Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Some Mathematical Theories of MHD Boundary Layers Tong Yang City University of Hong Kong Workshop on Kinetic and Fluid PDEs Paris, March 7-9, 2018 Joint work with Chengjie Liu and Feng Xie Research Supported by GRF 11320016 Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Prandtl system for classical fluid

In 1904, Prandtl developed the boundary layer theory by resolving the difference between viscous and inviscid flow near a boundary with no-slip boundary condition: √ • outside a layer of thickness of ε, convection dominates so that the flow is governed by Euler equations; √ • inside a layer (boundary layer) of thickness of ε, convection and viscosity balanced so that the flow is governed by the Prandtl layer equation. Well-posedness? Justification?

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Prandtl system for classical fluid

(Incompressible Navier-Stokes equations)  ε ε ε ε ε   ∂t u + (u · ∇)u + ∇p − ε∆u = 0 ∇ · uε = 0   ε u |z=0 = 0 noslip boundary condition with flat boundary {(x, y) ∈ D, z = 0}, set uε = (uε , vε , wε )T :    uε (t, x, y, z) = u(t, x, y, √zε ) + o(1)   vε (t, x, y, z) = v(t, x, y, √zε ) + o(1)    wε (t, x, y, z) = √εw(t, x, y, √z ) + o(√ε) ε

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Prandtl system for classical fluid

(Prandtl boundary layer equations)   ∂t u + (u∂x + v∂y + w∂Z )u + ∂x pE (t, x, y, 0) = ∂z2 u      ∂t v + (u∂x + v∂y + w∂Z )v + ∂y pE (t, x, y, 0) = ∂z2 v  ∂x u + ∂y v + ∂z w = 0      (u, v, w)|z=0 = 0,

lim (u, v) = (uE , vE )(t, x, y, 0),

z→+∞

with pE and uE = (uE , vE , 0)(t, x, y, 0) satisfy (Bernoulli’s law) ∂t uE + (uE · ∇)uE + ∇pE = 0.

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

(Oleinik’s monotonicity condition (2D)) • Coordinate transformations von Mise transformation for steady layer (Oleinik): (x, y) → (x, φ ), w = u2 , √ wx = wwyy − 2PEx . Crocco transformation for unsteady layer (Oleinik): (t, x, y) → (t, x, u),

w = uy ,

wt + uwx = w2 wuu .

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

• Cancellations ∂t u + u∂x u + v∂y u + ∂x PE = ∂y2 u ···+( · · · + u(

u∂x u + v)y + · · · = · · · ∂y u

∂x u )y + ∂x u + ∂y v + · · · = · · · . ∂y u

Denote w = ∂y u, then f0 = (

w2 − wy u u )y = . (Alexandre-Wang-Xu-Y.) ∂y u w2

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Another way to look at the cancellation: wt + u∂x w + v∂y w = ∂y2 w, ut + u∂x u1 + v∂y u = ∂y2 u. Set

wy u = wf0 . (Masmoudi-Wong) w A similar function f2 = wwx − wy u1x satisfying an equation without loss of derivative is used to study solutions in Gevrey function space without monotonicity condition, cf. Li-Y. f1 = w −

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

(Well-posedness with finite order of regularity) Oleinik (’60): local existence of classical solutions; Xin-Zhang (’04): existence of global weak solution with additional favorable pressure ∂x pE (t, x, 0) ≤ 0; Alexandre-Wang-Xu-Y.(’12), Masmoudi-Wong (’12): local existence in Sobolev spaces. ··· How about 3D?

