doi: 10.3934/dcdsb.2020237

Global existence and Gevrey regularity to the Navier-Stokes-Nernst-Planck-Poisson system in critical Besov-Morrey spaces

1. 

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, China

2. 

ICT School, The University of Suwon, Wau-ri, Bongdam-eup, Hwaseong-si, Gyeonggi-do, 445-743, South Korea

3. 

School of Mathematical Sciences, Qufu Normal University, Qufu, 273100, China

4. 

Department of Mathematics, Jiangxi University of Finance and Economics, Nanchang, 330032, China

*Corresponding author: Zunwei Fu

Received  March 2020 Revised  May 2020 Published  August 2020

Fund Project: This paper was partially supported by the National Natural Science Foundation of China (Grant No. 11801236), the Postdoctoral Science Foundation of China (Grant Nos. 2018M632593, 2019M660555), the Natural Science Foundation of Gansu Province for Young Scholars (Grant No. 18JR3RA102), the innovation capacity improvement project for colleges and universities of Gansu Province (Grant No. 2019A-011), the Natural Science Foundation of Jiangxi Province for Young Scholars (Grant No. 20181BAB211001), the Postdoctoral Science Foundation of Jiangxi Province (Grant No. 2017KY23) and Educational Commission Science Programm of Jiangxi Province (Grant No. GJJ190272)

The paper is concerned with the Navier-Stokes-Nernst-Planck-Poisson system arising from electrohydrodynamics in $ \mathbb{R}^d $. By means of the implicit function theorem, we prove the global existence of mild solutions for Cauchy problem of this system with small initial data in critical Besov-Morrey spaces. In comparison to the previous works, our existence result provides a new class of initial data, for which the problem is global solvability. Meanwhile, based on the so-called Gevrey estimates, we verify that the obtained mild solutions are analytic in the spatial variables. As a byproduct, we show the asymptotic stability of solutions as the time goes to infinity. Furthermore, decay estimates of higher-order derivatives of solutions are deduced in Morrey spaces.

Citation: Jinyi Sun, Zunwei Fu, Yue Yin, Minghua Yang. Global existence and Gevrey regularity to the Navier-Stokes-Nernst-Planck-Poisson system in critical Besov-Morrey spaces. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020237
References:
[1]

D. R. Adams, A note on Riesz potentials, Duke Math. J., 42 (1975), 765-778.  doi: 10.1215/S0012-7094-75-04265-9.  Google Scholar

[2]

H. BaeA. 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.  Google Scholar

[3]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, vol. 343, Springer-Verlag, Berlin, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[4]

M. Z. Bazant, K. Thornton and A. Ajdari, Diffuse-charge dynamics in electrochemical systems, Phys. Rev. E, 70 (2004), 021506. doi: 10.1103/PhysRevE.70.021506.  Google Scholar

[5]

M. Cannone, Y. Meyer and F. Planchon, Solutions auto-similaires des équations de Navier-Stokes(French), Séminaire sur les équations aux Dérivées Partielles, 1993–1994.  Google Scholar

[6]

M. Cannone and G. Wu, Global well-posedness for Navier-Stokes equations in critical Fourier-Herz spaces, Nonlinear Anal., 75 (2012) 3754–3760. doi: 10.1016/j.na.2012.01.029.  Google Scholar

[7]

C. DengJ. Zhao and S. Cui, Well-posedness of a dissipative nonlinear electrohydrodynamic system in modulation spaces, Nonlinear Anal., 73 (2010), 2088-2100.  doi: 10.1016/j.na.2010.05.037.  Google Scholar

[8]

C. DengJ. 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.  Google Scholar

[9]

C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369. doi: 10.1016/0022-1236(89)90015-3.  Google Scholar

[10]

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.  Google Scholar

[11]

C. Huang and B. Wang, Analyticity for the (generalized) Navier-Stokes equations with rough initial data, arXiv: 1310.2141. Google Scholar

[12]

T. Iwabuchi and R. Takada, Global well-posedness and ill-posedness for the Navier-Stokes equations with the Coriolis force in function spaces of Besov type, J. Funct. Anal., 267 (2014), 1321-1337.  doi: 10.1016/j.jfa.2014.05.022.  Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

