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

Liouville theorem for MHD system and its applications

Xian-gao Liu and  Xiaotao Zhang

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 $\mathbb{R}^3× (0, T)$.

Keywords: Incompressible magnetohydrodynamics(MHD), Liouville theorem, singularties.
Mathematics Subject Classification: 35Q35, 76W05, 35B65.
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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[4]

G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279.   Google Scholar

[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. Google Scholar

[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.   Google Scholar

[7]

Cheng He and Zhouping Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal., 227 (2005), 113-152.   Google Scholar

[8]

Cheng He and Zhouping Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254.   Google Scholar

[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.   Google Scholar

[10]

Tosio Kato, Strong Lp-solutions of the avier-tokes equation in Rm, with applications to weak solutions, Math. Z., 187 (1984), 471-480.   Google Scholar

[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.   Google Scholar

[12]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa. Izdat. Nauka, Moscow, (1967), 736. Google Scholar

[13]

Zhen Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[16]

Vladimir Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math., 66 (1976), 535-552.   Google Scholar

[17]

Michel Sermange and Roger Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.   Google Scholar

[18]

James Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195.   Google Scholar

[19]

Michael Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 437-458.   Google Scholar

[20]

Roger Temam. Navier-Stokes Equations, AMS Chelsea Publishing, Providence, RI, 2001, xiv+408. Google Scholar

[21]

Gang Tian and Zhouping Xin, Gradient estimation on Navier-Stokes equations, Comm. Anal. Geom., 7 (1999), 221-257.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

show all references

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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[4]

G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279.   Google Scholar

[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. Google Scholar

[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.   Google Scholar

[7]

Cheng He and Zhouping Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal., 227 (2005), 113-152.   Google Scholar

[8]

Cheng He and Zhouping Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254.   Google Scholar

[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.   Google Scholar

[10]

Tosio Kato, Strong Lp-solutions of the avier-tokes equation in Rm, with applications to weak solutions, Math. Z., 187 (1984), 471-480.   Google Scholar

[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.   Google Scholar

[12]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa. Izdat. Nauka, Moscow, (1967), 736. Google Scholar

[13]

Zhen Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[16]

Vladimir Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math., 66 (1976), 535-552.   Google Scholar

[17]

Michel Sermange and Roger Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.   Google Scholar

[18]

James Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195.   Google Scholar

[19]

Michael Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 437-458.   Google Scholar

[20]

Roger Temam. Navier-Stokes Equations, AMS Chelsea Publishing, Providence, RI, 2001, xiv+408. Google Scholar

[21]

Gang Tian and Zhouping Xin, Gradient estimation on Navier-Stokes equations, Comm. Anal. Geom., 7 (1999), 221-257.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

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