# American Institute of Mathematical Sciences

November 2018, 17(6): 2329-2350. doi: 10.3934/cpaa.2018111

## Liouville theorem for MHD system and its applications

 School of Mathematic Sciences, Fudan University, Shanghai, China

Received  June 2017 Revised  February 2018 Published  June 2018

In this paper, we construct Liouville theorem for the MHD system and apply it to study the potential singularities of its weak solution. And we mainly study weak axi-symmetric solutions of MHD system in $\mathbb{R}^3× (0, T)$.

Citation: Xian-gao Liu, Xiaotao Zhang. Liouville theorem for MHD system and its applications. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2329-2350. doi: 10.3934/cpaa.2018111
##### References:
 [1] H. Beirão da Veiga, A new regularity class for the Navier-Stokes equations in Rn, Chinese Ann. Math. Ser. B, 16 (1995), 407-412. [2] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. [3] Dongho Chae, Pierre Degond and Jian-Guo Liu, Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 555-565. [4] G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279. [5] Giovanni P. Galdi. An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. II, Springer-Verlag, New York, 39(1994), xii+323. [6] Yoshikazu Giga, Katsuya Inui and Shin'ya Matsui, On the Cauchy problem for the Navier-Stokes equations with nondecaying initial data, Dept. Math., Seconda Univ. Napoli, Caserta, 4 (1999), 27-68. [7] Cheng He and Zhouping Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal., 227 (2005), 113-152. [8] Cheng He and Zhouping Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254. [9] L. Iskauriaza, G. A. Serëgin and V. Shverak, L3, ∞-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk, 58 (2003), 3-44. [10] Tosio Kato, Strong Lp-solutions of the avier-tokes equation in Rm, with applications to weak solutions, Math. Z., 187 (1984), 471-480. [11] Gabriel Koch, Nikolai Nadirashvili, Gregory A. Seregin and Vladimir Šverák, Liouville theorems for the Navier-Stokes equations and applications, Acta Math., 203 (2009), 83-105. [12] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa. Izdat. Nauka, Moscow, (1967), 736. [13] Zhen Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215. [14] A. Mahalov, B. Nicolaenko and T. Shilkin, L3, ∞-solutions to the MHD equations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 336 (2006), 112-276. [15] J. Nečas, M. Růžička and V. Šverák, On Leray's self-similar solutions of the Navier-Stokes equations, Acta Math., 176 (1996), 283-294. [16] Vladimir Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math., 66 (1976), 535-552. [17] Michel Sermange and Roger Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. [18] James Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195. [19] Michael Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 437-458. [20] Roger Temam. Navier-Stokes Equations, AMS Chelsea Publishing, Providence, RI, 2001, xiv+408. [21] Gang Tian and Zhouping Xin, Gradient estimation on Navier-Stokes equations, Comm. Anal. Geom., 7 (1999), 221-257. [22] Tai-Peng Tsai, On Leray's self-similar solutions of the Navier-Stokes equations satisfying local energy estimates, Arch. Rational Mech. Anal., 143 (1998), 29-51. [23] Zujin Zhang, Xian Yang and Shulin Qiu, Remarks on Liouville type result for the 3D Hall-MHD system, J. Partial Differ. Equ., 28 (2015), 286-290. [24] Yong Zhou and Milan Pokorny, On the regularity of the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097-1107.

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##### References:
 [1] H. Beirão da Veiga, A new regularity class for the Navier-Stokes equations in Rn, Chinese Ann. Math. Ser. B, 16 (1995), 407-412. [2] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. [3] Dongho Chae, Pierre Degond and Jian-Guo Liu, Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 555-565. [4] G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279. [5] Giovanni P. Galdi. An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. II, Springer-Verlag, New York, 39(1994), xii+323. [6] Yoshikazu Giga, Katsuya Inui and Shin'ya Matsui, On the Cauchy problem for the Navier-Stokes equations with nondecaying initial data, Dept. Math., Seconda Univ. Napoli, Caserta, 4 (1999), 27-68. [7] Cheng He and Zhouping Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal., 227 (2005), 113-152. [8] Cheng He and Zhouping Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254. [9] L. Iskauriaza, G. A. Serëgin and V. Shverak, L3, ∞-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk, 58 (2003), 3-44. [10] Tosio Kato, Strong Lp-solutions of the avier-tokes equation in Rm, with applications to weak solutions, Math. Z., 187 (1984), 471-480. [11] Gabriel Koch, Nikolai Nadirashvili, Gregory A. Seregin and Vladimir Šverák, Liouville theorems for the Navier-Stokes equations and applications, Acta Math., 203 (2009), 83-105. [12] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa. Izdat. Nauka, Moscow, (1967), 736. [13] Zhen Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215. [14] A. Mahalov, B. Nicolaenko and T. Shilkin, L3, ∞-solutions to the MHD equations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 336 (2006), 112-276. [15] J. Nečas, M. Růžička and V. Šverák, On Leray's self-similar solutions of the Navier-Stokes equations, Acta Math., 176 (1996), 283-294. [16] Vladimir Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math., 66 (1976), 535-552. [17] Michel Sermange and Roger Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. [18] James Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195. [19] Michael Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 437-458. [20] Roger Temam. Navier-Stokes Equations, AMS Chelsea Publishing, Providence, RI, 2001, xiv+408. [21] Gang Tian and Zhouping Xin, Gradient estimation on Navier-Stokes equations, Comm. Anal. Geom., 7 (1999), 221-257. [22] Tai-Peng Tsai, On Leray's self-similar solutions of the Navier-Stokes equations satisfying local energy estimates, Arch. Rational Mech. Anal., 143 (1998), 29-51. [23] Zujin Zhang, Xian Yang and Shulin Qiu, Remarks on Liouville type result for the 3D Hall-MHD system, J. Partial Differ. Equ., 28 (2015), 286-290. [24] Yong Zhou and Milan Pokorny, On the regularity of the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097-1107.
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