Incompressible Navier-Stokes Equations in R 3

Incompressible Navier-Stokes Equations in R 3 Zhen Lei ( ) School of Mathematical Sciences Fudan University Incompressible Navier-Stokes Equations in R 3 p. 1/5

Contents Fundamental Work by Jean Leray Key Question and Its Main Difficulties Existence Theory under Small Data (Large Viscosity) Criterions, Caffarell-Kohn-Nirenberg No Self-similar Blowup Axially Symmetric Navier-Stokes Without Swirl, Partial Results with Swirl Gap between Known and Needed Role of Convection: A 3D Model Theoretical Similarities Between Model and NS Singularities of Model Equations Incompressible Navier-Stokes Equations in R 3 p. 2/5

Those are joint works with: Collaborators Thomas Y. Hou (Caltech) Congming Li (University of Colorado) Fang-Hua Lin (Courant Institute) Shu Wang (BJU of Technology) Qi S. Zhang (UC Riverside) etc. Incompressible Navier-Stokes Equations in R 3 p. 3/5

Derivations of Navier-Stokes Equations The Navier-Stokes equations were first derived by Claude-Louis Navier on 1822, based on a molecular theory of attraction and repulsion between neighbouring molecules. Navier Incompressible Navier-Stokes Equations in R 3 p. 4/5

Navier-Stokes Equations George Gabriel Stokes re-derived the Navier-Stokes equations on 1845 Stokes Incompressible Navier-Stokes Equations in R 3 p. 5/5

Navier-Stokes Equations Navier s form of Navier-Stokes Equations, the incompressible case: "Newton s second law" + "stress = ν( u + u )": u t + u u + p = ν u, u = 0, where ν is viscosity. There are other formulations by Cauchy (1828), Poisson (1829), etc. Incompressible Navier-Stokes Equations in R 3 p. 6/5

Leray s Contributions Leray s Contributions: For smooth, divergence-free initial data u(0,x) L 2 (R 3 ), there is a unique local solution There is at least a global weak solution, which coincides with the local smooth one before singularity time. The weak solutions satisfy the natural energy inequality Leray u(t, ) 2 L 2 +2ν t 0 u(s, ) 2 L 2 ds u(0, ) 2 L 2. Incompressible Navier-Stokes Equations in R 3 p. 7/5

Leray s Contributions Leray s Contributions: Let T be the lifespan, p > 3. If T <, then u(t, ) Lp (R 3 ) ǫ p (T t) 1 2 (1 3 p ) as t T. For a given weak solution, there exists a closed set S (0, ) of measure zero such that the solution is smooth on R 3 (0, ) S c. Incompressible Navier-Stokes Equations in R 3 p. 8/5

Open Problems Key question in mathematics: Is the weak solution smooth or not? One of the seven Millennium Prize problems Long standing open problems: Is the weak solution unique? Does a weak solution satisfy the energy identity? Is the solution smooth at (0, 0) if the velocity field satisfies sup u(t,x) x 2 + t <. t<0,x R 3 Incompressible Navier-Stokes Equations in R 3 p. 9/5

Main Difficulties in Mathematics Natural Scaling: If (u,p) solves the Navier-Stokes equations, then so does the solution pair (u λ,p λ ) for any λ > 0 which is defined by u λ (t,x) = λu(λ 2 t,λx), p λ (t,x) = λ 2 p(λ 2 t,λx). Remark: u λ (t,x) 2 L 2 = λ 1 u(t,x) 2 L 2. Incompressible Navier-Stokes Equations in R 3 p. 10/5

Main Difficulties in Mathematics Supercritical nature: Applying a "room-in procedure" to understand fine scales, which is equivalent to taking the limit λ 0, bounded quantities given by a priori estimate become worse. Unfortunately, the 3D Navier-Stokes equations fall into this kind of supercritical case. Heuristically, nonlinearities would dominate the dynamics which may develop turbulent behavior. This is the essence of difficulty of solving Navier-Stokes equations for mathematicians. Incompressible Navier-Stokes Equations in R 3 p. 11/5