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

(Ill-posedness without the monotonicity condition) E & Engquist (’97): construction of blowup solutions; Grenier (’00): unstable Euler shear flow yields instability of Prandtl; Gérard-Varet & Dormy (’10), Guo & Nguyen (’11): shear flow with non-degenerate critical point in 2D; Grenier-Nguyen (’17): unstable even for Rayleigh’s stable shear flow; Liu-Wang-Y. (’15): ill-posedness in 3D if U(z) 6≡ kV(z). Not know nonlinear stability even U(z) = kV(z) with Uz (z) > 0 and finite regularity. Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

(Well-posedness with infinite order of regularity) Sammartino & Caflisch (’98): Well-posedness of Prandtl system, and justification of the Prandtl ansatz when the data is analytic; Gérard-Varet & Masmoud: 2D with Gevrey index = 74 ; Li-Y.(’17): 2D optimal Gevrey index (1, 2]; Li-Y.(’18): 3D with index (1, 2] and monotonicity in one direction. Optimal Gevrey index 2 implied by the ill-posedness theory of Gérard-Varet & Dormy. Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

(High Reynolds number limit) Kato (’84): a necessary and sufficient condition; Bardos-Titi (’18): relation to Onsager conjecture; Maekawa (’14): initial vorticity is supported away from the boundary for 2D flow; Gérard-Varard, Maekawa & Masmoudi(’15): Gevrey stability of Prandtl expansion in 2D; Guo-Nguyen(’17): steady fow over a moving plate; ···

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness with Gevrey regularity in 2D Defintion Let ` > 3/2. For (ρ, σ ), ρ > 0, σ ≥ 1, Xρ,σ is a Gevrey function space with the norm kf kρ,σ

ρ m−5 = sup  σ < y >`−1 ∂xm f L2 m≥6 (m − 6)!

ρ m−5 + sup  σ < y >` ∂xm (∂y f ) L2 m≥6 (m − 6)!

ρ i+j−5 + sup  σ < y >`+1 ∂xi ∂yj (∂y f ) L2 1≤j≤4 (i + j − 6)! i+j≥6

+ ... Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

(Theorem Li-Y., JEMS, ’17) Let σ ∈ (1, 2], u0 ∈ X2ρ0 ,σ with ku0 kρ0 ,σ ≤ η0 . Suppose u0 satisfies the compatibility conditions, then there is a
unique solution u ∈ L∞ [0, T]; Xρ,σ for some T > 0 and some 0 < ρ < ρ0 , provided η0 is sufficiently small.

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Three types of cancellations The first type cancellation: Consider   ∂t + (us + u)∂x + v∂y − ∂y2 ∂xm u + (∂xm v)(ω s + ω) = · · · and vorticity equation   ∂t + (us + u)∂x + v∂y − ∂y2 ∂xm ω + (∂xm v)(∂y ω s + ∂y ω) = · · · In the region of monotonicity, introduce fm = ∂xm ω −

∂y ω s + ∂y ω m ∂x u; ωs + ω

cf. Masmoudi-Wong, Alexandre-Wang-Xu-Y. Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

The second type cancellation: For estimation on ∂xm ω in the neighborhood of the critical point, use the vorticity equation   ∂t + (us + u)∂x + v∂y − ∂y2 ∂xm ω + (∂xm v)(∂y ω s + ∂y ω) = · · · with inner product with ∂xm ω ; ∂y ω s + ∂y ω and use  (∂xm v)(∂y ω s + ∂y ω),

 ∂xm ω = (∂xm+1 u, ∂xm u)L2 = 0, ∂y ω s + ∂y ω L2

cf. Gérard-Varet and Masmoudi. Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

The third type cancellation: Note that the equation for ∂y ω: ∂t (∂y ω) + (us + u)∂x (∂y ω) + v(∂y2 ω s + ∂y2 ω) − ∂y2 (∂y ω) = −g1 , with g1 = (ω s + ω)∂x ω − (∂y ω s + ∂y ω)∂x u, we have   ∂t + (us + u)∂x + v∂y − ∂y2 ∂xm ∂y ω + (∂xm v)(∂y2 ω s + ∂y2 ω) = −∂xm g1 + · · · In the neighborhood of the critical point, consider hm = ∂xm ∂y ω −

∂y2 ω s + ∂y2 ω m ∂ ω. ∂y ω s + ∂y ω x

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

(Property of g1 ) Note that g1 = (ω s + ω)f1 satisfies:   ∂t + (us + u)∂x + v∂y − ∂y2 g1 = 2(∂y2 ω s + ∂y2 ω)∂x ω − 2(∂y ω s + ∂y ω)∂x ∂y ω. • The order of derivative in x on the RHS is the same as g1 , an extra m in front of gm can be added to its Gevrey norm.

