On 2-D Boussinesq equations for MHD convection with stratification effects

Section: Introductionmentioning

confidence: 78%

“…In the first equation of , the θ e 2 represents the buoyancy force. The equation of θ describes the temperature fluctuation in which the term

- u_{2} {scriptN}^{2} (x_{2})

stands for stratification effects about a positive linear mean temperature profile

{scriptN}^{2} (x_{2})

in the direction of gravity ; here, the real quantity

scriptN (x_{2})

is called the buoyancy or Brunt–Väisärä frequency (stratification parameter). Recall that

normal Δ : = (\partial_{1}^{2} + \partial_{2}^{2}), \nabla : = (\partial_{1}, \partial_{2}), \partial_{i} : = \partial_{x_{i}}

with i =1,2, and e 2 :=(0,1).…”

Section: Introductionmentioning

confidence: 99%

“…Compared with the case in , the energy estimates of ( u , θ ) only yield that ( u , θ ) is uniformly bounded in

L^{\infty} (0, T; L^{2} ({double-struckR}^{2})) \times (L^{\infty} (0, T; L^{2} ({double-struckR}^{2})) \cap L^{2} (0, T; H^{1} ({double-struckR}^{2}))

, which is not enough to extend continuously the solution ( u , θ ) beyond the time T . Fortunately, the 2D Euler equations in provide the uniformly estimates (Theorem ) in

L^{\infty} (0, T; L^{2} ({double-struckR}^{2}))

L^{2} (0, T; L^{\infty} ({double-struckR}^{2}))

in order to extend the local solution to the global one (Theorem ), which will be guaranteed by the fact that θ is Hölder continuous as shown previously (Proposition ).…”

Section: Introductionmentioning

confidence: 99%

“…Without loss of generality, we take

scriptN = 1

, which does not change the results of this paper. Then, the system is reformulated as

({, \begin{matrix} array \partial_{t} u + u \cdot \nabla u + \nabla p = θ e_{2}, (t, x) \in (0, \infty) \times R^{2}, \\ array \partial_{t} θ + u \cdot \nabla θ - \nabla \cdot (κ (θ) \nabla θ) = - u_{2}, (t, x) \in (0, \infty) \times R^{2}, \\ array div u = 0, \\ array u (0, x) = u_{0} (x), θ (0, x) = θ_{0} (x), x \in R^{2} . \end{matrix})

Motivated by , our goal is to prove the global well‐posedness of 2D Euler–Boussinesq system , which is stated as follows.…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by [1,18,19], our goal is to prove the global well-posedness of 2D Euler-Boussinesq system (1.1), which is stated as follows.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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2021

Self Cite

In this paper, we establish the nonlinear stability and large time behavior of hydrostatic equilibrium in a uniform magnetic field for the 2D Boussinesq system with magnetohydrodynamics convection on spatial domain normalΩ=𝕋×ℝ with partial dissipation. This is the important part of further work about the stability/instability of shear flow which is the combination of hydrostatic equilibrium and Couette/shear flow in a uniform magnetic field with partial dissipation.

On the global well‐posedness and striated regularity of the 2D Boussinesq‐MHD system

Niu,

Peng,

Wang

2023