Almost Global Existence for the Prandtl Boundary Layer Equations

Ignatova, Mihaela; Vicol, Vlad

doi:10.1007/s00205-015-0942-2

Cited by 112 publications

(101 citation statements)

References 39 publications

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“…Motivated by Ignatova and Vicol, to deal with better, we write u 1 ( t , x , y ) and b 1 ( t , x , y ) as perturbations u ( t , x , y ) and b ( t , x , y ) of the lifts κ 1 ϕ ( t , y ) and κ 2 ϕ ( t , y ), respectively, via

u_{1} (t, x, y) = κ_{1} ϕ (t, y) + u (t, x, y), b_{1} (t, x, y) = κ_{2} ϕ (t, y) + b (t, x, y),

where

ϕ (t, y) = \frac{1}{\sqrt{π μ}} \int_{0}^{y / \sqrt{⟨ t ⟩}} \exp (- \frac{z^{2}}{4 μ}) d z

(please see the details in Appendix ). Denote by

w (t, x, y) = \partial_{y} u (t, x, y), h (t, x, y) = \partial_{y} b (t, x, y), ⟨ t ⟩ = 1 + t .

Then we introduce new linearly good unknowns

g_{1} (t, x, y) = w (t, x, y) + \frac{y}{2 μ ⟨ t ⟩} u (t, x, y), g_{2} (t, x, y) = h (t, x, y) + \frac{y}{2 μ ⟨ t}

…”

Section: Resultsmentioning

confidence: 99%

“…In order to define the functional spaces of the solution, motivated by Ignatova and Vicol and Zhang and Zhang, it is convenient to define the following spaces. Firstly, let us recall from Zhang and Zhang that

{normalΔ}_{k}^{h} u {= ℱ}^{- 1} false(ϕ false(2^{- k} false| ξ false| false) \overset{u}{true} false), S_{k}^{h} u {= ℱ}^{- 1} false(χ false(2^{- k} false| ξ false| false) \overset{u}{true} false),

where and in all that follows

ℱ u

and

\overset{u}{true}

always denote the partial Fourier transform of the distribution u with respect to x variable,

\overset{u}{true} false(ξ, y false) {= ℱ}_{x \to ξ} false(u false) false(ξ, y false)

, χ ( τ ), ϕ ( τ ) are smooth functions such that

\begin{matrix} Supp ϕ \subset ({}, τ \in double-struckR false/ \frac{3}{4} \leq false| τ false| \leq \frac{8}{3}) and \forall τ > 0, \sum_{j \in double-struckZ} ϕ false(2^{- j} τ false) = 1, \\ Supp χ \subset ({}, τ \in double-struckR false/ false| τ false| \leq \frac{4}{3}) and χ false(τ false) + \sum_{j \geq 0} ϕ false(2^{- j} τ false) = 1 . \end{matrix}

…”

Section: Resultsmentioning

confidence: 99%

“…Recently, P Zhang and ZF Zhang proved that the d ‐dimensional Prandtl system has a unique solution on the time interval that is greater than

ϵ^{- \frac{4}{3}}

for d = 2 or d = 3. Later, the life span can be extended at least up to time

T_{ϵ} \geq \exp false(ϵ^{- 1} false/ \ln false(ϵ^{- 1} false) false)

for d = 2 in Ignatova and Vicol . Therefore, it is interesting to study whether the 2D MHD boundary layer system can also exist almost globally.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Almost global existence for 2D magnetohydrodynamics boundary layer system

Lin

Zhang

2018

Math Methods in App Sciences

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In this paper, we study the almost global existence and uniqueness of the small solution for the two‐dimensional (2D) magnetohydrodynamics (MHD) boundary layer system with small initial data, which are analytic in the x variable. If the initial data lie within ϵ of a stable shear flow, then we can extend the life span at least up to time Tϵ≥expfalse(ϵ−1false/lnfalse(ϵ−1false)false). The analytical estimates and the nonlinear terms of this system with the special structure play essential roles in proving this result.

show abstract

u_{1} (t, x, y) = κ_{1} ϕ (t, y) + u (t, x, y), b_{1} (t, x, y) = κ_{2} ϕ (t, y) + b (t, x, y),

where

ϕ (t, y) = \frac{1}{\sqrt{π μ}} \int_{0}^{y / \sqrt{⟨ t ⟩}} \exp (- \frac{z^{2}}{4 μ}) d z

(please see the details in Appendix ). Denote by

w (t, x, y) = \partial_{y} u (t, x, y), h (t, x, y) = \partial_{y} b (t, x, y), ⟨ t ⟩ = 1 + t .

