Energy dissipation for weak solutions of incompressible MHD equations

In this article, we mainly study the local equation of energy for weak solutions of 3D MHD equations. We define a dissipation term D(u, B) that stems from an eventual lack of smoothness in the solution, and then obtain a local equation of energy for weak solutions of 3D MHD equations. Finally, we consider the 2D case at the end of this article.     

Dissipation anomaly and energy cascade in 3D incompressible flows

The purpose of this Note is to present a mathematical evidence of the dissipation anomaly in 3D turbulent flows within a general setting for the study of energy cascade in physical scales of 3D incompressible flows recently introduced by the authors.     

Sobolev空间中非电阻MHD方程局部适定性

本文研究了R^d,d=2,3上的非电阻磁流体力学方程的柯西问题.通过建立一个交换子估计,我们在Sobolev空间H_s-1×H_s,s > d/2中证明了该方程组解的局部适定性.     

Local Existence of Strong Solutions to the Generalized MHD Equations

This paper devotes to consider the local existence of the strong solutions to the generalized MHD system with fractional dissipative terms $\Lambda^{2\alpha}u$ for the velocity field and $\Lambda^{2\alpha}b$ for the magnetic field, respectively. We construct the approximate solutions by the Fourier truncation method, and use energy method to obtain the local existence of strong solutions in $H^s(\mathbb{R}^n)\,(s>\max \left\{\frac{n}{2}+1-2\alpha, 0\right\})$ for any $\alpha\geq0$.     

Global regularity to the 2D incompressible MHD with mixed partial dissipation and magnetic diffusion in a bounded domain

This article considers the global regularity to the initial–boundary value problem for the 2D incompressible MHD with mixed partial dissipation and magnetic diffusion.To overcome the difficulty caused by the vanishing viscosities,we first establish the elliptic system for uxand by,which are estimated by▽×u_x and▽×b_y,respectively.Then,we establish the global estimates for▽×u and▽×b.     

Local Well-posedness for Linearized Degenerate MHD Boundary Layer Equations in Analytic Setting

In this note we derive MHD boundary layer equations according to viscosity and resistivity coefficients. Especially, when these viscosity and resistivity coefficients are of different orders, it leads to degenerate MHD boundary layer equations. We prove these degenerate boundary layers are stable around a steady solution.     

Shear layer solutions of incompressible MHD and dynamo effect

This paper is devoted to the equations of incompressible magnetohydrodynamics (MHD). Its general concern is the “dynamo effect”, i.e. the growth of magnetic field through the movement of a conducting fluid. Motivated by the so-called “stretch-diffuse mechanism”, we study the nonlinear stability of solutions (u,b=0)$(u,b=0)$ where the velocity u is a regularized vortex sheet, and the magnetic component b is zero. We prove that dynamo effect is possible when both curvature of the sheet and magnetic diffusion are non-zero, and impossible otherwise.     

Local and parallel finite element algorithm based on the partition of unity method for the incompressible MHD flow

Based on the partition of unity method (PUM), a local and parallel finite element method is designed and analyzed for solving the stationary incompressible magnetohydrodynamics (MHD). The key idea of the proposed algorithm is to first solve the nonlinear system on a coarse mesh, divide the globally fine grid correction into a series of locally linearized residual problems on some subdomains derived by a class of partition of unity, then compute the local subproblems in parallel, and obtain the globally continuous finite element solution by assembling all local solutions together by the partition of unity functions. The main feature of the new method is that the partition of unity provide a flexible and controllable framework for the domain decomposition. Finally, the efficiency of our theoretical analysis is tested by numerical experiments.     

Patterned turbulence and relaminarization in MHD pipe and duct flows

We present results of a numerical analysis of relaminarization processes in MHD duct and pipe flows. It is motivated by Julius Hartmann's classical experiments on flows of mercury in pipes and ducts under the influence of magnetic fields. The simulations, conducted both in periodic and non-periodic settings, provide a first detailed view of flow structures that have not been experimentally accessible. The main novelty of the analysis is very long (tens to hundreds of hydraulic diameters) computational domains that allows to discover new flow regimes with localized turbulent spots near the side walls parallel to the magnetic field. The computed critical parameters for transition as well as the friction coefficients are in good agreement with Hartmann's data. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Energy dissipation in the Smagorinsky model of turbulence

The Smagorinsky model often severely over-dissipates flows and, consistently, previous estimates of its energy dissipation rate blow up as $Re\to \infty$. This report estimates time averaged model dissipation, $〈{\epsilon }_S〉$, under periodic boundary conditions as$〈{\epsilon }_S〉\le 2\frac{U^3}{L}+{\mathrm{Re}}^{-1}\frac{U^3}{L}+\frac{32}{27}{C}_S^2{\left(\frac{\delta }{L}\right)}^2\frac{U^3}{L}\text{,}$ where $U,L$ are global velocity and length scales and $C_S\simeq 0.1,\delta <1$ are model parameters. Thus, in the absence of boundary layers, the Smagorinsky model does not over dissipate.     