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Sketch of Proof Define three auxilliary functions fm , hm and gm : fm = χ1 ∂xm ω − χ1

∂y ω s + ∂y ω m ∂xm u s ∂ u = χ (ω + ω)∂ ( ), y 1 x ω s + ωε ωs + ω

hm = χ2 ∂xm ∂y ω − χ2

∂y2 ω s + ∂y2 ω m ∂ ω, ∂y ω s + ∂y ω x

and   gm = ∂xm−1 (ω s + ωε )∂x ω − (∂y ω s + ∂y ω)∂x u .

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

(Define an equivalent norm)  

m kgm kL2 + < y >` fm L2 + khm kL2 1≤m≤5  + sup kχ2 ∂y ∂xm ωkL2

|u|ρ,σ = kukρ,σ + sup

1≤m≤5



ρ m−5  + sup  σ m kgm kL2 + < y >` fm L2 + khm kL2 m≥6 (m − 6)! ρ m−5 + sup  σ kχ2 ∂y ∂xm ωkL2 . m≥6 (m − 6)!

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

(Uniform estimate in Gevrey space)

|u(t)|2ρ,σ

.

|u0 |2ρ,σ

Z t

+ 0

(|u|2ρ,σ

+ |u|4ρ,σ ) ds +

Z t |u|2 ˜ ρ(s),σ 0

˜ −ρ ρ(s)

ds.

Define def

kuk(λ ,T) = sup( ρ,t

ρ0 + ρ − λ s ρ0 − ρ − λ t 1/2 ˜ ) |u(t)|ρ,σ , ρ(s) = , ρ0 − ρ 2

where the supremum is over ρ > 0, 0 ≤ t ≤ T and ρ + λ t < ρ0 . For small ku0 k2ρ0 ,σ , there exists R and λ such that |ku|k(λ , ρ0 ) ≤ R. 4λ

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness High Reynolds numbers limit

(2D incompressible MHD)  2
1  ∂t u + (u · ∇)u + ∇p = ν10 (H · ∇)H − ∇ |H|  2ν0 + Re 4u,    ∂t H − ∇ × (u × H) = Re1m 4H, in Ω,  ∇ · u = 0, ∇ · H = 0,     u| = 0, H| = Perfect conducting. Γ

Γ

u = (u1 , u2 ) : velocity, H = (h1 , h2 ) : magnetic filed, p : pressure, ν0 : permeability, Re = ν −1 : Reynolds number, ν : viscosity, Rem = ν0 σ : magnetic Reynolds number, σ : electrical conductivity, Rem magnetic Prandtl number : Prm := = νν0 σ . Re Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness High Reynolds numbers limit

Case 1. Prm  1 or σ  Re

The velocity has characteristic boundary layer with thickness √1 with leading order of the magnetic field unchanged. Re Denote H ·~n|Γ = B. There is an extra term σ B2 (U − u1 ) in the equation for u1 and it is similar to the classical Prandtl equations.

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness High Reynolds numbers limit

Case 2. Prm = O(1) or σ = O(Re) The boundary layer for both the velocity and magnetic fields. H ·~n|Γ = B 6= 0 : The non-characteristic boundary layer 1 1 (u1 , Re u2 , h1 , B + Re h2 )(x, Re y) is called Hartmann layer:   ∂ 2 u1 + B ∂Y h1 = 0, Y µ  ∂x u1 + ∂Y u2 = 0, =⇒ =⇒

−B∂Y u1 =

1 2 Prm ∂Y h1 ,

∂x h1 + ∂Y h2 = 0,

lim u1 = U.

