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Positive solutions for quasilinear Schrödinger equations in $\mathbb{R}^N$
Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces
1. | School of Information Technology, Jiangxi University of Finance and Economics, Nanchang 330032, China |
2. | Department of Mathematics, Northwest Normal University, Lanzhou 730070, China |
$\mathbb{C}$ |
$(u_{0}, n_{0}, c_{0})=(u_{0}^{h}, u_{0}^{3}, n_{0}, c_{0})$ |
$\begin{aligned}&\|(n_{0}, c_{0},u_{0}^{h})\|_{\dot{B}^{-2+3/q}_{q, 1}× \dot{B}^{-2+3/q}_{q, 1}×\dot{B}^{-1+3/p}_{p, 1}}+\|u_{0}^{h}\|_{\dot{B}^{-1+3/p}_{p, 1}}^{α}\|u_{0}^{3}\|_{\dot{B}^{-1+3/p}_{p, 1}}^{1-α}≤q1/\mathbb{C}\end{aligned}$ |
$p, q, α$ |
$1<p<q≤ 2p<\infty, \frac{1}{p}+\frac{1}{q}>\frac{1}{3}, 1< q<6, \frac{1}{p}-\frac{1}{q}≤\frac{1}{3}$ |
References:
[1] |
A. Biswas, V. Martinez and P. Silva,
On Gevrey regularity of the supercritical SQG equation in critical Besov spaces, J. Funct. Anal., 269 (2015), 3083-3119.
doi: 10.1016/j.jfa.2015.08.010. |
[2] |
H. Bae,
Existence and analyticity of Lei-Lin Solution to the Navier-Stokes equations, Proc. Amer. Math. Soc., 143 (2015), 2887-2892.
doi: 10.1090/S0002-9939-2015-12266-6. |
[3] |
H. Bae, A. Biswas and E. Tadmor,
Analyticity and decay estimates of the Navier-Stokes equations in critical Besov spaces, Arch. Ration. Mech. Anal., 205 (2012), 963-991.
doi: 10.1007/s00205-012-0532-5. |
[4] |
A. Biswas,
Gevrey regularity for a class of dissipative equations with applications to decay, J. Differ. Equ., 253 (2012), 2739-2764.
doi: 10.1016/j.jde.2012.08.003. |
[5] |
A. Biswas and D. Swanson,
Gevrey regularity of solutions to the 3D Navier-Stokes equations with weighted $\ell^{p}$ initial data, Indiana Univ. Math. J., 56 (2007), 1157-1188.
doi: 10.1512/iumj.2007.56.2891. |
[6] |
J. Y. Chemin,
Théorémes dunicité pour le systéme de Navier-Stokes tridimensionnal, J. Anal. Math., 77 (1999), 27-50.
doi: 10.1007/BF02791256. |
[7] |
J. Y. Chemin, M. Paicu and P. Zhang,
Global large solutions to 3-D inhomogeneous Navier-Stokes system with one slow variable diffusion, J. Differ. Equ., 256 (2014), 223-252.
doi: 10.1016/j.jde.2013.09.004. |
[8] |
H. Kozono and M. Yamazaki,
Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differ. Equ., 19 (1994), 959-1014.
doi: 10.1080/03605309408821042. |
[9] |
M. Cannone,
Harmonic analysis tools for solving the incompressible Navier-Stokes equations, Handbook of Mathematical Fluid Dynamics, 3 (2004), 161-244.
|
[10] |
R. Danchin,
Fourier Analysis Methods for PDEs, Lecture Notes, 14 November (2005). |
[11] |
R. Danchin,
Local theory in critical spaces for compressible viscous and heat-conducting gases, Comm. Partial Differ. Equ., 26 (2001), 1183-31233.
doi: 10.1081/PDE-100106132. |
[12] |
C. Deng, J. Zhao and S. Cui,
Well-posedness of a dissipative nonlinear electrohydrodynamic system in modulation spaces, Nonlinear Anal. Theory Methods Appl., 73 (2010), 2088-2100.
doi: 10.1016/j.na.2010.05.037. |
[13] |
C. Deng, J. Zhao and S. Cui,
Well-posedness for the Navier-Stokes-Nernst-Planck-Poisson system in Triebel-Lizorkin space and Besov space with negative indices, J. Math. Anal. Appl., 377 (2011), 392-405.
doi: 10.1016/j.jmaa.2010.11.011. |
[14] |
H. Fujita and T. Kato,
On the Navier-Stokes initial value problem Ⅰ, Arch. Ration. Mech. Anal., 16 (1964), 269-315.
doi: 10.1007/BF00276188. |
[15] |
C. Foias and R. Temam,
Gevrey class regularity for the solutions of the Navier-Stokes equations, J. of Funct. Anal., 87 (1989), 359-369.