Classical Known Theory: Smallness For data which is sufficiently smaller than viscosity, global well-posedness is true. Basically, nonlinearities are small in various critical space-time spaces. Sobolev space Ḣ 1 2, Fujita-Kato (1964, ARMA) Legesgue space L 3, Kato (1984, Math. Zeit.) Besov space Ḃ 1+3/p p,, Cannone-Meyer-Planchon, (1994) BMO 1, Koch-Tataru (2001, AM) X 1, Lei-Lin (2011, CPAM) P.S. Lei-Lin s result only requires that u 0 X 1 < ν. Incompressible Navier-Stokes Equations in R 3 p. 12/5

Classical Known Theory: Criteria Ladyzhenskaya (1967) [uniqueness by Prodi 1959 and Serrin 1963]: [Prodi-Serrin-Ladyzhenskaya s criterion] Let u 0 L 2 (R 3 ) and u 0 = 0. For two Leray-Hopf weak solutions u, v, if u L q( [0,T],L p (R 3 ) ) for some p,q satisfying 2 q + 3 p = 1, p > 3, then u = v and u,v are smooth on (0,T] R 3. Local version: Serrin (1962, ARMA), Struwe (1988 CPAM) The case p = 3: Escauriaza, Seregin and Sverak (2003). Incompressible Navier-Stokes Equations in R 3 p. 13/5

Beale-Kato-Majda s Criterion Beale-Kato-Majda (1984): If T 0 u(s, ) L (R 3 )ds <, Then u is smooth up to t = T. Beale-Kato-Majda s Criterion has been slightly improved by Kozono who proved that on the left-hand side in the above inequality, the L -norm can be replaced by the BMO norm. Incompressible Navier-Stokes Equations in R 3 p. 14/5

Classical Known Theory: CKN localization in space of Leray s partial regularity results of Leray-Hopf weak solutions: started by Scheffer and developed further by Caffarelli-Kohn- Nirenberg (1982, CPAM): The one-dimensional Hausdorff measure of space-time singularity set of any suitable weak solution is 0. F.-H. Lin (1998, CPAM): A new and simpler proof. lim sup r 0 1 r implies the regularity of z 0. Q r (z 0 ) u 2 dyds < ǫ Incompressible Navier-Stokes Equations in R 3 p. 15/5

Classical Known Theory: CKN Tian-Xin s local criterion (1999, CAG): 1 lim sup u 2 dy < ǫ r 0 r sup r 2 +t 0 <t<t 0 implies the regularity of z 0. B r (x 0 ) Incompressible Navier-Stokes Equations in R 3 p. 16/5

No Self-similar Blowup [Non-existence of self-similar solutions] Necas-Ruzicka-Sverak (1996, Acta Math.) If u(t,x) = and U Hloc 1, then u = 0. 1 x U( ), U L 3 (R 3 ), T t T t A local version by Tsai (1998, ARMA) Incompressible Navier-Stokes Equations in R 3 p. 17/5

Axially Symmetric Navier-Stokes Let r = x 2 1 + x2 2 and x 1 /r e r = x 2 /r, e θ = 0 x 2 /r x 1 /r, e z = e 3. 0 Axially symmetric solutions: u(t,x) = u r (t,r,z)e r + u θ (t,r,z)e θ + u z (t,r,z)e z, p(t,x) = p(t,r,z). Incompressible Navier-Stokes Equations in R 3 p. 18/5

Axially Symmetric Navier-Stokes The equations are t u r + u r r u r + u z z u r + p r (uθ ) 2 = ( 1 )u r, r r 2 t u θ + u r r u θ + u z z u θ + ur u θ = ( 1 )u θ, r r 2 t u z + u r r u z + u z z u z + p z = u z. The incompressible constraints are r u r + ur r + zu z = 0. Incompressible Navier-Stokes Equations in R 3 p. 19/5

Axially Symmetric Navier-Stokes Vorticity formula: Let ψ θ be the angular stream function so that Then u r = z ψ θ, u z = 1 r r(rψ θ ). t u θ + u r r u θ + u z z u θ + ur u θ = ( 1 )u θ, r r 2 t ω θ + u r r ω θ + u z z ω θ ur ω θ = z(u θ ) 2 + ( 1 )ω θ. r r r 2 Incompressible Navier-Stokes Equations in R 3 p. 20/5

