Differential and Integral Equations

Horizontal Biot-Savart law in general dimension and an application to the 4D magneto-hydrodynamics

Kazuo Yamazaki

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Abstract

We derive a Biot-Savart law type identity for the horizontal components of the solution to the fluid system of equations with incompressibility in general dimension. Along with another new decomposition of non-linear terms, we give its application to derive two regularity criteria for the four-dimensional magneto-hydrodynamics system, in particular a criteria in terms of two velocity field components, two magnetic field components and two partial derivatives of the other two magnetic field components in a scaling-invariant norm. It is an open problem to obtain a criterion in terms of just two velocity field components and two partial derivatives of two magnetic field components in a scaling-invariant norm; an analogous criterion in the three-dimensional case has already been established.

Article information

Source
Differential Integral Equations Volume 31, Number 3/4 (2018), 301-328.

Dates
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.die/1513652428

Subjects
Primary: 35B65, 35Q35, 35Q86

Citation

Yamazaki, Kazuo. Horizontal Biot-Savart law in general dimension and an application to the 4D magneto-hydrodynamics. Differential Integral Equations 31 (2018), no. 3/4, 301--328. https://projecteuclid.org/euclid.die/1513652428


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