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- Title
Linear and nonlinear analysis of the viscous Rayleigh–Taylor system with Navier-slip boundary conditions.
- Authors
Nguyễn, Tiến-Tài
- Abstract
In this paper, we are interested in the linear and the nonlinear Rayleigh–Taylor instability for the gravity-driven incompressible Navier–Stokes equations with Navier-slip boundary conditions around a smooth increasing density profile ρ 0 (x 2) in a slab domain 2 π L T × (- 1 , 1) ( L > 0 , T is the usual 1D torus). The linear instability study of the viscous Rayleigh–Taylor model amounts to the study of the following ordinary differential equation on the finite interval (- 1 , 1) , 0.1 - λ 2 [ ρ 0 k 2 ϕ - (ρ 0 ϕ ′) ′ ] = λ μ (ϕ (4) - 2 k 2 ϕ ′ ′ + k 4 ϕ) - g k 2 ρ 0 ′ ϕ , with the boundary conditions 0.2 ϕ (- 1) = ϕ (1) = 0 , μ ϕ ′ ′ (1) = ξ + ϕ ′ (1) , μ ϕ ′ ′ (- 1) = - ξ - ϕ ′ (- 1) , where λ > 0 is the growth rate in time, g > 0 is the gravity constant, k is the wave number and two Navier-slip coefficients ξ ± are nonnegative constants. For each k ∈ L - 1 Z \ { 0 } , we define a k-supercritical regime of viscosity coefficient, i.e. μ > μ c (k , Ξ) with Ξ = (ξ + , ξ -) to describe a spectral analysis by adapting an operator method of Lafitte and Nguyễn (Water Waves 4:259–305, 2022) and to prove that there are infinite nontrivial solutions (λ n , ϕ n) n ≥ 1 of (0.1)–(0.2) with λ n → 0 as n → ∞ and ϕ n ∈ H 4 ((- 1 , 1)) . Hence, we prove the linear Rayleigh–Taylor instability for any viscosity coefficient μ > 0 . As a by-product, based on the existence of infinitely many normal modes of the linearized problem, we construct a wide class of initial data to the nonlinear equations, being inspired by the previous framework of Guo–Strauss (Commun Pure Appl Math 48:861–894, 1995) and of Grenier (Commun Pure Appl Math 53:1067–1091, 2000) with a refinement, to prove the nonlinear Rayleigh–Taylor instability in a high regime of viscosity coefficient, namely μ > 3 sup k ∈ L - 1 Z \ { 0 } μ c (k , Ξ) .
- Subjects
RAYLEIGH-Taylor instability; NONLINEAR analysis; NAVIER-Stokes equations; WATER waves; LINEAR statistical models; ORDINARY differential equations; C*-algebras
- Publication
Calculus of Variations & Partial Differential Equations, 2024, Vol 63, Issue 2, p1
- ISSN
0944-2669
- Publication type
Article
- DOI
10.1007/s00526-023-02634-z