A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations

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Barbara Janik
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1 A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations G. Seregin & W. Zajaczkowski A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 1/2

2 Motivation I e 1, e 2, e 3 x 1, x 2, x 3 and e, e ϕ, e 3, ϕ, x 3 e = cosϕe 1 + sinϕe 2, e ϕ = sin ϕe 1 + cosϕe 2, e 3 = e 3 v = v i e i = v 1 e 1 + v 2 e 2 + v 3 e 3 = v e + v ϕ e ϕ + v 3 e 3 t v + v v v + p = 0, divv = 0 v,ϕ = v / ϕ = 0, v ϕ,ϕ = 0, v 3,ϕ = 0 p,ϕ = 0 A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 2/2

3 Motivation II Ladyzhenskaya (1968), Ukhovskij & Yudovich (1968), Leonardi, Malek, Necas, & Pokorny (1999), Neustupa & Pokorny (2001), Pokorny (2001), Chae & Lee (2002), Zajaczkowski (2004, 2005), Wiegner & Zajaczkowski (2005), and others. For the Cauchy problem on ]0,T[, Chae & Lee (2002) proved T 0 ( dt R 3 1 v γ dx ) α γ < + regularity if 1/α + 1/γ 1/2, 2 < γ < +, 2 < α + A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 3/2

4 Questions The marginal case γ = 2 and α = +? which is an analogue of L 3, -case in the absence of axial symmetry. Local Version? Notation C(x 0,R) = {x R 3 x = (x,x 3 ), x = (x 1,x 2 ), x x 0 < R, x 3 x 03 < R}, C(R) = C(0,R), C = C(1); z = (x,t), z 0 = (x 0,t 0 ), Q(z 0,R) = C(x 0,R) ]t 0 R 2,t 0 [, Q(R) = Q(0,R), Q = Q(1). A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 4/2

5 Suitable Weak Solutions v L 2, (Q) W 1,0 2 (Q), p L3(Q); 2 v and p satisfy NSE s in weak sense; ϕ(x,t) v(x,t) 2 dx + 2 t ϕ v 2 dxdt B 1 B t ( ) v 2 ( ϕ + t ϕ) + v ϕ( v 2 + 2p) dxdt 1 B for a.a t ] 1, 0[ and for all functions 0 ϕ C 0 (R3 R 1 ) vanishing in a neighborhood of the parabolic boundary of Q. A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 5/2

6 Main Result Theorem 0.1 Let v and p be an axially symmetric suitable weak solution to the Navier-Stokes equations in Q. Assume that 1 (0.1) A 0 = ess sup v(x,t) 2 dx < +. 1 t 0 C Then the point (x,t) = (0, 0) is a regular point of v, i.e., there exists r ]0, 1] such that v is Hölder continuous in the closure of the cylinder Q(r). A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 6/2

7 Preliminaries I A(z 0,r;v) = ess Invariant Functionals 1 sup t 0 r 2 <t<t 0 r C(z 0,r;v) = 1 r 2 C(x 0,r) v 3 dz, v(x,t) 2 dx, E(z 0,r;v) = 1 r Q(z 0,r) v 2 dz, D(z 0,r;p) = 1 r 2 p 3 2 dz. Q(z 0,r) Q(z 0,r) A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 7/2

8 Preliminaries II Energy Inequality A(z 0,R/2;v) + E(z 0,R/2;v) c(c 2 3 (z0,r;v)+ +C(z 0,R;v) + D(z 0,R;p)) Decay Estimate for Pressure D(z 0,r;p) c[ r r 1 D(z 0,r 1 ;p) + ( r1 r ) 2C(z0,r 1 ;v)], which is valid for all 0 < r r 1 R A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 8/2

9 Preliminaries III Lemma 0.2 Let v and p be a suitable weak solution to the Navier-Stokes equations in Q and let A 0 = sup A(0,r;v) < +. 0<r<1 Then, for any r ]0, 1/2[, we have C 4 3 (0,r;v) + D(0,r;p) + E(0,r;v) c ((A 0 + 1)r 1 2 (D(0, 1;p)+ +E(0, 1;v)) + A A A 0 ). A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 9/2

