Math 497 R1 Winter 2018 Navier-Stokes Regularity

Size: px

Start display at page:

Download "Math 497 R1 Winter 2018 Navier-Stokes Regularity"

Betty Harvey
5 years ago
Views:

1 Math 497 R Winter 208 Navier-Stokes Regularity Lecture : Sobolev Spaces and Newtonian Potentials Xinwei Yu Jan. 0, 208 Based on..2 of []. Some ne properties of Sobolev spaces, and basics of Newtonian potentials. Table of contents. Sobolev Spaces Lebesgue spaces Sobolev spaces Newtonian Potentials Bibliography

2 2 Navier-Stokes Regularity.. Lebesgue spaces. Sobolev Spaces Lemma. (Lemma. of []) Let 6 p <, then L p () is the completion of C 0 () in L p (). Proof.. We have = [B n where B n has center x n 2 rational and radius r n = 2 dist(x n; c ). 2. We also have can be approximated from within by compact sets C m such that the measure of the dierence C m goes to zero. 3. Each C m can be covered by nitely many B n 's. Denote the union of these balls by m. We now have dist ( m ; c ) > 0 and kuk Lp ( m )! Now mollify u m to get u m 2 C 0 (). Remark 2. i. Note that is only required to be a domain (open connected set). No regularity is needed ii. It is obvious that L is not the completion of C 0. Question 3. What about Lorentz spaces?.2. Sobolev spaces Definition 4. (Sobolev spaces) W k s (). L s integrability of weak derivatives. k kuk Ws k () = X i=0 kr i uk s; : () W s k (). W k s completion of C 0 (). L k s (). L s integrability of r k u. kuk Ls k () = krk uk s; : (2) L s k (). L s k completion of C 0 () in the following sense. [u] 2 L s k () is an equivalence class of functions satisfying a) 8v; w 2 [u], r k (v w) = 0, and b) 9u 0 2 [u] such that there exists u m 2 C 0 () with kr k (u m u 0 )k s;! 0. It turns out that integrability of derivative implies local integrability of the function. Theorem 5. (Theorem.) u 2 L s k () =) u 2 L s;loc (). Proof. It suces to prove for k =. Once this is done, the general case follows easily from induction. By assumption, for any ' 2 C 0 () we have hu; r'i = R g ' for some g 2 L s(). We need to prove, for any 0 b, u 2 L s ( 0 ). The main diculty lies in construction of the function u which is well-dened in the whole.

3 X Yu 3 Now x one such 0. Let 0 < " < dist( 0 ; c ). Let u " be the standard mollication of the distribution u. i. We rst show that u " 2 L ( 0 ) for each ". To see this, dene l: L ( 0 ) 7! R by l( ) := hu; " i: (3) As u 2 D 0 (), there is m 2 N [ f0g such that jl( )j 6 C k " k C m () 6 C(") k k L ( 0 ). Thus by Riesz representation theorem we have u " 2 L ( 0 ). ii. Now we easily see that g " = ru " in 0. iii. Next let u 0;" := u " [u " ] 0 where [u " ] 0 is the average of u " in 0. By Poincare's inequality we have u 0;"! u 0 in L s ( 0 ).??? Naturally g = ru 0 in 0. iv. Now we need to construct u, dened on, such that uj 0 =u 0. Let c 0 and b. Repeating the above we have u 2 L s ( ) such that g = ru in. As r(u u 0 ) = 0 in 0, u u 0 = C 0, a constant, in 0. We re-dene u C 0 2 L s ( ) as the new u. This can be repeated for a nested sequence of sets 0 b b 2 b b. Note that in each m, we have u m = u m+ = u m+2 =. Thus convergence is not an issue. Finally, any other v with rv = g is just a constant away from the u just constructed. Remark 6. It is clear that we cannot expect u 2 L s (). For example let = R and s =, and u = jxj. Corollary 7. If u m 2 C 0 () is Cauchy in L s k, then there is u 2 L s k () such that ku m uk Ls k ()! 0. Remark 8. The proof is similar to that of Theorem 5. Also note that does not need to be bounded here. Corollary 9. (Proposition.2) For bounded domain, L s k () = W s k (). Proof. Clearly W s k () L s k (). For the other direction, we need to show that for every [u] 2 L s k (), there must exist a v 2 [u] such that v 2 W s k (). Let [u] 2 L s k (). By denition there is w 2 L s;loc () such that i. For every other w 0 2 [u], r k (w w 0 ) = 0; ii. There is a sequence w m 2 C 0 () such that r k w m! r k w in L s (). As is bounded, we have Friedrichs inequality: kw m w m 0k Ls () 6 c kr k w m r k w m 0k Ls () (4) which can be obtained through repeated application of Poincaré. Thus w m converges in L s, to some function v. It is clear that v 2 [u]. Remark 0. When n > 3, thanks to the Gagliardo-Nirenberg inequality kuk p; 6 c(n) kruk 2; where p = 2 n n 2, we see that L 2 () L p (), in the sense that for every [u] 2 L 2 (), there is a w 2 [u] such that w 2 L p. When n = 2 and = R + 2, we can still select a good representative for every [u] 2 L 2 (R + 2 ) by the criterion kvk L2 () < where = R (0; ). =)Any with good boundary would be OK?. is bounded.

4 4 Navier-Stokes Regularity 2. Newtonian Potentials Recall the fundamental solutions 8 >< E(x) = >: 2 ln jxj! n n (n 2) We can dene the Newtonian potential of a function f: n = 2 : (5) jxj n 2 n > 3 u = E f: (6) Proposition. (Proposition 2.3) Let f 2 L p (R n ) with < p < and u =E f. Then 4u = f in R n, and u 2 L p 2 (R n ). Furthermore R R n jr 2 uj p dx 6 c(n; p) R R n jf j p dx. Proof. Note that R R n jr 2 uj p dx 6 c(n; p) R R n jf j p dx follows from the theory of singular integral operators. Next notice that for f 2 C 0, 4u = f holds by direct calculation. The general situation now follows from the above estimate and approximation argument. Thus all we need to prove is the existence of u m 2 C 0 such that kr 2 (u m u)k L2 (R n )! 0. Let f m 2 C 0 (R n ), f m! f in L p (R n ). Dene v m = E f m. As f m has compact support, we have jr i v m j 6 c(m; i) jxj n 2+i; x 2 Rn : (7) Now consider R > 0 and let ' R be the standard cut-o function. We calculate Z " Z jr 2 (' R v m v m )j p dx 6 c R n Z 6 c B R c B R c jr 2 v m j p + R p Z B 2R nb R jrv m j p + R 2p Z B 2R nb R jr 2 v m j p jr 2 v m j p + C(m) R n( p) : (8) # Thus for each m, we take R m such that R R n nishes the proof. jr 2 (' Rm v m v m )j p dx < m. Dening u m = ' Rm v m

5 X Yu 5 Bibliography [] Gregory Seregin. Lecture Notes on Regularity Theory for the Navier-Stokes Equations. World Scientic, 205.

ON TRIVIAL GRADIENT YOUNG MEASURES BAISHENG YAN Abstract. We give a condition on a closed set K of real nm matrices which ensures that any W 1 p -grad

ON TRIVIAL GRADIENT YOUNG MEASURES BAISHENG YAN Abstract. We give a condition on a closed set K of real nm matrices which ensures that any W 1 p -grad ON TRIVIAL GRAIENT YOUNG MEASURES BAISHENG YAN Abstract. We give a condition on a closed set K of real nm matrices which ensures that any W 1 p -gradient Young measure supported on K must be trivial the