doi: 10.1016/0022-1236(89)90015-3. |
[16] |
G. Gui and P. Zhang,
Stability to the global large solutions of 3D Navier-Stokes equations, Adv. Math., 225 (2010), 1248-1284.
doi: 10.1016/j.aim.2010.03.022. |
[17] |
B. Hajer, Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Berlin, 2011.
doi: 10.1007/978-3-642-16830-7.![]() ![]() ![]() |
[18] |
J. Huang, M. Paicu and P. Zhang,
Global well-posedness of incompressible inhomogeneous fluid systems with bounded density or non-Lipschitz velocity, Arch. Ration. Mech. Anal., 209 (2013), 631-382.
doi: 10.1007/s00205-013-0624-x. |
[19] |
T. Kato,
Strong $L^p$-solutions of the Navier-Stokes equation in $\mathbb{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[20] |
H. Koch and D. Tataru,
Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.
doi: 10.1006/aima.2000.1937. |
[21] |
Z. Lei and F. Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math. , 64, 1297-1304.
doi: 10.1002/cpa.20361. |
[22] |
J. Leray,
Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.
doi: 10.1007/BF02547354. |
[23] |
E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.
![]() ![]() |
[24] |
M. Paicu,
équation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoam., 225 (2010), 1248-1284.
doi: 10.4171/RMI/420. |
[25] |
M. Paicu and P. Zhang,
Global solutions to the 3D incompressible anisotropic Navier-Stokes system in the critical spaces, Comm. Math. Phys., 307 (2011), 713-759.
doi: 10.1007/s00220-011-1350-6. |
[26] |
M. Paicu and P. Zhang,
Global solutions to the 3D incompressible inhomogeneous Navier-Stokes system, J. Funct. Anal., 262 (2012), 3556-3584.
doi: 10.1016/j.jfa.2012.01.022. |
[27] |
F. Planchon,
Sur un in$\acute{e}$galit$\acute{e}$ de type Poincar$\acute{e}$, C. R. Acad. Sci. Paris S$\acute{e}$r. Ⅰ Math., 330 (2000), 21-23.
doi: 10.1016/S0764-4442(00)88138-0. |
[28] |
B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Methods for Nonlinear Evolution Equations, World Scientific, 2011.
doi: 10.1142/9789814360746.![]() ![]() ![]() |
[29] |
J. Xiao,
Homothetic variant of fractional Sobolev space with application to Navier-Stokes system, Dyn. Partial Differ. Equ., 4 (2007), 227-245.
doi: 10.4310/DPDE.2007.v4.n3.a2. |
[30] |
M. Z. Bazant, K. Thornton and A. Ajdari,
Diffuse-charge dynamics in electrochemical systems, Phys. Rev. E., 70 (2004), 021506.
|
[31] |
J. W. Joseph,
Analytical approaches to charge transport in a moving medium, Transport Theory Statist. Phys., 31 (2002), 333-366.
doi: 10.1081/TT-120015505. |
[32] |
F. Li,
Quasineutral limit of the electro-diffusion model arising in electrohydrodynamics, J. Differ. Equ., 246 (2009), 3620-3641.
doi: 10.1016/j.jde.2009.01.027. |
[33] |
F. Lin,
Some analytical issues for elastic complex fluids, Comm. Pure Appl. Math., 65 (2012), 893-919.
doi: 10.1002/cpa.21402. |
[34] |
J. Newman and K. Thomas, Electrochemical Systems, thirded., John Wiley Sons, 2004.
![]() |
[35] |
R. Ryham, An energetic variational approach to mathematical modeling of charged fluids: charge phases, simulation and well posedness (Doctoral dissertation), The Pennsylvania State University, 2006, p. 83. |
[36] |
M. Schmuck,
Analysis of the Navier-Stokes-Nernst-Planck-Poisson system, Math. Models Methods Appl. Sci., 19 (2009), 993-1014.
doi: 10.1142/S0218202509003693. |
[37] |
C. Huang and B. Wang, Analyticity for the (generalized) Navier-Stokes equations with rough initial data,
http://arxiv.org/abs/1310.2141. |
[38] |
J. Xiao,
Homothetic variant of fractional Sobolev space with application to Navier-Stokes system revisited, Dyn. Partial Differ. Equ., 11 (2014), 167-181.
doi: 10.4310/DPDE.2014.v11.n2.a3. |
[39] |
C. Zhai and T. Zhang, Global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity J. Math. Phys. , 56 091512 (2015).
doi: 10.1063/1.4931467. |
[40] |
J. Zhao, C. Deng and S. Cui,
Global well-posedness of a dissipative system arising in electrohydrodynamics in negative-order Besov spaces, J. Math. Phys., 51 (2010), 093-101.