Axially Symmetric Navier-Stokes Significant natures: Define Γ = ru θ, then this dimensionless quantity satisfies t Γ + u r r Γ + u z z Γ = ( 2r 1 r )Γ which gives the maximum principle: Γ L Γ 0 L. Localized energy estimate away from the axis is critical. Incompressible Navier-Stokes Equations in R 3 p. 21/5

A Special Case: Without Swirl In the case of u θ = 0: global well-posedness is available due to Ladyzhenskaya (1968) and Ukhovskii-Yudovich (1968) based on: and t Ω + u r r Ω + u z z Ω = ( + 2r 1 r )Ω, Here Ω = r 1 ω θ. r 1 u r L Ω L 3,1. Incompressible Navier-Stokes Equations in R 3 p. 22/5

A Special Case: With Swirl In the case of u θ dx > 0: global well-posedness is available due to Hou-Lei-Li (2007, CPDE) for a class of initial data: { u θ (0,r,z) = ǫ 1+δ U 0 (ǫr,z), ω θ (0,r,z) = ǫ 1+δ W 0 (ǫr,z), (ψ θ 0,u θ 0,ω θ ) is periodic and odd in z. Incompressible Navier-Stokes Equations in R 3 p. 23/5

General Axisymmetric Case: Blow-up Rate By Leray s result, blow-up rate satisfies u(t, ) L ǫ T t Tian-Xin s improved CKN theory implies regularity if the natural blow-up rate is u(t,x) ǫ x x0 2 + T t A natural question: If u(t,x) C, x x0 2 +T t then u is smooth at t = T? Incompressible Navier-Stokes Equations in R 3 p. 24/5

General Axisymmetric Case: Blow-up Rate In the axi-symmetric case: progress was made by Chen-Strain-Tsai-Yau (2008 IMRN/2009 CPDE) u(t,x) C r then global smooth solution. Koch-Nadirashvili-Seregin-Sverak: A new proof (2009 Acta Math.) Lei-Q. Zhang (2011 JFA/2011 PJM), Seregin (2011, a new proof Lei-Zhang s result) and Wang-Zhang (2012, related result) Incompressible Navier-Stokes Equations in R 3 p. 25/5

General Axisymmetric Case: Blow-up Rate In Koch-Nadirashvili-Seregin-Sverak (2009 Acta Math.), a conjecture was made: Any bounded ancient solution of the axisymmetric Navier-Stokes equations is a constant if Γ L <. Partial results are also obtained: True if either Γ 0 or sup t,x r u <. Incompressible Navier-Stokes Equations in R 3 p. 26/5

Main Results of Lei-Zhang Theorem 1. Let Γ L < and u L (BMO 1 ). Then regularity. A corollary is that if sup t,x r u z <, then regularity. Another corollaries: improved the L (L 3 ) criterion by ESS and answered the open question by Koch-Tataru in the axisymmetric case. Theorem 2. Let Γ L < and u L (BMO 1 ). Then ancient solutions are zero. This partially answered the conjecture by Koch-Nadirashvili-Seregin-Sverak on Acta Math. Incompressible Navier-Stokes Equations in R 3 p. 27/5

A Sketch of Proof The first goal is Hölder regularity of Γ, which follows by iterating the oscillation decay estimate: OSC Pλ Γ (1 δ)osc P1 Γ. Let M 1 = max P1 Γ, m 1 = M 1 = min P1 Γ, J 1 = M 1 m 1 and define Φ = { 2(M1 Γ)/J 1, if M 1 m 1, 2(Γ m 1 )/J 1, if M 1 < m 1. It suffices to prove inf P λ Φ 2δ (or max P λ Φ 2 2δ). Incompressible Navier-Stokes Equations in R 3 p. 28/5

A Sketch of Proof Backbone is Nash-Moser s method: Moser s iteration method (under u L (BMO 1 )): max Γ 2 C 0 R 5 Γ 2 dyds. P R/2 P R Nash s method (under u L (BMO 1 )): ηr 2 (x) ln Φdx C 0 B R for all t [ c 0 R 2, 0]. Incompressible Navier-Stokes Equations in R 3 p. 29/5

