On the inviscid limit problem for viscous incompressible ows in the half plane - Approach from vorticity formulation

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1 On the inviscid limit problem for viscous incompressible ows in the half plane - Approach from vorticity formulation Yi-Hsuan Lin Abstract The note is mainly for personal record, if you want to read it, please be careful. This Notes was taken when Prof. Yasunori Maekawa ( Tohoku University, Japan ) visiting Taiwan. I followed his lecture and took the note. Overviews: (NS) t u u u ν u p = t >, x R 2 u = t, x R 2 u = t, x R 2 u = a t =, x R 2 where R 2 = {(x, ) > }, u = (u, u 2 ): velocity, a = (a, a 2 ): initial velocity, p: pressure and ν > : kinematic viscosity (constant). u = on R 2 : no-slip B.C.. ω = u 2 2 u := Rotu: vorticity. Note that Rot u= Rotu, Rot(u, u) = u Rotu (since u = ), Rot( p) =. Then these imply t ω u ω ν ω =, t >, x R 2. R 2, T 2 ), which is well- Vorticity is useful if there is no boundary (e.g. studied. Behavior near the boundary? No-slip B.C. on u is a source of the vorticity on the boundary. Basic question: B.C. on ω? (vorticity formulation of (NS)): Topic. For typical ows, ν is very small (for water 2 C, ν. 6 m 2 /s). Problem: Behavior of u = u (ν) in the limit ν. Formally we obtain the Euler equation: For u E = (u E,, u E,2 ), (E) t u E u E u E p E = t >, x R 2 u E = t, x R 2 u E,2 = t, x R 2. u E = a t =, x R 2 In general, u E, on R 2, while u (ν) = and a = on R 2. Boundary layer appears.

2 Formal estimate of the boundary layer thickness δ (Parndtl 94). t u u u ν u p =, u 2 u 2 =. Near the boundary, we expect u, u, 2 u, t u O(), u 2 O( δ ), 2 2u O( δ 2 ), p O(). From u =, we have 2 u 2 O() u 2 O(δ), u 2 2 u O(δ) O( δ ) = O(), u u O(). ν δ 2 O(), δ = ν/2. Near the boundary u (ν) u (ν) p = (v p, (t, x, (x, X 2 ) where X 2 = ν /2. (P) ν /2 ), u p,2(t, x, t v p, X 2 2 v p, v p x v p, π p =, t >, x R 2 x v p =, X2 π p =, v p t= = v p X2= =, lim X2 v p, = u E, x2=, lim X2 π p = p E x2= )), x = ν/2 u E, We want to justify u (ν) ν /2 u p (ν) at least locally in time. Sammartino-, ν/2 Caisch (CMP, 98): Given data (initial velocity) a is analytic. This result is not applicable for, e.g., a C,σ(R 2 ). u E, Now, we justify u (ν) ν /2 u (ν) when the initial vorticity b =Rota p, ν/2 satises dist( R 2,suppb)> and Sobolev regularity: Topic 2.. Vorticity formulation for NS equation in R 2. Derivation Vorticity eld ω = u 2 2 u and u = imply u = ω, where = ( 2, ). Biot-Savart { law gives u = ( D ) ω, where h = ( D ) f h = f in R 2 is the solution of, Note that h = on R 2 ( D ) ω =, Rot ( D ) ω = ω, γ R 2 ( D ) ω = (trace). (Derivation of vorticity B.C.) Requirement: γ R 2 2 ( D ) ω =. In R 2, ω satises t ω u ω ν ω = in R 2, therefore, = t γ R 2 e ( D ) ω = γ R 2 2 ( D ) t ω = γ R 2 2 ( D ) (ν ω u ω) = γ R 2 2 ( D ) (ν (ω ω har ) u ω) = νγ R 2 (ω ω har ) γ R 2 ( D ) (u ω) = νγ R 2 2 ω ν( 2 ) /2 γ R 2 ω γ R 2 2 ( D ) (u ω) 2