Y→+∞

∂Y2 u1 − µ −1 Prm B2 (u1 − U) = 0, ∂Y h1 = Prm B(U − u1 ). h i p u1 = U 1 − exp{− µ −1 Prm B2 Y} . Hartmann Layer Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness High Reynolds numbers limit

H ·~n|Γ = 0 : Inside the boundary layer, friction force ∼ inertia force ∼ Lorentz force. The velocity and magnetic field have characteristic √ boundary layer profile (u1 , √1Re u2 , h1 , √1Re h2 )(t, x, Re y). Note that this MHD boundary layer system behaves very differently from the classical Prandtl system.

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness High Reynolds numbers limit

Case 3. Prm  1 or σ  Re The magnetic field has characteristic boundary layer with the 1 boundary layer thickness √Re , while the leading order of the m velocity field in the layer remains unchanged. The magnetic √ 1 h2 )(t, x, Rem y) satisfies boundary layer profile (h1 , B + √Re m

∂t h1 = ∂Y2 h1 + µ −1 B2 Re (H − h1 ) + Ht ,

∂x h1 + ∂Y h2 = 0,

where H := lim h1 . Y→+∞

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness High Reynolds numbers limit

(Derivation of the boundary layer system in Case 2)  ε 2
 ∂t uε + (uε · ∇)uε − (Hε · ∇)Hε + ∇ pε + |H2 | = ε4uε ,     ∂ Hε − ∇ × (uε × Hε ) = κε4Hε , in {t, y > 0, x ∈ T}, t ε ε  ∇ · u = 0, ∇ · H = 0,     (uε , Hε · n, (∇ × Hε ) × n)| y=0 = 0. ( perfect conducting) Near the boundary as ε → 0:
√ √ y (uε , Hε , pε )(t, x, y) ∼ u1 , εu2 , h1 , εh2 , p (t, x, √ ). ε

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness High Reynolds numbers limit

MHD boundary layer system

 ∂t u1 + u1 ∂x u1 + u2 ∂y u1 − h1 ∂x h1 − h2 ∂y h1 = ∂y2 u1 − Px ,      ∂t h1 + ∂y (u2 h1 − u1 h2 ) = κ∂y2 h1 ,      ∂x u1 + ∂y u2 = 0, ∂x h1 + ∂y h2 = 0, (Main) u1 |t=0 = u10 (x, y), h1 |t=0 = h10 (x, y),      (u1 , u2 , ∂y h1 , h2 )|y=0 = 0,      lim (u1 , h1 ) = (ue , he )(t, x, 0) := (U, H)(t, x). 1 1 y→+∞

Theories in analytic and Gevrey function spaces hold because of the same singularity and degeneracy as Prandtl equations. Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness High Reynolds numbers limit

(Observation) Motivatd by von Mise and Crocco transforms The stream function of the magnetic field ψ(t, x, y) : h1 = ∂y ψ, h2 = −∂x ψ, ψ|y=0 = 0, satisfies ∂t ψ + u1 ∂x ψ + u2 ∂y ψ = κ∂y2 ψ. If h1 6= 0, then ψ is monotone in y.

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness High Reynolds numbers limit

(Coordinate transformation) τ = t, ξ = x, η = ψ(t, x, y), (Symmetric quasilinear system)   ∂τ u1 + u1 ∂ξ u1 − h1 ∂ξ h1 + (κ − 1)h1 ∂η h1 ∂η u1 = h21 ∂η2 u1 ,       ∂τ h1 − h1 ∂ξ u1 + u1 ∂ξ h1 = κh21 ∂η2 h1 ,        (u1 , h1 ∂η h1 )|y=0 = 0, lim (u1 , h1 ) = (U, H). η→+∞

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness High Reynolds numbers limit

(Cancellation) m um 1 := ∂x u1 −

∂y u1 m ∂y h1 m m ∂ ψ, hm ∂ ψ. 1 := ∂x h1 − h1 x h1 x

m (Symmetry system for (um 1 , h1 ))

  

m m 2 m ∂t um 1 + (u1 ∂x + u2 ∂y )u1 − (h1 ∂x + h2 ∂y )h1 = ∂y u1 + ...,

 

m m 2 m ∂t hm 1 + (u1 ∂x + u2 ∂y )h1 − (h1 ∂x + h2 ∂y )u1 = k∂y h1 + ...

