Abstract
In this paper, we consider the initial boundary value problem of two-dimensional isentropic compressible Boussinesq equations with constant viscosity and thermal diffusivity in a square domain. Based on the time-independent lower-order and time-dependent higher-order a priori estimates, we prove that the classical solution exists globally in time provided the initial mass |rho _{0}|_{L^{1}} of the fluid is small. Here, we have no small requirements for the initial velocity and temperature.
Highlights
In this paper, we consider the following two-dimensional isentropic compressible Boussinesq equations in the Eulerian coordinates:⎧ ⎪⎪⎨ρt + div(ρu) = 0,⎪⎪⎩(ρ θt u)t + div(ρu ⊗ + (u · ∇)θ – κ u) θ – =μ 0, u + ∇P = ρθ e2, (1.1)where ρ = ρ(x, t), u = (u1, u2)(x, t), θ = θ (x, t) are unknown functions denoting the density, velocity, and temperature of the fluid, respectively, t ≥ 0 is time, x ∈ is spatial coordinate
Remark 1.1 Cho and Kim [11] considered the full Navier–Stokes equations for viscous polytropic fluids with nonnegative thermal conductivity in a three-dimensional space. They proved the existence of unique local strong solutions for all initial data satisfying some compatibility condition, where the initial density need not be positive and may vanish in an open set
Similar to the procedure of [22], we are fortunate to obtain the following inequality under the smallness of ρ0 L1 =: m0, when the momentum equation is related to temperature: sup
Summary
We consider the following two-dimensional isentropic compressible Boussinesq equations in the Eulerian coordinates:. In 2013, Li and Xu [20] considered the Cauchy problem of an inviscid Boussinesq system with temperature-dependent thermal diffusivity They proved the global well-posedness of strong solutions for arbitrarily large initial data in Sobolev spaces. Remark 1.1 Cho and Kim [11] considered the full Navier–Stokes equations for viscous polytropic fluids with nonnegative thermal conductivity in a three-dimensional space They proved the existence of unique local strong solutions for all initial data satisfying some compatibility condition, where the initial density need not be positive and may vanish in an open set. Their results hold for both bounded and unbounded domains. Similar to the procedure of [22], we are fortunate to obtain the following inequality under the smallness of ρ0 L1 =: m0, when the momentum equation is related to temperature: 0≤t≤T
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