Then we introduce new linearly good unknowns

g_{1} (t, x, y) = w (t, x, y) + \frac{y}{2 μ ⟨ t ⟩} u (t, x, y), g_{2} (t, x, y) = h (t, x, y) + \frac{y}{2 μ ⟨ t}

…”

Section: Resultsmentioning

confidence: 99%

{normalΔ}_{k}^{h} u {= ℱ}^{- 1} false(ϕ false(2^{- k} false| ξ false| false) \overset{u}{true} false), S_{k}^{h} u {= ℱ}^{- 1} false(χ false(2^{- k} false| ξ false| false) \overset{u}{true} false),

where and in all that follows

ℱ u

and

\overset{u}{true}

always denote the partial Fourier transform of the distribution u with respect to x variable,

\overset{u}{true} false(ξ, y false) {= ℱ}_{x \to ξ} false(u false) false(ξ, y false)

, χ ( τ ), ϕ ( τ ) are smooth functions such that

\begin{matrix} Supp ϕ \subset ({}, τ \in double-struckR false/ \frac{3}{4} \leq false| τ false| \leq \frac{8}{3}) and \forall τ > 0, \sum_{j \in double-struckZ} ϕ false(2^{- j} τ false) = 1, \\ Supp χ \subset ({}, τ \in double-struckR false/ false| τ false| \leq \frac{4}{3}) and χ false(τ false) + \sum_{j \geq 0} ϕ false(2^{- j} τ false) = 1 . \end{matrix}

…”

Section: Resultsmentioning

confidence: 99%

“…Recently, P Zhang and ZF Zhang proved that the d ‐dimensional Prandtl system has a unique solution on the time interval that is greater than

ϵ^{- \frac{4}{3}}

for d = 2 or d = 3. Later, the life span can be extended at least up to time

T_{ϵ} \geq \exp false(ϵ^{- 1} false/ \ln false(ϵ^{- 1} false) false)

for d = 2 in Ignatova and Vicol . Therefore, it is interesting to study whether the 2D MHD boundary layer system can also exist almost globally.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Almost global existence for 2D magnetohydrodynamics boundary layer system

Lin

Zhang

2018

Math Methods in App Sciences

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show abstract

“…However, the class of initial data for blowup solutions in [38] must have O(1) size, and, hence, it is still not clear whether global existence holds for sufficiently small data or not. Recently an important progress has been achieved in this direction, and long-time well posedness is established in [154,72] for small solutions in the analytic functional framework. In [154] the life span of local solutions is estimated from below to be of order O(ǫ 3 ) and the initial data u P 0,1 is O(ǫ) in a suitable norm measuring tangential analyticity.…”

Section: Well-posedness Results For the Prandtl Equationsmentioning

confidence: 99%

“…In [154] the life span of local solutions is estimated from below to be of order O(ǫ 3 ) and the initial data u P 0,1 is O(ǫ) in a suitable norm measuring tangential analyticity. In [72], almost global existence is established if the smallness condition on the uniform Euler flow is removed, and the life span is then estimated from below as O(exp(− 1 ǫ log ǫ )), 0 < ǫ ≪ 1, when u P 0,1 is O(ǫ).…”

Section: Well-posedness Results For the Prandtl Equationsmentioning

confidence: 99%

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

Maekawa¹,

Mazzucato²

2016

Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

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The validity of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations modeling viscous incompressible flows converge to solutions of the Euler equations modeling inviscid incompressible flows as viscosity approaches zero, is one of the most fundamental issues in mathematical fluid mechanics. The problem is classified into two categories: the case when the physical boundary is absent, and the case when the physical boundary is present and the effect of the boundary layer becomes significant. The aim of this article is to review recent progress on the mathematical analysis of this problem in each category.

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MHD Boundary Layers Theory in Sobolev Spaces Without Monotonicity I: Well‐Posedness Theory

Liu

Xie

Yang

2018

Comm Pure Appl Math

104

120

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We study the well‐posedness theory for the MHD boundary layer. The boundary layer equations are governed by the Prandtl‐type equations that are derived from the incompressible MHD system with non‐slip boundary condition on the velocity and perfectly conducting condition on the magnetic field. Under the assumption that the initial tangential magnetic field is not zero, we establish the local‐i‐time existence, uniqueness of solutions for the nonlinear MHD boundary layer equations. Compared with the well‐posedness theory of the classical Prandtl equations for which the monotonicity condition of the tangential velocity plays a crucial role, this monotonicity condition is not needed for the MHD boundary layer. This justifies the physical understanding that the magnetic field has a stabilizing effect on MHD boundary layer in rigorous mathematics. © 2018 Wiley Periodicals, Inc.

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Almost Global Existence for the Prandtl Boundary Layer Equations

Cited by 112 publications

References 39 publications

Almost global existence for 2D magnetohydrodynamics boundary layer system

Almost global existence for 2D magnetohydrodynamics boundary layer system

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

MHD Boundary Layers Theory in Sobolev Spaces Without Monotonicity I: Well‐Posedness Theory

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