(t~(Zo~-- K~&(z~ c ~ ~ and p and Kz&(mo~ n(R3\e~#o z.

40) has a solution the f o l l o w i n g properties: T(R~I ,~,~C"ZiR, ~,I, ~o~ ~ * ~ " ~+~" - ~ 1) C P'+~'( ~eC~+~'~+~ (R~ ~ p_ C~+~ ' ~. - -_~~_% =,,. ) , , ~ vp ~ C A'~" (Re~ for any ~, ~(~+'~,~0~ ~=~,z,v%eC=. ~. ,, ; in addition The solution satisfies the inequality -.. moreover, if [ ~~]t,Rj "' + [ '%]'-'-'i-''' < ~ t,R~ i'~a ,. ff)]_ . ,_. ~. 29) +x~-~[~]~ , t,R~, The constants Remark. C4,Cz~C, do not depend on r ; ~4)~. 30) 367 valld for any Proof. ~ e C~(R+~). First . me that ~ , ~ , [ , BL, , ~ .