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness High Reynolds numbers limit

Theorem (Liu-Xie-Y., CPAM, 2017) Let m ≥ 5, l ≥ 0, and assume   u10 (x, y) − U(0, x), h10 (x, y) − H(0, x) ∈ Hl3m+2 (Ω), h10 (x, y) ≥ 2δ0 , and the compatibility conditions up to m-th order. Here,
1 kf kHlm (Ω) = ∑m1 +m2 ≤m khyil+m2 ∂xm1 ∂ym2 f k2L2 (Ω) 2 . Then, there exist a time 0 < T ≤ T0 , and a unique solution (u1 , u2 , h1 , h2 ) to (Main), such that h1 ≥ δ0 , (u1 − U, h1 − H) ∈

m \

  W i,∞ 0, T; Hlm−i (Ω) .

i=0

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness High Reynolds numbers limit

Justification of the high Reynolds numbers limit Consider    ∂t uε + (uε · ∇)uε − (Hε · ∇)Hε + ∇pε = µε4uε ,       ∂ Hε + (uε · ∇)Hε − (Hε · ∇)uε = κε4Hε ,   t ∇ · uε = 0, ∇ · Hε = 0,      uε |y=0 = 0, ∂y hε1 |y=0 = 0, hε2 |y=0 = 0,     (uε , Hε )|t=0 = (u0 , h0 )(x, y),

(x, y) ∈ R2+ ,

with no-slip boundary and perfect conducting boundary conditions. Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness High Reynolds numbers limit

Assume ue = (ue1 , ue2 ), He = (he1 , he2 ) and pe solve ideal MHD equations with the same initial data and corresponding BC:    ∂t ue + (ue · ∇)ue − (He · ∇)He + ∇pe = 0,     ∂ He + (ue · ∇)He − (He · ∇)ue = 0, t

  ∇ · ue = 0, ∇ · He = 0,     (ue , he )|y=0 = 0, (ue , He )|t=0 = (u0 , h0 )(x, y). 2 2

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness High Reynolds numbers limit

The boundary layer profile (ub1 , ub2 , hb1 , hb2 )(t, x, Y) is given by  (ub , hb )(t, x, Y) := (up , hp )(t, x, Y) − (ue , he )(t, x, 0), 1 1 1 1 1 1 ub (t, x, Y) := R ∞ ∂ ub (t, x, z)dz, hb (t, x, Y) := R ∞ ∂ hb (t, x, z)dz, 2

Y

x 1

2

Y

x 1

where (up1 , hp1 )(t, x, Y) solves  p p p p p p p 2 p e   ∂t u1 + (u1 ∂x + u2 ∂Y )u1 − (h1 ∂x + h2 ∂Y )h1 = µ∂Y u1 − ∂x p (t, x, 0),     ∂t hp + (up1 ∂x + up2 ∂Y )hp1 − (hp1 ∂x + hp2 ∂Y )up1 = κ∂Y2 hp1 ,    1 ∂x up1 + ∂Y up2 = 0, ∂x hp1 + ∂Y hp2 = 0,     (up1 , up2 , ∂Y hp1 , hp2 )|Y=0 = 0, lim (up1 , hp1 )(t, x, Y) = (ue1 , he1 )(t, x, 0),   Y→+∞    (up , hp )| = (ue , he )(0, x, 0); 1 1 1 1 t=0 Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness High Reynolds numbers limit

(Theorem Liu-Xie-Y., ’17) Let the initial data (u0 , h0 )(x, y) be compatible and satisfy h10 (x, 0) ≥ δ0

for some constant δ0 > 0.

There is a time T∗ > 0 independent of ε, such that k(uε , Hε )(t, x, y) − (ue , He )(t, x, y) √ √
1 y − ub1 , εub2 , hb1 , εhb2 (t, x, √ )kLxy∞ ∼ ε 2 → 0, ε

Tong Yang

Boundary Layers Theories

ε → 0.

