TY - GEN
T1 - On the global regularity issue of the two-dimensional magnetohydrodynamics system with magnetic diffusion weaker than a laplacian
AU - Yamazaki, Kazuo
N1 - Publisher Copyright:
© 2019 American Mathematical Society.
PY - 2019
Y1 - 2019
N2 - In this manuscript, we discuss the recent developments in the research direction concerning whether the solution to the two-dimensional magnetohydrodynamics system with certain velocity dissipation and magnetic diffusion strengths preserves the high regularity of a given initial data for all time or exhibits a blowup in finite time. In particular, we address an open problem in case the magnetic diffusion is weaker than a full Laplacian. In short, we consider this system with both velocity dissipation and magnetic diffusion strength measured in terms of fractional Laplacians with certain powers, and point out the following gap in the results among the current literature. In case the power of the fractional Laplacian representing the magnetic diffusion is one, the global well-posedness follows as long as the velocity dissipation is present regardless of how weak its strength may be, and hence the sum of the two powers need to only be more than one ([Monatsh. Math. 175 (2014), no. 1, 127–131], [Second proof of the global regularity of the two-dimensional MHD system with full diffusion and arbitrary weak dissipation, Methods Appl. Anal., to appear]). On the other hand, once the power of the fractional Laplacian representing the magnetic diffusion drops below one, in order to ensure the system’s global well-posedness, the sum of the powers from the dissipation and diffusion must be equal to or more than two, improved only logarithmically ([Nonlinear Anal. 85 (2013), 43–51], [J. Math. Fluid Mech. 13 (2011), no. 2, 295–305], [Appl. Math. Lett. 29 (2014), 46–51]). We discuss this issue in more detail, explain its reasoning, and provide some basic regularity criteria to gain better insight to this difficult direction of research.
AB - In this manuscript, we discuss the recent developments in the research direction concerning whether the solution to the two-dimensional magnetohydrodynamics system with certain velocity dissipation and magnetic diffusion strengths preserves the high regularity of a given initial data for all time or exhibits a blowup in finite time. In particular, we address an open problem in case the magnetic diffusion is weaker than a full Laplacian. In short, we consider this system with both velocity dissipation and magnetic diffusion strength measured in terms of fractional Laplacians with certain powers, and point out the following gap in the results among the current literature. In case the power of the fractional Laplacian representing the magnetic diffusion is one, the global well-posedness follows as long as the velocity dissipation is present regardless of how weak its strength may be, and hence the sum of the two powers need to only be more than one ([Monatsh. Math. 175 (2014), no. 1, 127–131], [Second proof of the global regularity of the two-dimensional MHD system with full diffusion and arbitrary weak dissipation, Methods Appl. Anal., to appear]). On the other hand, once the power of the fractional Laplacian representing the magnetic diffusion drops below one, in order to ensure the system’s global well-posedness, the sum of the powers from the dissipation and diffusion must be equal to or more than two, improved only logarithmically ([Nonlinear Anal. 85 (2013), 43–51], [J. Math. Fluid Mech. 13 (2011), no. 2, 295–305], [Appl. Math. Lett. 29 (2014), 46–51]). We discuss this issue in more detail, explain its reasoning, and provide some basic regularity criteria to gain better insight to this difficult direction of research.
KW - BMO space
KW - Fourier transform
KW - Fractional Laplacian
KW - Magnetohydrodynamics system
KW - Regularity
UR - http://www.scopus.com/inward/record.url?scp=85116117253&partnerID=8YFLogxK
U2 - 10.1090/conm/725/14552
DO - 10.1090/conm/725/14552
M3 - Conference contribution
AN - SCOPUS:85116117253
SN - 9781470441098
T3 - Contemporary Mathematics
SP - 251
EP - 264
BT - Nonlinear Dispersive Waves and Fluids
A2 - Zheng, Shijun
A2 - Bona, Jerry
A2 - Chen, Geng
A2 - Van Phan, Tuoc
A2 - Beceanu, Marius
A2 - Soffer, Avy
PB - American Mathematical Society
Y2 - 5 January 2017 through 7 January 2017
ER -