A single exponential BKM type estimate for the 3D incompressible ideal MHD equations

Liu, Jianli; Wei, Fenglun; Pan, Kejia

doi:10.1186/1687-2770-2014-96

Bound Value Probl

2014

DOI: 10.1186/1687-2770-2014-96

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A single exponential BKM type estimate for the 3D incompressible ideal MHD equations

Jianli Liu

Fenglun Wei

Kejia Pan

Abstract: In this paper, we give a Beale-Kato-Majda type criterion of strong solutions to the incompressible ideal MHD equations. Instead of double exponential estimates, we get a single exponential bound on (u, h) H s (s > 5 2 ). It can be applied to a system of an ideal viscoelastic flow. MSC: 35B65; 76W05

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Cited by 2 publications

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“…Nevertheless, as indicated in [SZ08a,SZ11], the Lagrangian map lacks maximal regularity because all the variables are defined on an evolutionary domain. In fact, the moving surface can also be described using alternative methods, such as the study of the Euler equations with surface tension [Sch05], the fluid interface problem [SZ11], the surface diffusion flow with elasticity [FJM20], the motion of charged liquid drop [JLM22], and the plasma-vacuum problem [LX23b], among others.…”

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On the Motion of Free Interface in Ideal Incompressible MHD

Hao

2017

Arch Rational Mech Anal

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ABSTRACT. For the free boundary problem of the plasma-vacuum interface to three-dimensional ideal incompressible magnetohydrodynamics (MHD), the a priori estimates of smooth solutions are proved in Sobolev norms by adopting a geometrical point of view and some quantities such as the second fundamental form and the velocity of the free interface are estimated. In the vacuum region, the magnetic fields are described by the div-curl system of pre-Maxwell dynamics, while at the interface the total pressure is continuous and the magnetic fields are tangent to the interface, but we do not need any restrictions on the size of the magnetic fields on the free interface. We introduce the "virtual particle" endowed with a virtual velocity field in vacuum to reformulate the problem to a fixed boundary problem under the Lagrangian coordinates. The L 2 -norms of any order covariant derivatives of the magnetic fields both in vacuum and on the boundaries are bounded in terms of initial data and the second fundamental forms of the free interface and the rigid wall. The estimates of the curl of the electric fields in vacuum are also obtained, which are also indispensable in elliptic estimates.

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confidence: 99%

On the Motion of Free Interface in Ideal Incompressible MHD

Hao

2017

Arch Rational Mech Anal

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Resistance Distances and Kirchhoff Index in Generalised Join Graphs

Chen

2017

Zeitschrift Für Naturforschung A

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The resistance distance between any two vertices of a connected graph is defined as the effective resistance between them in the electrical network constructed from the graph by replacing each edge with a unit resistor. The Kirchhoff index of a graph is defined as the sum of all the resistance distances between any pair of vertices of the graph. Let G=H[G1, G2, …, Gk ] be the generalised join graph of G1, G2, …, Gk determined by H. In this paper, we first give formulae for resistance distances and Kirchhoff index of G in terms of parameters of ${G'_i}s$ and H. Then, we show that computing resistance distances and Kirchhoff index of G can be decomposed into simpler ones. Finally, we obtain explicit formulae for resistance distances and Kirchhoff index of G when ${G'_i}s$ and H take some special graphs, such as the complete graph, the path, and the cycle.

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A single exponential BKM type estimate for the 3D incompressible ideal MHD equations

Cited by 2 publications

References 30 publications

On the Motion of Free Interface in Ideal Incompressible MHD

On the Motion of Free Interface in Ideal Incompressible MHD

Resistance Distances and Kirchhoff Index in Generalised Join Graphs

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