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

 (uε , Hε )(t, x, y) pε (t, x, y)

Well-posedness High Reynolds numbers limit

= (ua , Ha )(t, x, y) + ε(u, h)(t, x, y), = pa (t, x, y) + εp(t, x, y),

Then, the remainder (u, h)(t, x, y) satisfies    ∂t u1 + (uε · ∇)u1 − (Hε · ∇)h1 + (u · ∇)ua1 − (h · ∇)ha1 + ∂x p − µε4u1 = r1ε ,       ∂t u2 + (uε · ∇)u2 − (Hε · ∇)h2 + (u · ∇)ua2 − (h · ∇)ha2 + ∂y p − µε4u2 = r2ε ,     ∂ h + (uε · ∇)h − (Hε · ∇)u + (u · ∇)ha − (h · ∇)ua − κε4h = rε , t 1 1 1 1 1 1 3 ε ε a a   ∂t h2 + (u · ∇)h2 − (H · ∇)u2 + (u · ∇)h2 − (h · ∇)u2 − κε4h2 = r4ε ,       ∇ · u = 0, ∇ · h = 0,     (u1 , u2 , ∂y h1 , h2 )|y=0 = 0, (u, h)|t=0 = 0. Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness High Reynolds numbers limit

Here, riε , i = 1 ∼ 4 are the error terms determined by the approximate solution. By a careful construction of the approximate solution (ua , Ha ), we can have k∂txα riε (t, ·)kL2 ≤ C,

|α| ≤ 3, i = 1, · · · , 4

for some positive constant C independent of ε.

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness High Reynolds numbers limit

(Difficulty) The following four terms in the first and the third equations can not be estimated directly: 
b b a a −1   u2 ∂y u1 − h2 ∂y h1 = ε 2 u2 ∂Y u1 − h2 ∂Y h1 + O(1),   u ∂ ha − h ∂ ua = ε − 21 u ∂ hb − h ∂ ub
+ O(1), 2 y 1 2 y 1 2 Y 1 2 Y 1

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness High Reynolds numbers limit

Denote the stream function ψ of the magnetic field: h1 = ∂y ψ,

h2 = −∂x ψ,

ψ|y=0 ,

ψ|t=0 = 0.

Note that ψ satisfies ∂t ψ + (uε · ∇)ψ − ha2 u1 + ha1 u2 − κ4ψ = ∂y−1 r3ε , r5ε .

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness High Reynolds numbers limit

Set η0p (t, x, y) : = η2p (t, x, y)

: =

up1 t, x, √yε hp1 t, x, √yε



,


p √y ∂ u t, x, y 1 ε
, η1p (t, x, y) := hp1 t, x, √yε


∂y hp1 t, x, √yε
. hp1 t, x, √yε

√ √ Note η0p (t, x, Y), εηip (t, x, Y), εYηip (t, x, Y) = O(1), i = 1, 2 uniformly in ε so that √ η0p (t, x, y), ε∂y η0p (t, x, y), y∂y η0p (t, x, y), √ p εηi (t, x, y), yηip (t, x, y) = O(1), i = 1, 2. Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness High Reynolds numbers limit

(Cancellation)    u(t, x, y) := u1 (t, x, y) − ∂y (η0p · ψ)(t, x, y),   v(t, x, y) := u2 (t, x, y) + ∂x (η0p · ψ)(t, x, y),     h(t, x, y) := h1 (t, x, y) − (η p · ψ)(t, x, y), g(t, x, y) := h2 (t, x, y). 2 Then U(t, x, y) := (u, v, h, g)T (t, x, y) satisfies    ∂t U + A1 (U)∂x U + A2 (U)∂y U + C(U)U + ψD+      (p , p , 0, 0)T − εB4U = Eε , x

y

  ∂x u + ∂y v = 0,     (u, v, ∂y h, g)|y=0 = 0, Tong Yang

U|t=0 = 0. Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Well-posedness High Reynolds numbers limit