A Sketch of Proof Proof of oscillation decay lemma: C 0 ηr(x) 2 ln Φdx Φ δ ln δ ηr(x)dx 2 ln 2 B R/2,Φ δ δ Φ 2 η 2 R(x)dx δ Φ 2 η 2 R(x) ln Φdx R 3 ln δ meas{x B R/2 Φ δ} ln 2, giving that meas{x B R/2 Φ δ} (C 0 + ln 2)R 3 ln 1 δ 1. Incompressible Navier-Stokes Equations in R 3 p. 30/5

A Sketch of Proof Proof of oscillation decay lemma: Applying the mean value inequality for (δ Φ) + : sup(δ Φ) 2 + C 0 R 5 (δ Φ) 2 +dyds P R/4 P R/2 giving that C 0 R 5 δ 2 (C 0 + ln 2)R 5 ln 1 δ 1 δ 2 ln 1 δ 1, inf Φ δ/2 P R/4 provided that δ is sufficiently small. Incompressible Navier-Stokes Equations in R 3 p. 31/5

A Sketch of Proof The second goal is to prove a Liouville theorem under u L (BMO 1 ): For ancient solutions, using oscillation decay, one has Γ(t 0,x 0 ) Γ(0, 0) C 0 ( x 0 2 + t 0 /L) α OSC PL Γ. Taking L givs that Γ(t 0,x 0 ) Γ(0, 0) = 0. Hence, any ancient solution with Γ L < and u L (BMO 1 ) has no swirl. Incompressible Navier-Stokes Equations in R 3 p. 32/5

A Sketch of Proof For ancient solutions with no swirl, it is known by [KNSS, 2009] that u = (0, 0,c(t)). Consequently, the stream function Ψ(t,x) is harmonic. Hence, by mean value equality: B R Ψ(t,x)dx = B R Ψ(t, 0). Incompressible Navier-Stokes Equations in R 3 p. 33/5

A Sketch of Proof On the other hand, Ψ BMO gives that Ψ dx Ψ Ψ(t, 0) dx + B R Ψ(t, 0) B R B R Hence Ψ is bounded: C 0 R 3 ( Ψ BMO + Ψ(t, 0) ). Ψ(y) = B y 1 B y 1 B y (y) B 2 y (0) Ψdx Ψ dx C 0 ( Ψ BMO + Ψ(t, 0) ). Incompressible Navier-Stokes Equations in R 3 p. 34/5

A Sketch of Proof By classical Liouville theorem, Ψ is independent of x. Hence u 0. This is our Liouville theorem. Incompressible Navier-Stokes Equations in R 3 p. 35/5

Proof of regularity. A Sketch of Proof Facts: Blowup can only occur on the symmetric axis (being explained later). For each t, the maximum point x(t) of u are contained in P 1 (0, 0) if u tends to infinity (being explained later). Suppose that (0, 0) is a singular point. Then t k ր 0 and γ k ր 1 s.t. Q k = u(t k,x k ) = γ k sup u(t,x) ր. 1 t<t k Incompressible Navier-Stokes Equations in R 3 p. 36/5

A Sketch of Proof For Q 2 k (t k + 1) t 0, define v (k) = Q 1 k u(t k + Q 2 k t,x k + xq 1 k ). Clearly, v (k) and p (k) = 1 (v (k) v (k) ) solve NS. Moreover, up to a subsequence, v (k) (t,x) v(t,x) locally uniformly and v is a bounded ancient weak solutions to NS which satisfies v(t,x) 1, v(0, 0) = 1. Incompressible Navier-Stokes Equations in R 3 p. 37/5

A Sketch of Proof Two possibilities: r k Q k uniformly bounded, or r k Q k ր up to a subsequence. If r k Q k is uniformly bounded, then the limit solution v is still axisymmetric (whose axis is not far from the original). Moreover, rv θ is still bounded and v L (BMO 1 ). Using Theorem 2, v 0 which gives a contradiction. Incompressible Navier-Stokes Equations in R 3 p. 38/5

A Sketch of Proof If r k Q k ր, then the limit solution v is two-dimensional (Lei-Zhang, 2011). By known Liouville theorem (KNSS, 2009), v is a constant. Similar argument gives that v 0 which yields a contradiction. The regularity theorem is proved. Incompressible Navier-Stokes Equations in R 3 p. 39/5

