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. CaffarelliR. 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 GigaKatsuya 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. IskauriazaG. 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 KochNikolai NadirashviliGregory 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. MahalovB. 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. NečasM. 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 ZhangXian 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.

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.

[2]

L. CaffarelliR. 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 GigaKatsuya 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. IskauriazaG. 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 KochNikolai NadirashviliGregory 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. MahalovB. 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. NečasM. 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 ZhangXian 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.

[1]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1337-1345. doi: 10.3934/cpaa.2014.13.1337

[2]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1553-1561. doi: 10.3934/cpaa.2014.13.1553

[3]

Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855

[4]

Gyungsoo Woo, Young-Sam Kwon. Incompressible limit for the full magnetohydrodynamics flows under Strong Stratification on unbounded domains. Communications on Pure & Applied Analysis, 2014, 13 (1) : 135-155. doi: 10.3934/cpaa.2014.13.135

[5]

Genggeng Huang. A Liouville theorem of degenerate elliptic equation and its application. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4549-4566. doi: 10.3934/dcds.2013.33.4549

[6]

Shigeru Sakaguchi. A Liouville-type theorem for some Weingarten hypersurfaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 887-895. doi: 10.3934/dcdss.2011.4.887

[7]

Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155

[8]

Anh Tuan Duong, Quoc Hung Phan. A Liouville-type theorem for cooperative parabolic systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 823-833. doi: 10.3934/dcds.2018035

[9]

Gaocheng Yue, Chengkui Zhong. On the global well-posedness to the 3-D incompressible anisotropic magnetohydrodynamics equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5801-5815. doi: 10.3934/dcds.2016055

[10]

Ovidiu Savin. A Liouville theorem for solutions to the linearized Monge-Ampere equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 865-873. doi: 10.3934/dcds.2010.28.865

[11]

Frank Arthur, Xiaodong Yan, Mingfeng Zhao. A Liouville-type theorem for higher order elliptic systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3317-3339. doi: 10.3934/dcds.2014.34.3317

[12]

Zhenjie Li, Ze Cheng, Dongsheng Li. The Liouville type theorem and local regularity results for nonlinear differential and integral systems. Communications on Pure & Applied Analysis, 2015, 14 (2) : 565-576. doi: 10.3934/cpaa.2015.14.565

[13]

Lizhi Zhang, Congming Li, Wenxiong Chen, Tingzhi Cheng. A Liouville theorem for $\alpha$-harmonic functions in $\mathbb{R}^n_+$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1721-1736. doi: 10.3934/dcds.2016.36.1721

[14]

Xiaohui Yu. Liouville type theorem for nonlinear elliptic equation with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4947-4966. doi: 10.3934/dcds.2014.34.4947

[15]

Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511

[16]

Begoña Barrios, Leandro Del Pezzo, Jorge García-Melián, Alexander Quaas. A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5731-5746. doi: 10.3934/dcds.2017248

[17]

Xinjing Wang, Pengcheng Niu, Xuewei Cui. A Liouville type theorem to an extension problem relating to the Heisenberg group. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2379-2394. doi: 10.3934/cpaa.2018113

[18]

Fei Chen, Yongsheng Li, Huan Xu. Global solution to the 3D nonhomogeneous incompressible MHD equations with some large initial data. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2945-2967. doi: 10.3934/dcds.2016.36.2945

[19]

Xiaoping Zhai, Yongsheng Li, Wei Yan. Global well-posedness for the 3-D incompressible MHD equations in the critical Besov spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1865-1884. doi: 10.3934/cpaa.2015.14.1865

[20]

Frank Arthur, Xiaodong Yan. A Liouville-type theorem for higher order elliptic systems of Hé non-Lane-Emden type. Communications on Pure & Applied Analysis, 2016, 15 (3) : 807-830. doi: 10.3934/cpaa.2016.15.807

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (53)
  • HTML views (76)
  • Cited by (0)

Other articles
by authors

[Back to Top]