Estimates on the coefficients: √ Ai (U) = Aai + εApi + ε A˜ i (U), ˜ C(U) = Ca + ε C(U),

i = 1, 2,

D = Da + εψDp ,

For |α| ≤ 2, i = 1, 2, k∂txα Aai (t, ·)kL∞ , k∂txα Api (t, ·)kL∞ , ky2 ∂txα Dp (t, ·)kL∞ , k∂txα B(t, ·)kL∞ ≤ C, k∂txα Ca (t, ·)kL∞ + ky∂txα Da (t, ·)kL∞ ≤ C, k∂txα Eε (t, ·)kL2 ≤ C.

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Global existence of smooth solutions? (Classical Prandtl equations) 4

Zhang-Zhang (’14): lower bound of life span ε − 3 for 5 outflow velocity of the order ε 3 and perturbation ε; Ignatova-Vicol (’16): almost global with lower bound of the −1 −1 life span e(ε log(ε )) for ε order perturbation of shear flow in the form Guassian error function. Observation: For shear flow as Guassian error function, a g damping can be obtained by using cancellation. Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

MHD boundary layer (Notations) αz2 θα (t, y) = exp ( ), 4

y z= √ ,

Xm (f , τ) = kθα ∂xm f kL2 τ m Mm , kf kXτ,α =

∑ Xm (f , τ),

√ m+1 Mm = . m!

Ym (f , τ) = kθα ∂xm f kL2 τ m−1 mMm .

kf kDτ,α = k∂y f kXτ,α ,

kf kYτ,α =

m≥0

∑ Ym (f , τ). m≥1

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

A family of solutions ¯ 0), (¯uφ (t, y), 0, b,

1 φ (t, y) = √ π

Z y/√hti

exp(− 0

z2 )dz. 4

Theorem ( Xie-Y., ’18) Suppose ¯ X ku0 − u¯ φ (0, y)kXτ0 ,1/2 , kb0 − bk ≤ ε, τ0 ,1/2

ε − 14 X(s)+ < s > 14 D(s))ds. 0 Z t

1

X(t) .< t >− 4 ε, 3 2

1

< s >− 4 D(s)ds ≤ Cε.

0 3 2

1

τ (t) ≥ τ (0) − C < t > 2 ε. • Stabilizing effect of the magnetic field on life span?

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

(Boundary layer in the Prandtl-Hartmann regime) Derived by Gerard-Varet & Prestipino  2   ∂t u1 + u1 ∂x u1 + u2 ∂y u1 = ∂y b1 + ∂y u1 , ∂y u1 + ∂y2 b1 = 0,   ∂x u1 + ∂y u2 = 0, admits the classical Hartmann layer solution u1 = (1 − e−y )¯u,

Tong Yang

u2 = 0.

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

Denote the perturbation by (u, v), note that ω = uy satisfies ∂t ω + (¯u(1 − e−y ) + u)∂x ω−ve−y + v∂y ω = −ω + ∂y2 ω. Corresponding to the cancellation f1 for Prandtl equations by u noting u1yy = −1, set 1y g = u+ω to have ∂t g + (¯u(1 − e−y ) + u)∂x g + v∂y g = −g + ∂y2 g.

Tong Yang

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

(Theorem Xie-Y., ’18) Assume k∂y u10 + u10 − u¯ kX r,β ≤ δ0 1. Then there exists a unique global solution (u1 , u2 ) satisfying kgk2 r,β

Xτ(t),α

Here, Mm =

(m+1)r (m!)β

< c(t),

τ(t) >

τ0 . 2

with r > 1 and β ≥ 1,

Xm = keαy ∂xm gkL2 τ m Mm ,

Tong Yang

kgk2 r,β = Xτ,α

∑ Xm2 . m≥0

Boundary Layers Theories

Prandtl Ansatz Oleinik’s monotonicity condition (2D) Gevrey regularity MHD boundary systems Life span of the solution

THANK YOU!

Tong Yang

Boundary Layers Theories