Abstract
In this paper, we consider the global existence of strong solutions to the three-dimensional Boussinesq equations on the smooth bounded domain Ω. Based on the blow-up criterion and uniform estimates, we prove that the strong solution exists globally in time if the initial L^{2}-norm of velocity and temperature are small. Moreover, an exponential decay rate of the strong solution is obtained.
Highlights
In this paper, we consider the following three-dimensional incompressible Boussinesq equations in the Eulerian coordinates: ⎧⎪⎪⎪⎪⎪⎨θutt μ κ u + u · ∇u + ∇P = θ e3, θ + u · ∇θ = 0,⎪⎪⎪⎪⎪⎩∇u(x·,u0=) =0,u0(x), θ (x, 0) = θ0(x), (1.1)where u = (u1, u2, u3)(x, t), θ = θ (x, t), P(x, t) are unknown functions denoting fluid velocity vector field, absolute temperature, and scalar pressure, t ≥ 0 is time, x ∈ Ω is spatial coordinate. μ is the kinematic viscosity, κ is the thermal diffusivity, and e3 = (0, 0, 1) is the unit vector in the x3 direction
Boussinesq system (1.1) has been widely used in atmospheric sciences and oceanic fluids [9], and there is a huge amount of literature on the well-posedness theory of strong and week solutions for the three-dimensional Boussinesq equations
Theorem 1.2 implies that Navier–Stokes equations admit a unique global strong solution on Ω × [0, T] for any T > 0, provided that there exists a constant ε0 > 0 such that u0
Summary
We consider the following three-dimensional incompressible Boussinesq equations in the Eulerian coordinates:. Theorem 1.2 (Global strong solution) For any given Ki > 0 (i = 1, 2), suppose that the initial data satisfy (u0, θ0) ∈ H01 ∩ H2, div u0 = 0 in Ω, and. Theorem 1.2 implies that Navier–Stokes equations admit a unique global strong solution on Ω × [0, T] for any T > 0, provided that there exists a constant ε0 > 0 such that (1.8). 3, we show that ∇u L∞(0,T;L2) will never blow up in finite time, which combines the blow-up criterion in Theorem 1.1, global existence of strong solution is proved in Theorem 1.2 provided the initial data of velocity and temperature are suitably small under the L2-norm. From Lemmas 2.1–2.3, we can see that Theorem 1.1 is proved
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call