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Solvability of nonlocal systems related to peridynamics
Some Remarks on regularity criteria of Axially symmetric Navier-Stokes equations
1. | Department of Mathematics and IMS, Nanjing University, Nanjing, 210093, China |
2. | Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China |
Two main results will be presented in our paper. First, we will prove the regularity of solutions to axially symmetric Navier-Stokes equations under a $log$ supercritical assumption on the horizontally radial component $u^r$ and vertical component $u^z$, accompanied by a $log$ subcritical assumption on the horizontally angular component $u^θ$ of the velocity. Second, the precise Green function for the operator $-(Δ-\frac{1}{r^2})$ under the axially symmetric situation, where $r$ is the distance to the symmetric axis, and some weighted $L^p$ estimates of it will be given. This will serve as a tool for the study of axially symmetric Navier-Stokes equations. As an application, we will prove the regularity of solutions to axially symmetric Navier-Stokes equations under a critical (or a subcritical) assumption on the angular component $w^θ$ of the vorticity.
References:
[1] |
M. Abramowitz and I. A. Stegun,
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Reprint of the 1972 edition. Dover Publications, Inc., New York, 1992. |
[2] |
H. Abidi and P. Zhang,
Global smooth axisymmetric solutions of 3-D inhomogeneous incompressible Navier-Stokes system, Calc. Var. Partial Differential Equations, 54 (2015), 3251-3276.
doi: 10.1007/s00526-015-0902-6. |
[3] |
L. J. Burke and Q. S. Zhang,
A priori bounds for the vorticity of axially symmetric solutions to the Navier-Stokes equations, Adv. Differential Equations, 15 (2010), 531-560.
|
[4] |
D. Chae and J. Lee,
On the regularity of the axisymmetric solutions of the Navier-Stokes equations, Math. Z., 239 (2002), 645-671.
doi: 10.1007/s002090100317. |
[5] |
E. A. Carlen and M. Loss,
Optimal smoothing and decay estimates for viscously damped conservation laws, with applications to the 2-D Navier-Stokes equation, Duke Math. J., 81 (1995), 135-157.
doi: 10.1215/S0012-7094-95-08110-1. |
[6] |
C. C. Chen, R. M. Strain, H. Z. Yau and T. P. Tsai, Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations,
Int. Math. Res. Not. (IMRN), (2008), Art. ID rnn016, 31 pp.
doi: 10.1093/imrn/rnn016. |
[7] |
C. C. Chen, R. M. Strain, T. P. Tsai and H. Z. Yau,
Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations Ⅱ, Comm. Partial Differential Equations, 34 (2009), 203-232.
doi: 10.1080/03605300902793956. |
[8] |
H. Chen, D. Fang and T. Zhang,
Regularity of 3D axisymmetric Navier-Stokes equations, Discrete and Continuous Dynamical Systems, 37 (2017), 1923-1939.
doi: 10.3934/dcds.2017081. |
[9] |
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.
doi: 10.1002/cpa.3160350604. |
[10] |
C. L. Fefferman, Existence and Smoothness of the Navier-Stokes Equation. The Millennium Prize Problems, Clay Math. Inst., Cambridge, MA, (2006), 57-67. |
[11] |
A. Grigor'yan, Heat kernels on weighted manifolds and applications, The Ubiquitous Heat Kernel, Amer. Math. Soc., Providence, RI, Contemp. Math., 398 (2006), 93-191.
doi: 10.1090/conm/398/07486. |
[12] |
D. Gilbarg and N. S. Trudinger,
Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, Berlin, (1983).
doi: 10.1007/978-3-642-61798-0. |
[13] |
T. Gallay and V. Sverak, Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations, preprint, arXiv: 1510.01036. |
[14] |
G. Seregin and D. Zhou, Regularity of solutions to the Navier-Stokes equations in $ \dot{B}^{-1}_{\infty, \infty }$ preprint, arXiv: 1802.03600.
doi: 10.1007/s00021-002-8533-z. |
[15] |
T.Y. Hou, Z. Lei and C. Li,
Global regularity of the 3D axi-symmetric Navier-Stokes equations with anisotropic data, Comm. Partial Differential Equations, 33 (2008), 1622-1637.
doi: 10.1080/03605300802108057. |
[16] |
T. Y. Hou and C. Li,
Dynamic stability of the three-dimensional axisymmetric Navier-Stokes equations with swirl, Comm. Pure Appl. Math., 61 (2008), 661-697.
doi: 10.1002/cpa.20212. |
[17] |
Q. Jiu and Z. Xin,
Some regularity criteria on suitable weak solutions of the 3-D incompressible axisymmetric Navier-Stokes equations, New Stud. Adv. Math., 2 (2003), 119-139.