3 where ω har satises { ω har = in R 2 ω har = ω on R 2 and the last equation comes from the Dirichlet-to-Neumann map on the half-plane property. ν( 2 ω( ) 2 /2 ω) = ( D ) (u ω) on R 2. (V) t ω M(ω, ω) ν ω = t >, x R 2 ν( 2 ω ( ) 2 /2 ω) = N(ω, ω) t >, x R 2 ω = Rota t =, x R 2 where M(f, g) = J(f) g, J(f) = ( D ) f, N(f, g) = γ R 2 J (M(f, g)). Then (NS) (V) γ R 2 a = (Note that we have used the fractional Laplacian, so the nonlocal property automatically holds)..2 Solution formula for linearization t ω ν ω = f t >, x R 2 (LV) ν( 2 ω ( ) 2 /2 ω = g t >, x R 2 ω = b y =, x R 2 and we have ( 2 ) /2 ω(ξ) = ξ ω(ξ, ) and b = for simplicity, where ω(s, ξ, ) = (2π) /2 R ω(t, x)e ixξ ts dx dt. Then we have 2 2 ω ( s ν ξ2 ) ω = ν f > 2 ω ξ ω = g ν =, ω ( ). G(t, x) = /4t, E(x) = 4πt e x 2 2π log x, (h h 2 ) is the standard convolution, and (h h 2 ) = h R 2 x (x y )h 2 (y)dy, where y = (y, y 2 ). h (gh = R )(x) 2 R h(x y, )g(y )dy, Γ(t, x) = 2( 2 ( ) 2 /2 2 )(E G(t))(x), e t N f = G(t) f G(t) f, Γ() f = lim t Γ(t) f = 2( 2 ( ) 2 /2 2 )E f. Theorem.. The solution formula for (LV) is given by ω(t) = e νtb b Γ() b ˆ t where e νtb := e νt N Γ(νt). ˆ t Γ() (f(s) g(s)h R 2 )ds e ν(t s)b (f(s) g(s)h R 2 )ds Refer to Solution formula for the Stokes equation: Solonnikov (AMS Transl 68(, Ukai (CPAM 87). Proposition.2. If g = γ R 2 J (f) = γ R 2 2 ( D ) f, then Γ() (f gh R 2 ) = in R 2. In particular, if γ R 2 J (b) =, then Γ() b = in R 2. (Note: a =, b=rota, γ R 2 a = γ R 2 J(b) =. 3

4 Proof. (Prop.2) E (f gh R 2 ) = if x R 2. = ˆ ˆ R 2 R 2 ˆ = ˆ E(x y )f(y)dy E(x y ) ( D ) fdνy R 2 y E(x y ) y ( D ) fdy R 2 y E(x y )f(y)dy = Proposition.3. Γ(νt) b Γ() b = ν t IG(νs)ds b in R2 and I = 2( 2 ( 2 ) /2 2 ). Proof. RHS= t I( R 2) ξ (G(νs))ds b = IE G(νt) b IE b..3 Properties of {e tb } t Set Ẇ.q,σ (R2 ) = C,σ (R2 ) Ẇ,q,σ (R2 )}. a Lq, < q <. X q = {Rota L q (R 2 ) a Theorem.4. {e tb } t> denes a C -analytic semigroup in X q. Moreover, its generator B q is given by D(B q ) = {f X q W 2,q (R 2 ) γ R 2 ( 2 f ( 2 ) /2 f) = }, and B q f = f when f D(B q ). Moreover, 2 f L q C B q f Lq, f D(B q ). 2 Inviscid limit problem 2. Result t u (ν) u (ν) u (ν) ν u (ν) p (ν) = t (, T ), x R 2 u = t (, T ), x R 2 (NS) u = t (, T ), x R 2 u = a t =, x R 2 v p : velocity of the Prandtl ow, ṽ p = (ṽ p,, ṽ p,2 ), ṽ p, = v p, γ R 2 u E.,ṽ p.2 = ṽ p, dy 2. (Velocity of the modied Prandtl ow) ũ (ν) p, (t, x) = ṽ p,(t, x, ), ũ(ν) ν/2 p,2 (t, x) = ν/2 ṽ p,2 (t, x, ). ν/2 Theorem 2.. (M., to appear in CPAM) Let a Ẇ,p,σ (R2 ) W 4.2 (R 2 ) for some < p < 2 and b = a 2 2 a W 4, (R 2 ) W 4,2, (R 2 ). Assume that d =dist( R 2,suppb)>. Then T, C > such that sup u (ν) (t) u E (t) ũ (ν) p (t) L Cν /2 <t<t for suciently small ν >. The time T is estimated from below as T c min{d, } with c > depending only on b W 4, W,4,2. 4

