Análise e Teoria de Equações de Navier-Stokes Incompressíveis em R3, Notas de estudo de Engenharia Mecânica

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Incompressible Navier-StokesEquations in

3 R Zhen Lei (X )

School of Mathematical SciencesFudan University

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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

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Derivations of Navier-StokesEquations

Navier The Navier-Stokesequations were first derived byClaude-Louis Navier on 1822,based on a molecular theory ofattraction and repulsion betweenneighbouring molecules.

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Navier-Stokes Equations

Stokes George Gabriel Stokesre-derived the Navier-Stokesequations on 1845

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Leray’s Contributions

Leray Leray’s Contributions:^ √^ For smooth, divergence-freeinitial data^ u(0, x

(^23) ) ∈ L(R), there is a unique local solution √^ There is at least a global weaksolution, which coincideswith the local smooth onebefore singularity time. Theweak solutions satisfy thenatural energy inequality^ ∫^2 ‖u(t,^ ·)‖+2ν^2 L

t^2 ‖∇u(s,^ ·)‖ds^ ≤ ‖^2 L 0

(^2) u(0, ·)‖.^2 L^3 Incompressible Navier-Stokes Equations in^ R

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Leray’s Contributions Leray’s Contributions: √ Let Tbe the lifespan, p > 3. If⋆

T<^ ∞, then⋆^ ‖u(t,^ ·)‖p^3 ≥^ L(R)^

(^13) − (1−^ ) 2 p^ ǫ(T− t)p⋆ as^ t^ →^ T.⋆ √ For a given weak solution, there exists a closedset^ S ⊂^ (0,^ ∞)^ of measure zero such that thesolution is smooth on

3 R×^ (0,^ ∞)^ ∩ S

c.^3 Incompressible Navier-Stokes Equations in^ R– p. 8/

Main Difficulties in Mathematics Natural Scaling: If^ (u, p

)^ solves the Navier-Stokes equations, then so does the solution pair

λλ (u, p)^ for any^ λ >^0 which is defined byλ^ u(t, x) =^ λu(λ

2 λt, λx),^ p(t, x) =

(^22) λp(λt, λx). λ Remark: ‖u(t, x 2 −^1 )‖=^ λ‖u(t, x^2 L

(^2) )‖.^2 L^ Incompressible Navier-Stokes Equations in (^3) R– p. 10/

Main Difficulties in Mathematics Supercritical nature: Applying a "room-in procedure"to understand fine scales, which is equivalent to takingthe limit^ λ^ →^0 , bounded quantities given by

a priori estimate become worse.Unfortunately, the 3D Navier-Stokes equations fallinto this kind of supercritical case. Heuristically,nonlinearities would dominate the dynamics whichmay develop turbulent behavior.This is the essence of difficulty of solvingNavier-Stokes equations for mathematicians.

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Classical Known Theory: Criteria Ladyzhenskaya (1967) [uniqueness by Prodi 1959 andSerrin 1963]:[Prodi-Serrin-Ladyzhenskaya’s criterion] Let^23 u∈^ L(R)^ and^ ∇ · 0

u= 0. For two Leray-Hopf^0 weak solutions^ u,

(q v, if u ∈ L[0, T )^ p (^3) ], L(R)for some^ p, q^ satisfying

23 +^ = 1,^ p >^3 q^ p^ , then^ u^ =^ v^ and u, v^ are smooth on

3 (0, T ] × R.

Local version: Serrin (1962, ARMA), Struwe (1988CPAM)The case^ p^ = 3: Escauriaza, Seregin and Sverak(2003).^

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Beale-Kato-Majda’s Criterion Beale-Kato-Majda (1984): If^ ∫^ T^ ‖∇ ×^ u^0

∞^3 (s, ·)‖ds <^ L(R)

Then^ u^ is smooth up to

t^ =^ T^. Beale-Kato-Majda’s Criterion has been slightlyimproved by Kozono who proved that on the left-handside in the above inequality, the

∞ L-norm can be replaced by the BMO norm.

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Classical Known Theory: CKN Tian-Xin’s local criterion (1999, CAG):^1 lim supsup^2 r^ r→^0 −r+

∫^2 |u| t<t<t^ B(x) 00 r^0 dy < ǫ∗ implies the regularity of

z.^0 Incompressible Navier-Stokes Equations in

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No Self-similar Blowup [Non-existence of self-similar solutions]Necas-Ruzicka-Sverak (1996, Acta Math.) If^1 √ u(t, x) =^ U^ (^ T^ −^ t

x^3 √),^ U^ ∈^ L( T^ −^ t

3 R),

(^1) and U ∈ H, thenloc u^ = 0. A local version by Tsai (1998, ARMA)

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Axially Symmetric Navier-Stokes The equations are^ r^ rr^ ^ ∂u+^ u∂u+^ utr 

θ^2 (u)z r (^) ∂u+ p− = (∆zr r^ 1 r − )u,^2 r θ^ rθ^ z∂u+^ u∂u+^ u∂tr

r^ θuu^1 θ (^) u+ = (∆^ −^ z r^ r θ)u, 2 z^ rz^ z^ ∂u+^ u∂u+^ u∂tr z^ zu+^ p= ∆u.zz^

The incompressible constraints arer^ ∂ur

ru + +^ ∂uz r^

z^ = 0.^ Incompressible Navier-Stokes Equations in

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Axially Symmetric Navier-Stokes Vorticity formula: Let

θ^ ψbe the angular stream function so thatr^ u=^ −

(^1) θz (^) ∂ψ, u= ∂zr^ r θ(rψ). Then^ θ^ rθ^ ∂u+^ u∂u+^ tr 

r^ θuuz θ (^) u∂u+ = (∆^ z^ r^ 1 θ− )u,^2 r θ^ rθ^ z^ ∂ω+^ u∂ω+^ u∂tr

θ^2 r θ∂(u)uωz (^) θ (^) ω− = (^) z r r 1 θ+ (∆ − )ω.^2 r^3 Incompressible Navier-Stokes Equations in^ R– p. 20/