Calculation of the SDE-contribution of the dissipation operator to the energy spectrum of developed turbulence

The nonanalytic correction to the energy spectrum of the developed turbulence in the one-loop approximation of ε-expansion is calculated. It has the form y⁴ ln u (u≡kL, L is the external scale of the turbulence), which is in agreement with Wilson's short distance expansion prediction. The amplitude of the contribution of the dissipation operator to the energy spectrum is determined. Bibliography: 7 titles. Dedicated to the memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 36–42. Translated by N. Yu. Netsvetaev.     

Well-Posedness of MHD Equations in Sobolev-Gevery Space

This paper is devoted to the study of the 3D incompressible magnetohydrodynamic system. We prove the local in time well-posedness for any large initial data in $\dot{H}_{a,1}^{1}({\mathbb{R}}^{3})$ or $H_{a,1}^{1}({\mathbb{R}}^{3})$. Furthermore, the global well-posedness of a strong solution in $\tilde{L}^{\infty}(0,T;H_{a,1}^{1}({\mathbb{R}}^{3}))\cap L^{2}(0,T;\dot{H}_{a,1}^{1}({\mathbb{R}}^{3})\cap \dot{H}_{a,1}^{2}({\mathbb{R}}^{3}))$ with initial data satisfying a smallness condition is established.     

带真空无磁扩散不可压磁流体方程柯西问题的局部适定性

该文讨论了在真空远场的密度条件下,二维不可压零磁耗散磁流体力学方程组柯西问题的局部适定性.在初始密度和磁场具有一定的衰减性时,证明了磁流体方程具有唯一的局部强解.当初值满足兼容性条件和适当的正则性条件时,该强解就是经典解.除此之外,文中还给出了一个仅与磁场有关的爆破准则.     

On the infra-red asymptotic behavior of the pair correlator of the energy dissipation rate for well-developed turbulence

The infra-red asymptotic of the pair correlator of the energy dissipation rate for well-developed turbulence is described by the sum of two close contributions (constant and linear in r=⊢x−y⊢). This possibly explains the experimentally observed nonuniversality of the “intermittency exponent” μ and its deviation from the Kolmogorov value μ=0. Bibliography: 12 titles. Dedicated to the memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 81–86. Translated by B. M. Bekker.     

Non-Uniform Decay of MHD Equations With and Without Magnetic Diffusion

We consider the long time behavior of solutions to the magnetohydrodynamics equations in two and three spatial dimensions. It is shown that in the absence of magnetic diffusion, if strong bounded solutions were to exist their energy cannot present any asymptotic oscillatory behavior, the diffusivity of the velocity is enough to prevent such oscillations. When magnetic diffusion is present and the data is only in L ², it is shown that the solutions decay to zero without a rate, and this nonuniform decay is optimal.     

半空间MHD方程组弱解的L~2衰减

该文利用Stokes算子的谱表示及能量估计的方法研究半空间MHD方程组初边值问题弱解的L~2衰减.     

DNS of MHD turbulence in liquid metals based on the least dissipative modes

We investigate numerically the incompressible Navier-Stokes equations under an externally imposed magnetic field. The results obtained for the simplified geometry of a three dimensional periodic box and show a lower computational complexity and stronger relationship between the physical reality and its numerical representation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational and reciprocal principles in thermoelasticity without energy dissipation

In the context of the linear theory of thermoelasticity without energy dissipation for homogeneous and isotropic materials, variational principles of Biot-and Hamilton-types and a reciprocal principle of Betti-Rayleigh-type are presented.     

PL~n∩PL~p空间中MHD方程组强解的存在唯一性及衰减性质

本文运用半群理论和Kato的方法,研究了MHD方程组在PL~n∩PL~p(1相似文献