Uniqueness and regularity for the 3D Boussinesq system with damping
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iqueness and regularity for the 3D Boussinesq system with damping Yong-Ho Kim1 · Kwang-Ok Li1 · Chol-Ung Kim2 Received: 4 May 2020 / Accepted: 10 October 2020 © Università degli Studi di Ferrara 2020
Abstract This paper is concerned with the Boussinesq system with a damping term and the homogeneous Dirichlet boundary conditions in 3D bounded domains. For a certain range of parameters, we prove that the weak solution is unique if the temperature belongs to L ∞ (0, T ; L 3 ()). Also, the global existence of strong solutions to the problem is proved. Keywords Boussinesq system with damping · Uniqueness · Regularity Mathematics Subject Classification 35Q30 · 76D05
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Yong-Ho Kim [email protected] Kwang-Ok Li [email protected] Chol-Ung Kim [email protected]
1
Department of Mathematics, University of Science, Pyongyang, Democratic People’s Republic of Korea
2
Department of Mathematics, Wonsan University of Agriculture, Wonsan, Democratic People’s Republic of Korea
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ANNALI DELL’UNIVERSITA’ DI FERRARA
1 Introduction In this paper we consider the Boussinesq system with a nonlinear damping term ⎧ β−1 u − ∇ p + f (θ ), x ∈ , t ∈ (0, T ], ⎪ ⎪u t + (u · ∇)u − νu = −α|u| ⎪ ⎪ ⎪ ⎪θt + u · ∇θ − κθ = h, x ∈ , t ∈ (0, T ], ⎨ (1.1) divu = 0, x ∈ , t ∈ (0, T ], ⎪ ⎪ ⎪ u(x, t) = 0, θ (x, t) = 0, x ∈ ∂, t ∈ (0, T ], ⎪ ⎪ ⎪ ⎩u(x, 0) = u (x), θ (x, 0) = θ (x), x ∈ , 0 0 where T > 0, α > 0 and β > 0 are constants and h(x, t) is the heat source. ⊂ R3 is an open bounded set with the boundary ∂ smooth enough. The unknown functions u(x, t), p(x, t) and θ (x, t) are the velocity field, the pressure and the absolute temperature, respectively. ν is the constant kinematic viscosity and κ is the thermal conductivity. The forcing term is given by f (θ ) = −ρ(θ )g/ρ0 , where g is the acceleration due to gravity and ρ(θ ) is the varying density in the forcing term and ρ0 stands for a reference density constant corresponding to a reference temperature, which can be taken to be the mean temperature in the flow or the temperature at the boundary. When α = 0, the system (1.1) is known in the literature as the Oberbeck– Boussinesq, or Boussinesq system which describes the motion of a fluid driven by buoyancy forces and simplified by assuming the motion is isochoric. For a detailed discussion on the Boussinesq system, see e.g. [8,15] and the references therein. The Boussinesq system has been widely studied by several authors from a theoretical interest. First, in [4], they proved the existence of a unique, local in time, weak solution in Rn ×(0, T ], where the advection for the temperature equation is satisfied with an extra term which is given. These results were improved or generalized by many authors (see e.g. [5,7,9,13]). The term −α|u|β−1 u in (1.1) is said to be a damping term or a absorption term. When α > 0, the Navier-Stokes equations with damping describe the flow with the resistance to the motion such as porous media flow and friction or absorption effects (see e.g. [1,3] and references therein). From a mathematical vi
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