|
[18] |
G. Koch, N. Nadirashvili, G. A. Seregin and V. Sverak,
Liouville theorems for the Navier-Stokes equations and applications, Acta Math., 203 (2009), 83-105.
doi: 10.1007/s11511-009-0039-6. |
[19] |
O. A. Ladyzenskaja, Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968) 155-177 (Russian). |
[20] |
Z. Lei and Q. S. Zhang,
A Liouville theorem for the axially-symmetric Navier-Stokes equations, J. Funct. Anal., 261 (2011), 2323-2345.
doi: 10.1016/j.jfa.2011.06.016. |
[21] |
Z. Lei and Q. S. Zhang,
Structure of solutions of 3D axisymmetric Navier-Stokes equations near maximal points, Pacific J. Math., 254 (2011), 335-344.
doi: 10.2140/pjm.2011.254.335. |
[22] |
Z. Lei and Q. S. Zhang, Notes on axially symmetric Navier-Stokes equations,
(2014) (private communications). |
[23] |
Z. Lei and Q. S. Zhang,
Criticality of the axially symmetric Navier-Stokes equations, Pacific J. Math., 289 (2017), 169-187.
doi: 10.2140/pjm.2017.289.169. |
[24] |
J. Neustupa and M. Pokorny,
An interior regularity criterion for an axially symmetric suitable weak solution to the Navier-Stokes equations, J. Math. Fluid Mech., 2 (2000), 381-399.
doi: 10.1007/PL00000960. |
[25] |
X. Pan,
Regularity of solutions to axisymmetric Navier-Stokes equations with a slightly supercritical condition, J. Differential Equations, 260 (2016), 8485-8529.
doi: 10.1016/j.jde.2016.02.026. |
[26] |
X. Pan,
A regularity condition of 3d axisymmetric Navier-Stokes equations, Acta Appl. Math., 150 (2017), 103-109.
doi: 10.1007/s10440-017-0096-3. |
[27] |
G. Tian and Z. Xin,
One-point singular solutions to the Navier-Stokes equations, Topol. Methods Nonlinear Anal., 11 (1998), 135-145.
doi: 10.12775/TMNA.1998.008. |
[28] |
M. R. Ukhovskii and V. I Iudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, Prikl. Mat. Meh., 32, 59-69 (Russian); translated as J. Appl. Math. Mech., 32 (1968), 52-61.
doi: 10.1016/0021-8928(68)90147-0. |
[29] |
G. N. Watson,
A Treatise on the Theory of Bessel Functions, Reprint of the second edition, 1994, Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1995. |
[30] |
D. Wei,
Regularity criterion to the axially symmetric Navier-Stokes equations, J. Math. Anal. Appl., 435 (2016), 402-413.
doi: 10.1016/j.jmaa.2015.09.088. |
show all references
References:
[1] |
M. Abramowitz and I. A. Stegun,
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Reprint of the 1972 edition. Dover Publications, Inc., New York, 1992. |
[2] |
H. Abidi and P. Zhang,
Global smooth axisymmetric solutions of 3-D inhomogeneous incompressible Navier-Stokes system, Calc. Var. Partial Differential Equations, 54 (2015), 3251-3276.
doi: 10.1007/s00526-015-0902-6. |
[3] |
L. J. Burke and Q. S. Zhang,
A priori bounds for the vorticity of axially symmetric solutions to the Navier-Stokes equations, Adv. Differential Equations, 15 (2010), 531-560.
|
[4] |
D. Chae and J. Lee,
On the regularity of the axisymmetric solutions of the Navier-Stokes equations, Math. Z., 239 (2002), 645-671.
doi: 10.1007/s002090100317. |
[5] |
E. A. Carlen and M. Loss,
Optimal smoothing and decay estimates for viscously damped conservation laws, with applications to the 2-D Navier-Stokes equation, Duke Math. J., 81 (1995), 135-157.
doi: 10.1215/S0012-7094-95-08110-1. |
[6] |
C. C. Chen, R. M. Strain, H. Z. Yau and T. P. Tsai, Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations,
Int. Math. Res. Not. (IMRN), (2008), Art. ID rnn016, 31 pp.
doi: 10.1093/imrn/rnn016. |
[7] |
C. C. Chen, R. M. Strain, T. P. Tsai and H. Z. Yau,
Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations Ⅱ, Comm. Partial Differential Equations, 34 (2009), 203-232.
doi: 10.1080/03605300902793956. |
[8] |
H. Chen, D. Fang and T. Zhang,
Regularity of 3D axisymmetric Navier-Stokes equations, Discrete and Continuous Dynamical Systems, 37 (2017), 1923-1939.
