Lagrangian Approach to Global Well-Posedness of the Viscous Surface Wave Equations Without Surface Tension
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Lagrangian Approach to Global Well‑Posedness of the Viscous Surface Wave Equations Without Surface Tension Guilong Gui1 Received: 23 November 2019 / Revised: 30 April 2020 / Accepted: 21 May 2020 © Peking University 2020
Abstract In this paper, we revisit the global well-posedness of the classical viscous surface waves in the absence of surface tension effect with the reference domain being the horizontal infinite slab, for which the first complete proof was given in Guo–Tice [Anal. PDE 6,1429–1533 (2013)] via a hybrid of Eulerian and Lagrangian schemes. The fluid dynamics are governed by the gravity-driven incompressible Navier– Stokes equations. Even though Lagrangian formulation is most natural to study free boundary value problems for incompressible flows, few mathematical works for global existence are based on such an approach in the absence of surface tension effect, due to breakdown of Beale’s transformation. We develop a mathematical approach to establish global well-posedness based on the Lagrangian framework by analyzing suitable “good unknowns” associated with the problem, which requires no nonlinear compatibility conditions on the initial data. Keywords Viscous surface waves · Lagrangian coordinates · Global well-posedness Mathematics Subject Classification 35Q30 · 35R35 · 76D03
1 Introduction 1.1 Formulation in Eulerian Coordinates We consider in this paper the global existence of time-dependent flows of a viscous incompressible fluid in a moving domain Ω(t) with an upper free surface ΣF (t) and a fixed bottom ΣB
* Guilong Gui [email protected] 1
Center for Nonlinear Studies, School of Mathematics, Northwest University, Xi’an 710069, China
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⎧𝜕t u + (u ⋅ ∇)u + ∇p − 𝜈Δ u = −g e1 in Ω(t), ⎪∇ ⋅ u = 0 in Ω(t), ⎪ ⎨(p 𝕀 − 𝜈𝔻(u))n(t) = patm n(t) on ΣF (t), ⎪V(ΣF (t)) = u ⋅ n(t) on ΣF (t), ⎪u� = 0, ⎩ ΣB
(1.1)
where we denote n(t) the outward-pointing unit normal on ΣF (t) , 𝕀 the 3 × 3 identity matrix, (𝔻u)ij = 𝜕i uj + 𝜕j ui twice the symmetric gradient of the velocity u = (u1 , u2 , u3 ) , ui = ui (x1 , x2 , x3 ) , i, j = 1, 2, 3 , the constant g > 0 stands for the strength of gravity, and 𝜈 > 0 is the coefficient of viscosity. We denote V(ΣF (t)) the outer-normal velocity of the free surface ΣF (t) . The tensor (p 𝕀 − 𝜈𝔻(u)) is known as the viscous stress tensor. Equation (1.1)1 is the conservation of momentum, where gravity is the only external force, which points in the negative x1 direction (as the vertical direction); the second equation in (1.1) means the fluid is incompressible; Eq. (1.1)3 means the fluid satisfies the kinetic boundary condition on the free boundary ΣF (t) , where patm stands for the atmospheric pressure, assumed to be constant. The kinematic boundary condition (1.1)4 states that the free boundary ΣF (t) is moving with speed equal to the normal component of the fluid velocity; (1.1)5 implies that the fluid is no-slip, no-penetrated on the fixed bottom boundary. Here the effect of surface tension is neglected on the free surface. If we denote 𝜁
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