5 (Away the boundary, (NS) is estimated by Euler equation,; Near the boundary, (NS) is estimated by Prandtl equation) We assume < d, for simplicity. Note: ω E =Rotu E C ([, T ] R 2 ) L (, T ; W 4, W 4,2 ) T > satises t ω E u E ω E =. In particular, <t<t suppω E(t) {x R 2 d 2 } for T = c d with some C > depending only on b W 4, W 4,2. Key observations for the proof of theorem 2.: () Analyticity of the data near the boundary (from the Prandtl) Local solvability of the Prandtl equation (cf. Sammartino-Caisch (CMP '98), Lombardo-Cannone-Sammartino (SIMA JMA '3), Kukavica-Vicol (CMS '3). (2) Exponential smallness in ν of the vorticity in the region between the boundary layer and suppω E Small direct interaction between the vorticity of the outer ow and the vorticity created on the boundary. We make the ansatz ω (ν) = ω E R ν ω p R remainder terms, (R f)(t, x) = ν (V E ) ω(ν) ν IP ). ν/2 ν f(t,, /2 { t ω E M(ω E, ω E ) = w E t= = Rota ω(ν) IE, the last two are where M(f, g) = J(f) g, J(f) = ( D ) f. t ω p X 2 2 ω p = v p X ω p t >, x R 2 (V P ) t ω p = v p X ω p dy 2 N(ω E, ω E ) on the boundary ω p t= = 2.2 Function spaces ϕ (µ,ρ) µ, ρ, θ : parameters. ξ : wave number in x direction. X 2 = ν /2. (α) = max(α, ), α R. ( (µ ν /2 X 2 ) ξ = exp ( 4 (µ x2 ) ϕ (µ,ρ) E,ν (ξ, X 2 ) = exp 4 ρx 2 2 ˆf(ξ, ) = F x ξ [f(, ](ξ ), ˆf L p f (µ,p) X = k=, ϕ(µ,ρ) X k/2 2 IP ν, regularity. f X (µ,p) p,2 := f X (µ,p) IP,2 ), ξ θ ν (d 8 ) 2 ξ L q x 2 = ( ˆf L 2 ξ L k ), R ( ˆf(ξ, ) q d ) p/q dξ ) /p,, f X (µ,p) IP ν,j, j =, 2: additional Sobolev (ν = ), f X (µ θ) IEν, ϕ (,θ) E,ν f L, f, j =, 2: additional Sobolev regularity. x X (µ,θ) IEν,j = ϕ (µ,θ) E,ν ˆf L 2 ξ,l 2 5