10 Preliminaries IV Lemma 0.3 Under the conditions of Theorem 0.1, we have A(z 0,r;v) + C(z 0,r;v) + D(z 0,r;p) + E(z 0,r;v) A < + for all z 0 = (x 0, 0), x 0 = (0,b), b 1/4, and for all 0 < r 1/4, where A depends on D(0, 1;p), E(0, 1;v), and A 0 only. P(R 1,R 2 ;a) = {x R 3 R 1 < x < R 2, x 3 < a} A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 10/2

11 Preliminaries V Lemma 0.4 Let v and p be a suitable weak solution to the NSE s in the set Q = P(3/4, 9/4; 3/2) ] (3/2) 2, 0[. Assume that v(z) 6 dz m < +. bq Then, there exists a function Φ 0 : R + R + R +, nondecreasing in each variables, such that v(z) + v(z) Φ 0 (m, A ) < +, A = p(z) 3 2 dz for any z P(1, 2; 1) ] 1, 0[. bq A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 11/2

12 Preliminaries VI Lemma 0.5 Assume that all conditions of Theorem 0.1 hold. Then 1 v(x,t) 2 dx A 0 for all t ] 1, 0[. C A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 12/2

13 Scaling and Blow up I Ad absurdum. Let z = 0 is a singular point of v. Then {R k } k=1 such that R k 0 as k + and 1 v 3 dz ε > 0 for all k N. Rk 2 Q(R k ) u k (y,s) = R k v(r k y,r 2 k s), qk (y,s) = R 2 k p(r ky,r 2 k s), where e = (y,s) Q(1/R k ). A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 13/2

14 Scaling and Blow up II x b k = (0,bR k), y b = (0,b), zk b = (xb k, 0), eb = (y b, 0) b R k < 1/4, ar k < 1/4 C(zk b,ar k;v) = C(e b,a;u k ) A, E(zk b,ar k;v) = E(e b,a;u k ) A, A(zk b,ar k;v) = A(e b,a;u k ) A, D(zk b,ar k;p) = D(e b,a;q k ) A for all k k 0 (a,b). A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 14/2

15 Scaling and Blow up III Limit functions u and q satisfy the NSE s on R 3 R, R = {s R s 0}. They are called an ancient solution to the NSE s. For any a > 0, u k u in W 1,0 2 (Q(a)), u k u in L2, (Q(a)), and u k u in L 3 (Q(a)), q k q in L3(Q(a)) 2 C(e b,a;u) A, A(e b,a;u) A, E(e b,a;u) A, D(e b,a;q) A A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 15/2

16 Scaling and Blow up IV 1 Rk 2 Q(R k ) ess sup <s 0 R 3 v 3 dz = Q u(y,t) 2 y dy A 0, u k 3 de u 3 de ε, Q and, for any a > 1, u k u in C([ 1, 0];L9(C(a))). 8 A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 16/2

17 Scaling and Blow up V ( u(y, 0) 9 8 dy )8 9 ( u k (y, 0) u(y, 0) 9 8 dy )8 9 + P(r 1,r 2 ;h) P(r 1,r 2 ;h) ( + u k (y, 0) 9 8 dy )8 9 = α k + β k, P(r 1,r 2 ;h) α k 0 A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 17/2

18 Scaling and Blow up VI β k = ( R 15 8 k P(R k r 1,R k r 2 ;R k h) ( 1 c(r 1,r 2,h) R k v(x, 0) 9 8 dx )8 9 )1 v(x, 0) 2 2 dx ( c(r 1,r 2,h) P(R k r 1,R k r 2 ;R k h) P(R k r 1,R k r 2 ;R k h) v(x, 0) 2 x )1 2 dx 0 u(, 0) = 0 in R 3. A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 18/2

