Global existence for two regularized MHD models in three space-dimension

Catania, Davide; Secchi, Paolo

doi:10.4171/pm/1880

Cited by 22 publications

(25 citation statements)

References 12 publications

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“…Voigt-regularization for MHD was first proposed and studied in [37], with further study in [6,7,38]. The MHD system, with Voigt-regularization added only to the momentum equation, is strikingly similar to system (1.3).…”

Section: Introductionmentioning

confidence: 99%

Global well-posedness of the velocity–vorticity-Voigt model of the 3D Navier–Stokes equations

Larios

Pei

Rebholz

2019

Journal of Differential Equations

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The velocity-vorticity formulation of the 3D Navier-Stokes equations was recently found to give excellent numerical results for flows with strong rotation. In this work, we propose a new regularization of the 3D Navier-Stokes equations, which we call the 3D velocity-vorticity-Voigt (VVV) model, with a Voigt regularization term added to momentum equation in velocity-vorticity form, but with no regularizing term in the vorticity equation. We prove global well-posedness and regularity of this model under periodic boundary conditions. We prove convergence of the model's velocity and vorticity to their counterparts in the 3D Navier-Stokes equations as the Voigt modeling parameter tends to zero. We prove that the curl of the model's velocity converges to the model vorticity (which is solved for directly), as the Voigt modeling parameter tends to zero. Finally, we provide a criterion for finite-time blow-up of the 3D Navier-Stokes equations based on this inviscid regularization.∇ · u = 0, u(·, 0) = u 0 and w(·, 0) = w 0 .where u = (u 1 , u 2 , u 3 ) represents an averaged velocity, w = (w 1 , w 2 , w 3 ), which plays the role of vorticity but for which we do not assume w = ∇ × u, and f is an external forcing term. Without loss of generality, we assume for our analysis in later sections that the viscosity ν = 1. Note that in the case where α = 0, the system formally reduces the velocity-vorticty formulation, while for α > 0, if one imposes w = ∇ × u, the system formally reduces to the Navier-Stokes-Voigt equations.The term −α 2 ∆∂ t u in (1.3a) is often referred to as the "Voigt-term", due to an application of modeling Kelvin-Voigt fluids by A.P. Oskolkov [49,50] (see also [28]). In the context of the velocity formulation use of the Voigt term was first proposed as a regularization for either the Navier-Stokes (for ν > 0) or Euler (for ν = 0) equations in [4], for small values of the regularization parameter α. This paper also proved global well-posedness of the Voigt-regularized versions of the 3D Euler and 3D Navier-Stokes equations.These equations have been studied analytically and extended in a wide variety of contexts (see, e.g.], and the references therein). Voigtregularizations of parabolic equations are a special case of pseudoparabolic equations, that is, equations of the form M u t + N u = f , where M and N are (possibly non-linear, or even non-local) operators. For more about pseudoparabolic equations, see, e.g., [15,52,58,59,57,5,55,56,3].

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Section: Introductionmentioning

confidence: 99%

Global well-posedness of the velocity–vorticity-Voigt model of the 3D Navier–Stokes equations

Larios

Pei

Rebholz

2019

Journal of Differential Equations

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“…The global existence of a unique strong solution has been obtained by Catania and Secchi in [10] when µ = ν = 0, but the same proof can be carried out when both ν and µ are positive (the forcing term f can be handled as in [9]) using a priori estimates provided in Section 2. In the ideal case, this result and higher order ones have been obtained independently by Larios and Titi [16].…”

Section: The Mhd-voight Viscous Modelmentioning

confidence: 87%

“…In [10], it is proved that there exists a unique global attractor for solutions to the considered model, and upper bounds for its Hausdorff and fractal dimensions in terms of relevant physical quantities (the Grashof number and the averaged dissipation length) are yielded. These bounds can be interpreted as estimates for the number of degrees of freedom of long-time dynamics of solutions.…”

Section: Downloaded By [Rmit University] At 03:40 13 August 2015mentioning

confidence: 99%

Global attractor and determining modes for a hyperbolic MHD turbulence Model

Catania

2011

Journal of Turbulence

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Magnetohydrodynamics (MHD)-α models are regularized systems used to study the turbulent flow of a magnetofluid, because of the impossibility to handle non-regularized models either analitically or numerically. In order to establish the validity of a model as a large eddy simulation model, one needs to test properties related to the number of degrees of freedom of long-time dynamics of the solutions. Preliminary information is provided by estimates concerning the number of determining modes. We consider the MHD-Voight model for an incompressible fluid and find estimates for a number of asymptotically determining modes and modes on global attractor.

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“…Other global existence results in a 3D periodic box are provided in [5] for two MHD-α models derived from Euler equation. In the first case, a model with no viscosity nor diffusivity is concerned, but both the velocity and the magnetic field are regularized.…”

Section: Introductionmentioning

confidence: 99%

Finite Dimensional Global Attractor for 3D MHD-α Models: A Comparison

Catania

2011

J. Math. Fluid Mech.

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We consider two magnetohydrodynamic-α (MHDα) models with kinematic viscosity and magnetic diffusivity for an incompressible fluid in a three-dimensional periodic box (torus). More precisely, we consider the Navier-Stokes-α-MHD and the Modified Leray-α-MHD models. Similar models are useful to study the turbulent behavior of fluids in presence of a magnetic field because of the current impossibility to handle non-regularized systems neither analytically nor via numerical simulations. In both cases, the global existence of the solution and of a global attractor can be shown. We provide an upper bound for the Hausdorff and the fractal dimension of the attractor. This bound can be interpreted in terms of degrees of freedom of the long-time dynamics of the involved system and gives information about the numerical stability of the model. We get the same bound that holds for the Simplified Bardina-MHD model, considered in a previous paper (this result provides, in some sense, an intermediate bound between the number of degrees of freedom for the Simplified Bardina model and the Navier-Stokes-α equation in the nonmagnetic case). However, the Navier-Stokes-α-MHD system is preferable since, in the ideal case, it conserves more quadratic invariants derived from the standard MHD model. Mathematics Subject Classification (2010). Primary 35Q35; Secondary 76D03.

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Global existence for two regularized MHD models in three space-dimension

Cited by 22 publications

References 12 publications

Global well-posedness of the velocity–vorticity-Voigt model of the 3D Navier–Stokes equations

Global well-posedness of the velocity–vorticity-Voigt model of the 3D Navier–Stokes equations

Global attractor and determining modes for a hyperbolic MHD turbulence Model

Finite Dimensional Global Attractor for 3D MHD-α Models: A Comparison

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