doi: 10.1063/1.3484184. |
[41] |
J. Zhao, C. Deng and S. Cui,
Well-posedness of a dissipative system modeling electrohydrodynamics in Lebesgue spaces, Differential Equations Appl., 3 (2011), 427-448.
doi: 10.7153/dea-03-27. |
[42] |
J. Zhao, T. Zhang and Q Liu,
Global well-posedness for the dissipative system modeling electro-hydrodynamics with large vertical velocity component in critical Besov space, Discrete Contin. Dyn. Syst., 35 (2015), 555-582.
doi: 10.3934/dcds.2015.35.555. |
show all references
References:
[1] |
A. Biswas, V. Martinez and P. Silva,
On Gevrey regularity of the supercritical SQG equation in critical Besov spaces, J. Funct. Anal., 269 (2015), 3083-3119.
doi: 10.1016/j.jfa.2015.08.010. |
[2] |
H. Bae,
Existence and analyticity of Lei-Lin Solution to the Navier-Stokes equations, Proc. Amer. Math. Soc., 143 (2015), 2887-2892.
doi: 10.1090/S0002-9939-2015-12266-6. |
[3] |
H. Bae, A. Biswas and E. Tadmor,
Analyticity and decay estimates of the Navier-Stokes equations in critical Besov spaces, Arch. Ration. Mech. Anal., 205 (2012), 963-991.
doi: 10.1007/s00205-012-0532-5. |
[4] |
A. Biswas,
Gevrey regularity for a class of dissipative equations with applications to decay, J. Differ. Equ., 253 (2012), 2739-2764.
doi: 10.1016/j.jde.2012.08.003. |
[5] |
A. Biswas and D. Swanson,
Gevrey regularity of solutions to the 3D Navier-Stokes equations with weighted $\ell^{p}$ initial data, Indiana Univ. Math. J., 56 (2007), 1157-1188.
doi: 10.1512/iumj.2007.56.2891. |
[6] |
J. Y. Chemin,
Théorémes dunicité pour le systéme de Navier-Stokes tridimensionnal, J. Anal. Math., 77 (1999), 27-50.
doi: 10.1007/BF02791256. |
[7] |
J. Y. Chemin, M. Paicu and P. Zhang,
Global large solutions to 3-D inhomogeneous Navier-Stokes system with one slow variable diffusion, J. Differ. Equ., 256 (2014), 223-252.
doi: 10.1016/j.jde.2013.09.004. |
[8] |
H. Kozono and M. Yamazaki,
Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differ. Equ., 19 (1994), 959-1014.
doi: 10.1080/03605309408821042. |
[9] |
M. Cannone,
Harmonic analysis tools for solving the incompressible Navier-Stokes equations, Handbook of Mathematical Fluid Dynamics, 3 (2004), 161-244.
|
[10] |
R. Danchin,
Fourier Analysis Methods for PDEs, Lecture Notes, 14 November (2005). |
[11] |
R. Danchin,
Local theory in critical spaces for compressible viscous and heat-conducting gases, Comm. Partial Differ. Equ., 26 (2001), 1183-31233.
doi: 10.1081/PDE-100106132. |
[12] |
C. Deng, J. Zhao and S. Cui,
Well-posedness of a dissipative nonlinear electrohydrodynamic system in modulation spaces, Nonlinear Anal. Theory Methods Appl., 73 (2010), 2088-2100.
doi: 10.1016/j.na.2010.05.037. |
[13] |
C. Deng, J. Zhao and S. Cui,
Well-posedness for the Navier-Stokes-Nernst-Planck-Poisson system in Triebel-Lizorkin space and Besov space with negative indices, J. Math. Anal. Appl., 377 (2011), 392-405.
doi: 10.1016/j.jmaa.2010.11.011. |
[14] |
H. Fujita and T. Kato,
On the Navier-Stokes initial value problem Ⅰ, Arch. Ration. Mech. Anal., 16 (1964), 269-315.
doi: 10.1007/BF00276188. |
[15] |
C. Foias and R. Temam,
Gevrey class regularity for the solutions of the Navier-Stokes equations, J. of Funct. Anal., 87 (1989), 359-369.
doi: 10.1016/0022-1236(89)90015-3. |
[16] |
G. Gui and P. Zhang,
Stability to the global large solutions of 3D Navier-Stokes equations, Adv. Math., 225 (2010), 1248-1284.
doi: 10.1016/j.aim.2010.03.022. |
[17] |
B. Hajer, Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Berlin, 2011.
doi: 10.1007/978-3-642-16830-7.![]() ![]() ![]() |
[18] |
J. Huang, M. Paicu and P. Zhang,
Global well-posedness of incompressible inhomogeneous fluid systems with bounded density or non-Lipschitz velocity, Arch. Ration. Mech. Anal., 209 (2013), 631-382.
doi: 10.1007/s00205-013-0624-x. |
[19] |
T. Kato,
Strong $L^p$-solutions of the Navier-Stokes equation in $\mathbb{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[20] |
H. Koch and D. Tataru,
Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.
doi: 10.1006/aima.2000.1937. |
[21] |
Z. Lei and F. Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math. , 64, 1297-1304.
doi: 10.1002/cpa.20361. |
[22] |
J. Leray,
Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.
doi: 10.1007/BF02547354. |
[23] |
E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.