doi: 10.3934/dcds.2017081. |
[9] |
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.
doi: 10.1002/cpa.3160350604. |
[10] |
C. L. Fefferman, Existence and Smoothness of the Navier-Stokes Equation. The Millennium Prize Problems, Clay Math. Inst., Cambridge, MA, (2006), 57-67. |
[11] |
A. Grigor'yan, Heat kernels on weighted manifolds and applications, The Ubiquitous Heat Kernel, Amer. Math. Soc., Providence, RI, Contemp. Math., 398 (2006), 93-191.
doi: 10.1090/conm/398/07486. |
[12] |
D. Gilbarg and N. S. Trudinger,
Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, Berlin, (1983).
doi: 10.1007/978-3-642-61798-0. |
[13] |
T. Gallay and V. Sverak, Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations, preprint, arXiv: 1510.01036. |
[14] |
G. Seregin and D. Zhou, Regularity of solutions to the Navier-Stokes equations in $ \dot{B}^{-1}_{\infty, \infty }$ preprint, arXiv: 1802.03600.
doi: 10.1007/s00021-002-8533-z. |
[15] |
T.Y. Hou, Z. Lei and C. Li,
Global regularity of the 3D axi-symmetric Navier-Stokes equations with anisotropic data, Comm. Partial Differential Equations, 33 (2008), 1622-1637.
doi: 10.1080/03605300802108057. |
[16] |
T. Y. Hou and C. Li,
Dynamic stability of the three-dimensional axisymmetric Navier-Stokes equations with swirl, Comm. Pure Appl. Math., 61 (2008), 661-697.
doi: 10.1002/cpa.20212. |
[17] |
Q. Jiu and Z. Xin,
Some regularity criteria on suitable weak solutions of the 3-D incompressible axisymmetric Navier-Stokes equations, New Stud. Adv. Math., 2 (2003), 119-139.
|
[18] |
G. Koch, N. Nadirashvili, G. A. Seregin and V. Sverak,
Liouville theorems for the Navier-Stokes equations and applications, Acta Math., 203 (2009), 83-105.
doi: 10.1007/s11511-009-0039-6. |
[19] |
O. A. Ladyzenskaja, Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 7 (1968) 155-177 (Russian). |
[20] |
Z. Lei and Q. S. Zhang,
A Liouville theorem for the axially-symmetric Navier-Stokes equations, J. Funct. Anal., 261 (2011), 2323-2345.
doi: 10.1016/j.jfa.2011.06.016. |
[21] |
Z. Lei and Q. S. Zhang,
Structure of solutions of 3D axisymmetric Navier-Stokes equations near maximal points, Pacific J. Math., 254 (2011), 335-344.
doi: 10.2140/pjm.2011.254.335. |
[22] |
Z. Lei and Q. S. Zhang, Notes on axially symmetric Navier-Stokes equations,
(2014) (private communications). |
[23] |
Z. Lei and Q. S. Zhang,
Criticality of the axially symmetric Navier-Stokes equations, Pacific J. Math., 289 (2017), 169-187.
doi: 10.2140/pjm.2017.289.169. |
[24] |
J. Neustupa and M. Pokorny,
An interior regularity criterion for an axially symmetric suitable weak solution to the Navier-Stokes equations, J. Math. Fluid Mech., 2 (2000), 381-399.
doi: 10.1007/PL00000960. |
[25] |
X. Pan,
Regularity of solutions to axisymmetric Navier-Stokes equations with a slightly supercritical condition, J. Differential Equations, 260 (2016), 8485-8529.
doi: 10.1016/j.jde.2016.02.026. |
[26] |
X. Pan,
A regularity condition of 3d axisymmetric Navier-Stokes equations, Acta Appl. Math., 150 (2017), 103-109.
doi: 10.1007/s10440-017-0096-3. |
[27] |
G. Tian and Z. Xin,
One-point singular solutions to the Navier-Stokes equations, Topol. Methods Nonlinear Anal., 11 (1998), 135-145.
doi: 10.12775/TMNA.1998.008. |
[28] |
M. R. Ukhovskii and V. I Iudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, Prikl. Mat. Meh., 32, 59-69 (Russian); translated as J. Appl. Math. Mech., 32 (1968), 52-61.
doi: 10.1016/0021-8928(68)90147-0. |
[29] |
G. N. Watson,
A Treatise on the Theory of Bessel Functions, Reprint of the second edition, 1994, Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1995. |
[30] |
D. Wei,
Regularity criterion to the axially symmetric Navier-Stokes equations, J. Math. Anal. Appl., 435 (2016), 402-413.
doi: 10.1016/j.jmaa.2015.09.088. |
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