Regularity criteria for 3D MHD flows in terms of spectral components

J. Bravo-Olivares; E. Fernández-Cara; E. Notte-Cuello; M.A. Rojas-Medar; J. Bravo-Olivares; E. Fernández-Cara; E. Notte-Cuello; M.A. Rojas-Medar

doi:10.3934/era.2022164

Electronic Research Archive

2022, Volume 30, Issue 9: 3238-3248. doi: 10.3934/era.2022164

Previous Article Next Article

Research article Special Issues

Regularity criteria for 3D MHD flows in terms of spectral components

1.
Departamento de Matemática, Universidad de La Serena, La Serena, Chile
2.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, IMUS, Universidad de Sevilla, Sevilla, Spain
3.
Departamento de Matemática, Universidad de Tarapacá, Arica, Chile

Academic Editor: Runzhang Xu

Received: 26 November 2021 Revised: 17 April 2022 Accepted: 20 April 2022 Published: 20 June 2022

We extend the spectral regularity criteria of the Prodi-Serrin kind for the Navier-Stokes equations in a torus to the MHD equations. More precisely, the following is established: for any $ N > 0 $, let $ {{\boldsymbol x}}_{N} $ and $ {{\boldsymbol y}}_{N} $ be the sum of all spectral components of the velocity and magnetic field whose wave numbers possess absolute value greater that $ N $; then, it is possible to show that for any $ N $ the finiteness of the Prodi-Serrin norm of $ {{\boldsymbol x}}_{N} $ implies the regularity of the weak solution $ ({{\boldsymbol u}}, {{\boldsymbol h}}) $; thus, no restriction on the magnetic field is needed.
- MHD equations,
- Prodi-Serrin criteria,
- spectral Galerkin approximations,
- regularity
Citation: J. Bravo-Olivares, E. Fernández-Cara, E. Notte-Cuello, M.A. Rojas-Medar. Regularity criteria for 3D MHD flows in terms of spectral components[J]. Electronic Research Archive, 2022, 30(9): 3238-3248. doi: 10.3934/era.2022164

Related Papers:

Abstract

We extend the spectral regularity criteria of the Prodi-Serrin kind for the Navier-Stokes equations in a torus to the MHD equations. More precisely, the following is established: for any $ N > 0 $, let $ {{\boldsymbol x}}_{N} $ and $ {{\boldsymbol y}}_{N} $ be the sum of all spectral components of the velocity and magnetic field whose wave numbers possess absolute value greater that $ N $; then, it is possible to show that for any $ N $ the finiteness of the Prodi-Serrin norm of $ {{\boldsymbol x}}_{N} $ implies the regularity of the weak solution $ ({{\boldsymbol u}}, {{\boldsymbol h}}) $; thus, no restriction on the magnetic field is needed.

