Energy dissipation for weak solutions of incompressible MHD equations

Gao, Zhensheng; Tan, Zhong; Wu, Guochun

doi:10.1016/s0252-9602(13)60046-6

Cited by 14 publications

(12 citation statements)

References 8 publications

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“…From (3.5), we can easily prove another conclusion about the total energy balance. Referring to Gao et al, 11 the total energy E(t) is Proof. To prove our corollary, we need to prove for arbitrary (t) ∈ C ∞ 0 (0, T), there is…”

Section: Then the System Satisfies The Following Local Energy Equatimentioning

confidence: 95%

“…From (), we can easily prove another conclusion about the total energy balance. Referring to Gao et al, 11 the total energy E ( t ) is

E (t) = \frac{1}{2} \int_{𝕋^{3}} ρ | u |^{2} d x + \int_{𝕋^{3}} P (ρ) d x + \frac{1}{2 μ} \int_{𝕋^{3}} | B |^{2} d x + \frac{1}{μ^{2} σ} \int_{0}^{t} \int_{𝕋^{3}} | \nabla B |^{2} d x .

…”

Section: Energy Conservation For Compressible Isentropic Mhd Equationsmentioning

confidence: 99%

“…In 2013, Gao et al 11 studied the local energy equation of the ideal incompressible MHD equation in

L^{2} false(0, T; H^{1} false(ℝ^{3} false) false) \cap L^{\infty} false(0, T; L^{2} false(ℝ^{3} false) false)

. In 2016, based on the energy dissipation method, Guo and Tan 12 considered the energy conservation equation of three‐dimensional incompressible MHD equations from the longitudinal and transverse directions and established the relationship between dissipation term D ( u , B ) and turbulence in the MHD equation.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Regularity and energy conservation for compressible isentropic magnetohydrodynamic equations

Tan

2020

Math Methods in App Sciences

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Section: Then the System Satisfies The Following Local Energy Equatimentioning

confidence: 95%

“…From (), we can easily prove another conclusion about the total energy balance. Referring to Gao et al, 11 the total energy E ( t ) is

E (t) = \frac{1}{2} \int_{𝕋^{3}} ρ | u |^{2} d x + \int_{𝕋^{3}} P (ρ) d x + \frac{1}{2 μ} \int_{𝕋^{3}} | B |^{2} d x + \frac{1}{μ^{2} σ} \int_{0}^{t} \int_{𝕋^{3}} | \nabla B |^{2} d x .

…”

Section: Energy Conservation For Compressible Isentropic Mhd Equationsmentioning

confidence: 99%

“…In 2013, Gao et al 11 studied the local energy equation of the ideal incompressible MHD equation in

L^{2} false(0, T; H^{1} false(ℝ^{3} false) false) \cap L^{\infty} false(0, T; L^{2} false(ℝ^{3} false) false)

Section: Introductionmentioning

confidence: 99%

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Regularity and energy conservation for compressible isentropic magnetohydrodynamic equations

Tan

2020

Math Methods in App Sciences

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“…Whether the energy equality holds for weak solutions or is connected to the fundamental problem is an open problem. There are also extensions, [10], of the results of Duchon and Robert [7] to MHD flows. These yield existence of a non negative operator (in effect) filling in possible gaps between the energy equality and inequality.…”

Section: The Ensemble Averaged Mhd Systemmentioning

confidence: 99%

“…Remark 3.1 Just as for non-conducting fluids, the energy equality is known to hold for strong solutions and for slightly more regular weak solutions [5], [10]. Whether the energy equality holds for weak solutions or is connected to the fundamental problem is an open problem.…”

Section: The Ensemble Averaged Mhd Systemmentioning

confidence: 99%

Magnetohydrodynamic flows: Boussinesq conjecture

Labovsky

Layton

2016

Journal of Mathematical Analysis and Applications

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The Boussinesq assumption that turbulent fluctuations have a dissipative effect on the mean flow is the basis for most turbulence models used in practical flow simulations. Data from computational tests and experiments has indicated that in 2d fluid flows an inverse energy cascade is expected. However, the Boussinesq assumption has recently been proven to hold in a time averaged sense for the Navier-Stokes equations. In Magnetohydrodynamic (MHD) flows, the dynamo effect suggests an inverse cascade of energy from small to large scales in both 2d and 3d, suggesting its re-evaluation for MHD flows. This article uses the MHD formulation in the Elsässer variables to show that the Boussinesq assumption also holds for MHD turbulence. A short discussion on the choice of eddy viscosity models is presented, as a consequence of the main result.

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Local 4/5-law and energy dissipation anomaly in turbulence of incompressible MHD Equations

Guo

Tan

2016

Z. Angew. Math. Phys.

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Energy dissipation for weak solutions of incompressible MHD equations

Cited by 14 publications

References 8 publications

Regularity and energy conservation for compressible isentropic magnetohydrodynamic equations

Regularity and energy conservation for compressible isentropic magnetohydrodynamic equations

Magnetohydrodynamic flows: Boussinesq conjecture

Local 4/5-law and energy dissipation anomaly in turbulence of incompressible MHD Equations

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