![]() ![]() |
[24] |
M. Paicu,
équation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoam., 225 (2010), 1248-1284.
doi: 10.4171/RMI/420. |
[25] |
M. Paicu and P. Zhang,
Global solutions to the 3D incompressible anisotropic Navier-Stokes system in the critical spaces, Comm. Math. Phys., 307 (2011), 713-759.
doi: 10.1007/s00220-011-1350-6. |
[26] |
M. Paicu and P. Zhang,
Global solutions to the 3D incompressible inhomogeneous Navier-Stokes system, J. Funct. Anal., 262 (2012), 3556-3584.
doi: 10.1016/j.jfa.2012.01.022. |
[27] |
F. Planchon,
Sur un in$\acute{e}$galit$\acute{e}$ de type Poincar$\acute{e}$, C. R. Acad. Sci. Paris S$\acute{e}$r. Ⅰ Math., 330 (2000), 21-23.
doi: 10.1016/S0764-4442(00)88138-0. |
[28] |
B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Methods for Nonlinear Evolution Equations, World Scientific, 2011.
doi: 10.1142/9789814360746.![]() ![]() ![]() |
[29] |
J. Xiao,
Homothetic variant of fractional Sobolev space with application to Navier-Stokes system, Dyn. Partial Differ. Equ., 4 (2007), 227-245.
doi: 10.4310/DPDE.2007.v4.n3.a2. |
[30] |
M. Z. Bazant, K. Thornton and A. Ajdari,
Diffuse-charge dynamics in electrochemical systems, Phys. Rev. E., 70 (2004), 021506.
|
[31] |
J. W. Joseph,
Analytical approaches to charge transport in a moving medium, Transport Theory Statist. Phys., 31 (2002), 333-366.
doi: 10.1081/TT-120015505. |
[32] |
F. Li,
Quasineutral limit of the electro-diffusion model arising in electrohydrodynamics, J. Differ. Equ., 246 (2009), 3620-3641.
doi: 10.1016/j.jde.2009.01.027. |
[33] |
F. Lin,
Some analytical issues for elastic complex fluids, Comm. Pure Appl. Math., 65 (2012), 893-919.
doi: 10.1002/cpa.21402. |
[34] |
J. Newman and K. Thomas, Electrochemical Systems, thirded., John Wiley Sons, 2004.
![]() |
[35] |
R. Ryham, An energetic variational approach to mathematical modeling of charged fluids: charge phases, simulation and well posedness (Doctoral dissertation), The Pennsylvania State University, 2006, p. 83. |
[36] |
M. Schmuck,
Analysis of the Navier-Stokes-Nernst-Planck-Poisson system, Math. Models Methods Appl. Sci., 19 (2009), 993-1014.
doi: 10.1142/S0218202509003693. |
[37] |
C. Huang and B. Wang, Analyticity for the (generalized) Navier-Stokes equations with rough initial data,
http://arxiv.org/abs/1310.2141. |
[38] |
J. Xiao,
Homothetic variant of fractional Sobolev space with application to Navier-Stokes system revisited, Dyn. Partial Differ. Equ., 11 (2014), 167-181.
doi: 10.4310/DPDE.2014.v11.n2.a3. |
[39] |
C. Zhai and T. Zhang, Global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity J. Math. Phys. , 56 091512 (2015).
doi: 10.1063/1.4931467. |
[40] |
J. Zhao, C. Deng and S. Cui,
Global well-posedness of a dissipative system arising in electrohydrodynamics in negative-order Besov spaces, J. Math. Phys., 51 (2010), 093-101.
doi: 10.1063/1.3484184. |
[41] |
J. Zhao, C. Deng and S. Cui,
Well-posedness of a dissipative system modeling electrohydrodynamics in Lebesgue spaces, Differential Equations Appl., 3 (2011), 427-448.
doi: 10.7153/dea-03-27. |
[42] |
J. Zhao, T. Zhang and Q Liu,
Global well-posedness for the dissipative system modeling electro-hydrodynamics with large vertical velocity component in critical Besov space, Discrete Contin. Dyn. Syst., 35 (2015), 555-582.
doi: 10.3934/dcds.2015.35.555. |
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