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Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian
Liouville theorem for MHD system and its applications
School of Mathematic Sciences, Fudan University, Shanghai, China |
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 $\mathbb{R}^3× (0, T)$.
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.
|
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. 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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