Global solvability to the high-dimensional inhomogeneous Boussinesq equations with zero thermal diffusion
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Global solvability to the high-dimensional inhomogeneous Boussinesq equations with zero thermal diffusion Zhuan Ye Abstract. In this paper, we prove the global existence and uniqueness of the strong solution to the high-dimensional inhomogeneous incompressible Boussinesq equations with zero thermal diffusion. Mathematics Subject Classification (2010). 35Q35, 35B65, 76N10, 76D05. Keywords. Boussinesq equations, Vacuum, Inhomogeneous, Global strong solution.
1. Introduction In this paper, we consider the Cauchy problem of the following high-dimensional inhomogeneous incompressible Boussinesq equations with zero thermal diffusion ⎧ ∂t ρ + div(ρu) = 0, ⎪ ⎪ ⎪ ⎪ ⎨ ∂t (ρu) + div(ρu ⊗ u) + (−Δ)α u + ∇p = ρθen , ∂t θ + (u · ∇)θ = 0, (1.1) ⎪ ⎪ ∇ · u = 0, ⎪ ⎪ ⎩ ρ(x, 0) = ρ0 (x), u(x, 0) = u0 (x), θ(x, 0) = θ0 (x), where x ∈ Rn , t > 0, the unknowns are the density ρ, the velocity u = (u1 , u2 , . . . , un ), the temperature θ and the scalar pressure p. Here ρ0 , u0 and θ0 are the prescribed initial data for the density, the velocity and the temperature. en = (0, . . . , 1)⊥ is the unit vertical vector. The power α > 0 and the fractional Laplacian operator (−Δ)α is defined via the Fourier transform α f (ξ) = |ξ|2α f(ξ), (−Δ)
where f is the Fourier transform of f . The Boussinesq equations describe the motion of lighter or denser incompressible fluid under the influence of gravitational forces and have important roles in the atmospheric sciences [1,2]. When ρ is a constant, (1.1) reduces to the homogeneous incompressible Boussinesq equations with zero thermal diffusion, which possess a unique global smooth solution as long as α ≥ 12 for n = 2 (see [3]) and α ≥ 12 + n4 for n ≥ 3 (see [4–6]). When ρ is not a constant, we call (1.1) as the density-dependent Boussinesq equations or inhomogeneous Boussinesq equations. Under the assumption that the initial density is bounded away from zero, Qiu and Yao [7] showed the local existence and uniqueness of global strong solutions of (1.1) with α = 0. However, there are few results concerning strong solvability of the density-dependent Boussinesq equations with initial data allowing vacuum. Very recently, after adding the dissipation Δθ in the θ equation, Zhong [8] proved the global existence of strong solutions to the Cauchy problem of (1.1) with α = 1 and nonnegative density in two-dimensional space. Here we want to point out that with suitable modification of [8], the global existence of strong solutions to (1.1) with 0123456789().: V,-vol
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Z. Ye
ZAMP
α=1 still holds true in two-dimensional case. However, for the case α = 1, the global existence of strong or smooth solutions of (1.1) with the general initial data in higher dimensions remains open. One main difficulty is that the Laplacian dissipation is not enough to control the nonlinearity. Consequently, in order to obtain the global strong solution for the general initial data, a natural and interesting generalization is to replace the Laplacian operator by the
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