Boussinesq system with measure forcing
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Mathematische Annalen
Boussinesq system with measure forcing Piotr B. Mucha1 · Liutang Xue2 Received: 13 December 2019 / Revised: 13 August 2020 © The Author(s) 2020
Abstract The paper analyzes the Navier–Stokes system coupled with the convective-diffusion equation responsible for thermal effects. It is a version of the Boussinesq approximation with a heat source. The problem is studied in the two dimensional plane and the heat source is a measure transported by the flow. For arbitrarily large initial data, we prove global in time existence of unique regular solutions. Measure being a heat source limits regularity of constructed solutions and it requires a non-standard framework of inhomogeneous Besov spaces of the L ∞ (0, T ; B sp,∞ )-type. The uniqueness of solutions is proved by using the Lagrangian coordinates. Mathematics Subject Classification Primary 76D03 · 35Q35 · 35Q86
1 Introduction Heat conducting fluids systems are an important part of the fluid mechanics. For the most general models the total energy is usually conserved in time. Viscous fluids generate internal friction and produce thermal effects. Variability of the temperature, on the other hand, creates motion of the fluid. As a basic model the theory considers
Communicated by Y. Giga.
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Piotr B. Mucha [email protected] Liutang Xue [email protected]
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Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, Warsaw, Poland
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Laboratory of Mathematics and Complex Systems (MOE), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China
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P. B. Mucha, L. Xue
the Navier–Stokes–Fourier system for the compressible flows: ⎧ ⎪ ⎪ ⎨∂t ρ + div(ρu) = 0, ∂t (ρu) + div(ρu ⊗ u) − div S(θ, ∇u)+ ∇ p(ρ, θ) = ρ f , ⎪ ⎪ q(θ,∇θ) ⎩∂t ρs(ρ, θ ) + div ρs(ρ, θ )u + div = σ.
(1.1)
θ
In short, ρ, u, θ are sought quantities: the density, velocity and temperature of the fluid, respectively. Functions p(·, ·) and s(·, ·) are the pressure and entropy. The stress tensor S is given in the Newtonian form, the energy flux is given in the Fourier form 2 q = −κ(θ )∇θ and the entropy production σ = θ1 (S : ∇u + κ(θ)|∇θ| ) (for more θ details see [12]). Nowadays mathematics is able to deliver existence of weak solutions [12–14] for the system (1.1), but regular solutions can be obtained for small data [5,24,29] only. The structure of nonlinearities in (1.1) is complex, hence it is natural to look for a reduced version of the system. Taking a low Mach number limit (see [9,15]), we obtain an incompressible limit which takes into account weak thermal effects. This system is known as the Boussinesq approximation ⎧ ⎪ ⎨∂t θ + v · ∇θ − θ = μ, ∂t v + v · ∇v − v + ∇ p = θ ed , ⎪ ⎩ div v = 0,
(1.2)
where d = 2, 3, ed is the last canonical vector of Rd , v is the velocity and θ is the temperature. In simple words, the above system (1.2) is the incompressible Navier– Stokes equations coupled with the heat equation with a drift given by the velocity. Force in the momentum equation is defined by
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