6 Theorem 2.2. C, T, µ, ρ, θ > such that u (ν) = J(ω (ν) ). 2.3 Biot-Savart law sup ω p (t) (µ p C, <t<t X t ) p sup ( ω (ν) IP (t) <t<t X (µ, ρ ω (ν) t ) IE (t) ) Cν /2. IP ν, X (µ, θ t ) IE ν, Lemma 2.3. We have F( ( D ) f)(ξ, ) = iξ 2 ξ { e ξ (x2 z2) ( e 2 ξ z2 ) ˆf(ξ, z 2 )dz 2 e ξ (z2 x2) ( e 2 ξ x2 ) ˆf(ξ, z 2 )dz 2 }, F( 2 ( D ) f)(ξ, ) = ˆ x2 2 { e ξ (x2 z2) ( e 2 ξ z2 ) ˆf(ξ, z 2 )dz 2 e ξ (z2 x2) ( e 2 ξ x2 ) ˆf(ξ, z 2 )dz 2 }. Note: If suppf {x R 2 m > }, then for < m, we have F[ j ( d ) f](ξ, ) c m e ξ (z2 x2) ˆf(ξ, z 2 ) dz 2 ˆ e ξ (m x2) ˆf(ξ, z 2 ) dz 2. Key ingredient of the proof: () Solution formula for linearized (vorticity) problem. Weighted norm estimates for linear and bilinear maps. (2) Pointwise estimate for the Green function of t ν u under the Neumann boundary condition. (3) Abstract Cauchy-Kowalevsky theorem. Key propositions Proposition 2.4. Let <s<t, µ >, < ρ, θ. Then Proof. Set g(t, X 2 ) = Y 2 ). Then R ν e ν(t s) N R f ν X (µ, p f t ) (µ, p, IPν,o X s ) IP ν, e ν(t s) N f f X (µ, θ t ) IE ν, X (µ, θ s ) IEν,o (4πt) /2 e x2 2 /4t, g N (t, X 2, Y 2 ) = g(t, X 2 Y 2 ) g(t, X 2 F[R ν e ν(t s) N R ν f](ξ, X 2 ) = e ν(t s)ξ2 = e ν(t s)ξ2 e ξ ϕ (µ,) 6 ˆfdY 2.. g N (t s, X 2, Y 2 ) ˆf(ξ, Y 2 )dy 2 g N (t s, X 2, Y 2 )e 4 (µ ν./2 Y 2)

7 Therefore, ϕ (µ, p t ) F[R ν e ν(t s) N R ν f](ξ, X 2 ) By e p t X2 2 (t s) /2 e ν(t s)ξ2 X 2 Y 2 2 4(t s) e (µ ν /2 X2 ) (µ ν /2 Y 2 ) 4 ξ ϕ (µ,) ˆf dy 2. µ ν /2 X 2 ) ξ (µ ν /2 Y 2 ) ξ ν /2 X 2 Y 2 ξ (µ ν /2 Y 2 ) ξ ν(t s)ξ 2 X 2 Y 2 2 e ρ t X2 2 (t s) /2 Young-type inequality with weights gives 4(t s) e 3 4 ν(t s)ξ2 e X 2 Y 2 2 8(t s) ϕ ˆf dy2, ϕ (µ, p t ) F[R ν e ν(t s) N R f] L ϕ (ν, ρ s ) ˆf(ξ ) ν X L 2 X2 and ϕ (µ, p t ) F[R ν e ν(t s) N R ν f] L 2 ξ L X 2 ϕ (ν, ρ s ) ˆf(ξ ) L 2 ξ L X 2. ω (ν) = ω E R ω p R ω(ν) ν ν IP ω(ν) IE, the last term is the remainder of the outer ow. t ω IE ν ω IE M(ω (ν), ω IE ) = H (ν) (V IEν ) 2 ω IE = on the boundary ω IE t= = and M(ω (ν), ω IE ) = u (ν) ω IE, ν( 2 ω (ν) ( 2 ) /2 ω (ν) ) =Nonlinear 2 ω IE =. H (ν) = M(R ν ω(ν) IP ω(ν) IE, ω E) ν ω E = J(R ν ω(ν) IP ω(ν) IE ) ω E ν ω E, <t<t supph (ν) (t) {x R 2 d 2 }. Proposition 2.5. ω (ν) IE = t P (ν) (t, x; s, y)h (ν) (s, y)dyds, < P (ν) (t, x; s, y) R 2 u (ν) u ( (ν) 2πν(t s) exp ( x y ) t s u(ν) (t) L ) 2. 4ν(t s) (From Carlen-Loss '95 Duke) Exponential decay in ν of ω IE near boundary is observed as follows: ˆ t ˆ ( ) ω (ν) IE (t, x) d 2 ν(t s) exp H (ν) dyds. ( x y t s u(ν) (t) L ) 2 4ν(t s) 7

8 If d 4 and < t < T = O(d ), sup <t<t u (ν) (t) L C, then ω IE (t, x) d 2 ˆ d 2 t e 8νt H (ν) L ds. For the third part, Cauchy-Kowalevsky theorem. (Nirenberg '72, Nishida '77...) 8