What We Need: What We Have u z Cr 1. What We Have: A priori Estimate is u Cr 2. This is due to a recent work by Lei-Navas-Q. Zhang (preprint). Incompressible Navier-Stokes Equations in R 3 p. 40/5

Sketch of the Proof Proof of the a priori estimate u Cr 2. Lemma 1. The following a priori estimate holds: r 2 ω θ (t, ) L 2 + t Remark. Chae-Lee (2002): 0 (r 2 ω θ ) 2 L 2 ds C 0 e t. r 3 ω θ (t, ) L 2 + t 0 (r 3 ω θ ) 2 L 2 ds <. Incompressible Navier-Stokes Equations in R 3 p. 41/5

Sketch of the Proof Lemma 2. Let p = 2 + be any number which is bigger than and close to 2 and r 0 > 0. There holds b Lp (B r0 /2) C pr 3(1 2 1 p ) 0, where C p is a positive constant depending only on p. Main point of the proof: dimension reduction argument and using Lemma 1. Incompressible Navier-Stokes Equations in R 3 p. 42/5

Lemma 3. There holds Sketch of the Proof Ω L 4 t (L 4 (P(1/2))) Cr 11 4 0, where C depends only on v 0 L 2 and rv θ 0 L. Main point of the proof: dimension reduction argument and using Lemma 2. Incompressible Navier-Stokes Equations in R 3 p. 43/5

Moser s iteration: Sketch of the Proof sup P 1/2 ω θ Cr 3.5 0. Main point of the proof: dimension reduction argument, starting with Ω L 4 t (L 4 (P(1/2))) which is stronger than energy estimate. Incompressible Navier-Stokes Equations in R 3 p. 44/5

Sketch of the Proof Using b = curl (ω θ e θ ) to derive that sup B ρ/4 (x) b Cρ 3/2 b L2 (B ρ (x)) + Cρ sup B ρ (x) Using the axisymmetric nature to derive that b(x,t) Cr 2. ω θ. Incompressible Navier-Stokes Equations in R 3 p. 45/5

Model Equations To understand the global regularity of Navier-Stokes equations, people suggested a lot of model equations, which share part of the difficulties of Navier-Stokes. The most well-oiled one is the so-called 2D Quasi-Geostrophic equation: θ t + u θ = ( ) 1 2 θ, u = R θ. Disadvantages: dimension 2, scalar equation, non-local diffusion. Incompressible Navier-Stokes Equations in R 3 p. 46/5

Model Equations We hope to suggest a model which shares the following properties with Navier-Stokes equations: 3D Energy law nonlinearities (vortex stretching) incompressibility Incompressible Navier-Stokes Equations in R 3 p. 47/5

Model Equations Hou-Lei constructed such a 3D model (Hou-Lei, 2009 CPAM): t u 1 = ν ( r 2 + 3 r r + z) 2 u1 + 2 z ψ 1 u 1, t ω 1 = ν ( r 2 + 3 r r + z) 2 ω1 + z (u 2 1 ), ( r 2 + 3 r r + z) 2 ψ1 = ω 1. Difference between the 3D model and Navier-Stokes: part convective terms are ignored in the model equations. Incompressible Navier-Stokes Equations in R 3 p. 48/5

Model Equations The incompressible constraint can be recovered by introducing u = u r e r + u θ e θ + u z e z, where u r = (rψ 1 ) z, u θ = ru 1, u z = (r2 ψ 1 ) r r. A lot of known theories for Navier-Stokes are also valid for the model equations. For instance, CKN (Hou-Lei, 2009 CMP), Serrin criterion, BKM criterion, etc. Incompressible Navier-Stokes Equations in R 3 p. 49/5

Model Equations: Singularities For the model equation, numerically, we showed finite time singularity. Moreover, analytically, we proved that Theorem.(Hou-Lei-Wang, 2013, ARMA) Under the Neumann-Robin type boundary condition (ψ r + βψ) r=1 = 0, ψ z z=0 = ψ z z=1 = 0, the 3D inviscid model can develop a finite time singularity for certain large β > 0. Rem: The box domain was considered by Hou-Shi-Wang (2012, AM). Incompressible Navier-Stokes Equations in R 3 p. 50/5

Thank you very much!! Incompressible Navier-Stokes Equations in R 3 p. 51/5