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Almost Global Existence for the 3D Prandtl Boundary Layer Equations Xueyun Lin1,2 · Ting Zhang1

Received: 9 September 2018 / Accepted: 10 December 2019 © Springer Nature B.V. 2019

Abstract In this paper, we prove the almost global existence of classical solutions to the 3D Prandtl system with the initial data which lie within ε of a stable shear flow. Using anisotropic Littlewood-Paley energy estimates in tangentially analytic norms and introducing new linearly-good unknowns, we prove that the 3D Prandtl system has a unique solution with the lifespan of which is greater than exp(ε −1 / log(ε −1 )). This result extends the work obtained by Ignatova and Vicol (Arch. Ration. Mech. Anal. 2:809–848, 2016) on the 2D Prandtl equations to the three-dimensional setting. Keywords Prandtl equations · Almost global existence · Littlewood-Paley theory

1 Introduction We consider the following Prandtl boundary layer equations in R+ × R3+ : ⎧ p  p  ∂t u + u ∂x + v p ∂y + wp ∂z up + ∂x p E = ∂z2 up , ⎪ ⎪ ⎪ ⎪  p  p ⎪ ⎪ p p p E 2 p ⎪ ⎪ ∂t v + u ∂x + v ∂y + w ∂z v + ∂y p = ∂z v , ⎪ ⎪ ⎪ ⎪ p p p ⎪ ⎨ ∂x u + ∂y v + ∂z w = 0,  p p p  ⎪ u , v , w z=0 = (0, 0, 0), ⎪ ⎪ ⎪ ⎪     ⎪ ⎪ lim up , v p = U E (t, x, y), V E (t, x, y) , ⎪ ⎪ z→+∞ ⎪ ⎪ ⎪ ⎪  p p    ⎩ u , v t=0 = u0 (x, y, z), v0 (x, y, z) ,

B T. Zhang

[email protected] X. Lin [email protected]

1

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

2

Present address: College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350108, China

(1.1)

X. Lin, T. Zhang

here and in what follows, (t, x, y, z) ∈ R+ × R × R × R+ , (up , v p ) and wp denote the tangential and normal velocity of the boundary layer flow, the initial data (u0 , v0 ) := (u0 (x, y, z), v0 (x, y, z)) and the far-field (U E (t, x, y), V E (t, x, y)) are given. Furthermore, (U E (t, x, y), V E (t, x, y)) and the given scalar pressure p E (t, x, y) are the tangential velocity field and pressure on the boundary {z = 0} of the Euler flow, satisfying ∂t U E + U E ∂x U E + V E ∂y U E + ∂x p E = 0, (1.2) ∂t V E + U E ∂x V E + V E ∂y V E + ∂y p E = 0, t > 0, (x, y) ∈ R2 . This system (1.1) introduced by Prandtl [33] is a model for the first approximation of the velocity field near the boundary in the zero viscosity limit of the initial boundary value problem of the incompressible Navier-Stokes equations, with the non-slip boundary condition. It is then natural to ask whether solutions to the Navier-Stokes system with zero Dirichlet boundary condition do converge to a solution to the Euler system when the viscosity goes to zero. We refer to [13, 17, 35, 36] and references therein for this justification. One of the key steps to justify the zero viscosity limit is to deal with the well-posedness of the Prandtl system. Up to now, whether the Prandtl equation with general data is well-posed in Sobolev spaces or not is still open except for some special cases, for example, the initial data that are monotonic with respect to the normal variable [1, 32, 34, 43] and in