On the attractor for the semi-dissipative Boussinesq equations

Biswas, Animikh; Foiaş, Ciprian; Larios, Adam

doi:10.1016/j.anihpc.2015.12.006

Cited by 28 publications

(21 citation statements)

References 44 publications

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“…• [5,17,18,19,20,22,27,37,51,53] for local well-posedness, blowup criteria, explicit solutions and finite-time singularities for the degenerate case (i.e., ν = κ = 0); • [1,2,3,4,14,15,16,21,23,24,25,32,33,34,35,36,38,39,44,45,46,48,49,62] for global well-posedness and regularity for the non-degenerate and partially degenerate cases; • [41,42,47,52,59,60,61] for well-posedness and regularity with critical and supercritical dissipation; and • [10,26,44,54,58,62] for long-time behaviours.…”

Section: Siran LI Jiahong Wu and Kun Zhaomentioning

confidence: 99%

On the degenerate boussinesq equations on surfaces

Li¹,

Wu²,

Zhao³

2020

Journal of Geometric Mechanics

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In this paper we study the non-degenerate and partially degenerate Boussinesq equations on a closed surface Σ. When Σ has intrinsic curvature of finite Lipschitz norm, we prove the existence of global strong solutions to the Cauchy problem of the Boussinesq equations with full or partial dissipations. The issues of uniqueness and singular limits (vanishing viscosity/vanishing thermal diffusivity) are also addressed. In addition, we establish a breakdown criterion for the strong solutions for the case of zero viscosity and zero thermal diffusivity. These appear to be among the first results for Boussinesq systems on Riemannian manifolds.

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Section: Siran LI Jiahong Wu and Kun Zhaomentioning

confidence: 99%

On the degenerate boussinesq equations on surfaces

Li¹,

Wu²,

Zhao³

2020

Journal of Geometric Mechanics

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“…if no-penetration boundary condition v \cdot n| \Gamma = 0 is imposed on the velocity field [3,25]. Here n denotes the outward unit normal vector with respect to the domain \Omega .…”

Section: Preliminarymentioning

confidence: 99%

“…Due to the divergence-free and no-penetration boundary conditions imposed on the velocity field, it can be shown that any L p -norm of θ is conserved (cf. [20,2]), i.e.,…”

Section: Mix-norm and Optimal Control Formulationmentioning

confidence: 99%

Boundary Control for Optimal Mixing via Navier--Stokes Flows

Hu¹,

Wu²

2018

SIAM J. Control Optim.

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We discuss the problem of optimal mixing of an inhomogeneous distribution of a scalar field \theta via an active control of the flow velocity v, governed by the incompressible Navier--Stokes equations, in an open bounded and connected domain \Omega \subset \BbbR 2 . We consider the velocity field generated by a control input that acts tangentially on the boundary of the domain through the Navier slip boundary conditions. This problem is motivated by mixing the fluids within a cavity or vessel by moving the walls or stirring at the boundaries. Our main objective is to design an optimal Navier slip boundary control that optimizes mixing at a given final time T > 0. Nondissipative scalars, both passive and active, governed by the transport equation will be addressed. In the absence of diffusion, transport and mixing occur due to pure advection. This essentially leads to a nonlinear control problem of a semidissipative system. Sobolev norm for the dual space (H 1 (\Omega )) \prime of H 1 (\Omega ) is adopted to quantify mixing due to the property of weak convergence. The challenge arises from the vanishing diffusivity and nonlinear coupling of the system, which results in requiring the velocity field to satisfy \int T 0 \| \nabla v\| L \infty (\Omega ) d\tau < \infty . We present a rigorous proof to show the existence of an optimal controller for both passive and active scalars and that the compatibility conditions for initial and boundary data are not required for Navier slip boundary control. Finally, we establish the first-order necessary conditions for optimality for both cases by using a variational inequality.

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“…In a recent paper by Doering et al [DWZZ], the global existence, uniqueness, and regularity for the Boussinesq for the Lions boundary condition on a Lipschitz domain Ω, was proven along with the dissipation of the L 2 norm of the velocity and its gradient. For other papers on the global existence and the regularity in Sobolev and Besov spaces, see [ACW,ACS..,BFL,BS,BrS,CD,CG,CN,CW,DP,HK1,HK2,HKR,HKZ2,HS,JMWZ,KTW,KW2,KWZ,LPZ,SW].…”

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confidence: 99%

Asymptotic properties of the Boussinesq Equations with Dirichlet Boundary Conditions

Kukavica¹,

Massatt²,

Ziane³

2021

Preprint

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We address the asymptotic properties for the Boussinesq equations with vanishing thermal diffusivity in a bounded domain with no-slip boundary conditions. We show the dissipation of the L 2 norm of the velocity and its gradient, convergence of the L 2 norm of Au, and an o(1)-type exponential growth for A 3/2 u L 2 . We also obtain that in the interior of the domain the gradient of the vorticity is bounded by a polynomial function of time.

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On the attractor for the semi-dissipative Boussinesq equations

Cited by 28 publications

References 44 publications

On the degenerate boussinesq equations on surfaces

On the degenerate boussinesq equations on surfaces

Boundary Control for Optimal Mixing via Navier--Stokes Flows

Asymptotic properties of the Boussinesq Equations with Dirichlet Boundary Conditions

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