19 Estimates of Axially Sym Solutions I Proposition 0.6 Let V and P be a sufficiently smooth axially symmetric solution to the Navier-Stokes equations in Q = P ] 2 2, 0[, where P = P(1/4, 3; 2). Then, there exists a non-decreasing function Φ : R + R + such that sup z P(1,2;1) ] 1,0[ ( V (z) + V (z) ) Φ(A 2 ), where A 2 = sup 2 2 <t<0 ep V (x,t) 2 dx + eq ( ) V 2 + V 3 + P 3 2 dz. A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 19/2

20 Estimates of Axially Sym Solutions II Lemma 0.7 Under assumptions of Proposition 0.6, there exists a function Φ 1 : R + R + R +, non-decreasing in each variable, such that (0.2) sup V a (x,t) q dx Φ 1 (q, A 2 ), 1 q +. (7/4) 2 <t<0 ep 1 Here, V a = (V,V 3 ), V a = V 2 + V 3 2, P 1 = P(5/16, 11/4; 7/4), and Q 1 = P 1 ] (7/4) 2, 0[. A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 20/2

21 Estimates of Axially Sym Solutions III Lemma 0.8 Under assumptions of Proposition 0.6, there exists a non decreasing function Φ 5 : R + R + such that eq 2 V ϕ 6 dz Φ 5 (A 2 ), where Q 2 = P 2 ] (3/2) 2, 0[ and P 2 = P(3/8, 5/2; 3/2). Corollary 0.9 Under assumptions of Proposition 0.6, there exists a non-decreasing function Φ 6 : R + R + such that eq 2 V 6 dz Φ 6 (A 2 ). A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 21/2

22 Decay I Given R > 1, Q b R = P b R ] (2R)2, 0[, where b R and P b R = P R + be 3, PR = P(R/4, 3R; 2R). Scale ancient solution u and q in the following way u R (x,t) = Ru(Rx+be 3,R 2 t), q R (x,t) = R 2 q(rx+be 3,R 2 t) for z = (x,t) Q. A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 22/2

23 Decay II sup z e Q 0 { } u R (z) + u R (z) Φ(A 2 ), where Q 0 = P(1, 2; 1) and A 2 = u R (x,t) 2 dx + sup 2 2 t 0 ep eq ( ) u R 2 + u R 3 + q R 3 2 dz. A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 23/2

24 Decay III sup (y,s) Q b R { } R u(y,s) + R 2 u(y,s) Φ(cA), where Q b R = P b 0R ] R2, 0[, P b 0R = be 3 + P 0R, P 0R = P(R, 2R;R) y u(y,b,s) + y 2 u(y,b,s) Φ(cA) for any b R, for any y > 20, and for any s [ 20, 0] u(y,s) + u(y,s) cφ(ca) = c(a) for any y > 20 and for any s [ 20, 0] A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 24/2

25 Backward Uniqueness So, ω(u) = u t ω ω = ω u u ω t ω ω c(a)( ω + ω ), ω c(a) for any y > 20 and for any s [ 20, 0] and ω(, 0) = 0 in R 3 ω(y,s) = 0 for any y > 20 and for any s [ 20, 0]. A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 25/2

26 Unique Continuation I u 0 in R 3 \ {y 0} [ 8, 0] A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 26/2

27 Unique Continuation II A 0 ess sup 20 s 0 y 40 u(y,s) 2 y dy + dy 3 y 40 u(y,s) 2 y dy A 0 < +, for any s S [ 20, 0] and S = 20. for any s S. u(,s) 0 in R 3 A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 27/2

28 Unique Continuation III For any y 0 { y 30, y 3 R}, B(y 0, 1) { y 40, y 3 R} and, since u is harmonic, ( u(y 0,s) c )1 u(y,s) 2 2 dy ( c )1 u(y,s) 2 2 dy B(y 0,1) y 40 c 40A 0 for any s S [ 8, 0] u(,s) is bounded in R 3 for any s S [ 8, 0] u(,s) = 0 in R 3 for any s S [ 8, 0]. A sufficient condition of regularity for axially symmetric solutions to the Navier-Stokes equations p. 28/2

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