References

[1]	A. Schlüter, Dynamik des Plasma, I and II, Z. Naturforsch, 5a (1950), 72–78; 6a (1951), 73–79. https://doi.org/10.1515/zna-1950-0202 https://doi.org/10.1515/zna-1950-0202
[2]	S. B. Pikelner, Grundlagen der kosmischen Elektrodynamik, (1966), Moskau.
[3]	O. A. Ladyzhenkaya, The sixth millennium problem: Navier-Stokes equations, existence and smoothness, Uspekhi Mat. Nauk, 58 (2003), 45–78. https://doi.org/10.1070/RM2003v058n02ABEH000610 doi: 10.1070/RM2003v058n02ABEH000610
[4]	T. Buckmaster, V. Vicol, Nonuniqueness of weak solutions to the Navier-Stokes equation, Ann. Math., 189 (2019), 101–144. https://doi.org/10.4007/annals.2019.189.1.3 doi: 10.4007/annals.2019.189.1.3
[5]	G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173–182. https://doi.org/10.1007/BF02410664 doi: 10.1007/BF02410664
[6]	J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 9 (1962), 187–195. https://doi.org/10.1007/BF00253344 doi: 10.1007/BF00253344
[7]	J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauhtiers-Villars, Paris, 1969.
[8]	L. Escauriaza, G. Seregin, V. Sverak, Backward uniqueness for parabolic equations, Arch. Ration. Mech. Anal., 169 (2003), 147–157. https://doi.org/10.1023/B:JOTH.0000041475.11233.d8 doi: 10.1023/B:JOTH.0000041475.11233.d8
[9]	P.-L. Lions, N. Masmoudi, Uniqueness of mild solutions of the Navier-Stokes system in $L^N$, Commun. Partial. Differ. Equ., 26 (2001), 2211–2226. https://doi.org/10.1081/PDE-100107819 doi: 10.1081/PDE-100107819
[10]	N. Kim, M. Kwak, M. Yoo, Regularity conditions of 3D Navier-Stokes flow in terms of large spectral components, Nonlinear Anal. Theory Methods Appl., 116 (2015), 75–84. https://doi.org/10.1016/j.na.2014.12.011 doi: 10.1016/j.na.2014.12.011
[11]	A. M. Alghamdi, S. Gala, C. Qian, M. A. Ragusa, The anisotropic integrability logarithmic regularity criterion for the 3D MHD equations, Electron. Res. Arch., 28 (2020), 183–193. https://doi.org/10.3934/era.2020012 doi: 10.3934/era.2020012
[12]	P. Damázio, M. A. Rojas-Medar, On some questions of the weak solutions of evolution equations for magnetohydrodynamic type, Proyecciones, 16 (1997), 83–97. https://doi.org/10.22199/S07160917.1997.0002.00001 doi: 10.22199/S07160917.1997.0002.00001
[13]	C. He, Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differ. Equ., 213 (2005), 235–254. https://doi.org/10.1016/j.jde.2004.07.002 doi: 10.1016/j.jde.2004.07.002
[14]	G. Lassner, Über Ein Rand-Anfangswert - Problem der Magnetohydrodinamik, Arch. Rat. Mech. Anal., 25 (1967), 388–405. https://doi.org/10.1007/BF00291938 doi: 10.1007/BF00291938
[15]	Y. Li, C. Zhao, Existence, uniqueness and decay properties of strong solutions to an evolutionary system of MHD type in $\mathbb{R}^3$, J. Dyn. Differ. Equ., 18 (2006), 393–426. https://doi.org/10.1007/s10884-006-9012-7 doi: 10.1007/s10884-006-9012-7
[16]	A. Mahalov, B. Nicolaenko, T. Shilkin, $L_{3, \infty}$-Solution to the MHD Equations, J. Math. Sci., 143 (2007), 2911–2923. https://doi.org/10.1007/s10958-007-0175-5 doi: 10.1007/s10958-007-0175-5
[17]	M. A. Rojas-Medar, J. L. Boldrini, The weak solutions and reproductive property for a system of evolution equations of magnetohydrodynamic type, Proyecciones, 13 (1994), 85–97. https://doi.org/10.22199/S07160917.1994.0002.00002 doi: 10.22199/S07160917.1994.0002.00002
[18]	C. Zhao, Initial boundary value problem for the evolution system of MHD type describing geophysical flow in three-dimensional domains, Math. Methods Appl. Sci., 26 (2003), 759–781. https://doi.org/10.1002/mma.394 doi: 10.1002/mma.394
[19]	C. Zhao, K. Li, On existence, uniqueness and $L_r$-exponential stability for stationary solutions to the MHD equations in three-dimensional domains, ANZIAM J., 46 (2004), 95–109. https://doi.org/10.1017/S1446181100013705 doi: 10.1017/S1446181100013705
[20]	J. C. Robinson, J. L. Rodrigo, W. Sadowski, The three-Dimensional Navier-Stokes Equations, Studies in Advanced Mathematics, Vol 157. Cambridge University Press, Cambridge, 2016. https://doi.org/10.1017/CBO9781139095143
[21]	R. Temam, Navier-Stokes equations and nonlinear functional analysis, Society for industrial and applied mathemtics, 1995. https://doi.org/10.1137/1.9781611970050
[22]	H. B. da Veiga, J. Bemelmans, J. Brand, On a two components condition for regularity of the 3D Navier-Stokes equations under physical slip boundary conditions on non-flat boundaries, Math. Ann., 374 (2019), 1559–1596. https://doi.org/10.1007/s00208-018-1755-z doi: 10.1007/s00208-018-1755-z
[23]	S. Bosia, V. Pata, J. C. Robinson, A weak-$L^p$ Prodi-Serrin type regularity criterion for the Navier-Stokes equations, J. Math. Fluid Mech., 16 (2014), 721–725. https://doi.org/10.1007/s00021-014-0182-5 doi: 10.1007/s00021-014-0182-5
[24]	J. Neustupa, A refinement of the local Serrin-type regularity criterion for a suitable weak solution to the Navier-Stokes equations, Arch. Ration. Mech. Anal., 214 (2014), 525–544. https://doi.org/10.1007/s00205-014-0761-x doi: 10.1007/s00205-014-0761-x
[25]	B. Pineau, X. Yu, On Prodi- Serrin type conditions for the 3D Navier- Stokes equations, Nonlinear Anal., 190 (2020), 111612. https://doi.org/10.1016/j.na.2019.111612 doi: 10.1016/j.na.2019.111612
[26]	J. Wolf, A regularity criterion of Serrin-type for the Navier-Stokes equations involving the gradient of one velocity component, Analysis, 35 (2015), 259–292. https://doi.org/10.1515/anly-2014-1301 doi: 10.1515/anly-2014-1301

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)