T. Kato, Strong solutions of the Navier-Stokes equation in Morrey spaces, Bol. Soc. Brasil. Mat.(N.S.), 22 (1992), 127–155. doi: 10.1007/BF01232939.  Google Scholar

[16]

H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937.  Google Scholar

[17]

P. Konieczny and T. Yoneda, On dispersive effect of the Coriolis force for the stationary Navier-Stokes equations, J. Differential Equations, 250 (2011), 3859-3873.  doi: 10.1016/j.jde.2011.01.003.  Google Scholar

[18]

H. Kozono and M. Yamazaki, Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differential Equations, 19 (1994), 959-1014.  doi: 10.1080/03605309408821042.  Google Scholar

[19]

Z. Lei and F. Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 1297-1304. doi: 10.1002/cpa.20361.  Google Scholar

[20] P. G. Lemarié-Rieusset, The Navier-Stokes Problem in the 21st Century, CRC Press, Boca Raton, FL, 2016.  doi: 10.1201/b19556.  Google Scholar
[21]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[22]

Q. Liu and S. Cui, Regularizing rate estimates for mild solutions of the incompressible magneto-hydrodynamic system, Commun. Pure Appl. Anal., 11 (2012), 1643–1660. doi: 10.3934/cpaa.2012.11.1643.  Google Scholar

[23]

Q. LiuJ. Zhao and S. Cui, Existence and regularizing rate estimates of solutions to a generalized magneto-hydrodynamic system in pseudomeasure spaces, Ann. Mat. Pura Appl., 191 (2012), 293-309.  doi: 10.1007/s10231-010-0184-8.  Google Scholar

[24]

F. Li, Quasineutral limit of the electro-diffusion model arising in electrohydrodynamics, J. Differential Equations, 246 (2009), 3620-3641.  doi: 10.1016/j.jde.2009.01.027.  Google Scholar

[25]

A. L. Mazzucato, Besov-Morrey spaces function space theory and applications to non-linear PDE, Trans. Amer. Math. Soc., 355 (2003), 1297-1364.  doi: 10.1090/S0002-9947-02-03214-2.  Google Scholar

[26]

J. Newman and K. Thomas-Alyea, Electrochemical Systems(3rd Edition), J. Wiley, Hoboken, 2004. Google Scholar

[27]

M. Oliver and E. S. Titi, Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations in $\mathbb{R}^n$, J. Funct. Anal., 172 (2000), 1-18.  doi: 10.1006/jfan.1999.3550.  Google Scholar

[28]

R. J. Ryham, An energetic variational approach to mathematical modeling of charged fluids: Charge phases, simulation and well posedness (Doctoral dissertation), The Pennsylvania State University, 2006, 83pp.  Google Scholar

[29]

M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system, Math. Models Methods Appl. Sci., 19 (2009), 993-1014.  doi: 10.1142/S0218202509003693.  Google Scholar

[30]

J. Sun, M. Yang and S. Cui, Existence and analyticity of mild solutions for the 3D rotating Navier-Stokes equations, Ann. Mat. Pura Appl., 196 (2017), no. 4, 1203–1229. doi: 10.1007/s10231-016-0613-4.  Google Scholar

[31]

M. E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations, 17 (1992), 1407–1456. doi: 10.1080/03605309208820892.  Google Scholar

[32]

M. Yang, Z. Fu and J. Sun, Existence and Gevrey regularity for a two-species chemotaxis system in homogeneous Besov spaces, Sci. China Math., 60 (2017), 1837-1856. doi: 10.1007/s11425-016-0490-y.  Google Scholar

[33]

M. Yang and J. Sun, Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces, Commun. Pure Appl. Anal., 16 (2017), 1617-1639.  doi: 10.3934/cpaa.2017078.  Google Scholar

[34]

M. YangZ. Fu and J. Sun, Existence and large time behavior to coupled chemotaxis-fluid equations in Besov-Morrey spaces, J. Differential Equations, 266 (2019), 5867-5894.  doi: 10.1016/j.jde.2018.10.050.  Google Scholar

[35]

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), 093101. doi: 10.1063/1.3484184.  Google Scholar

[36]

J. ZhaoC. 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.  Google Scholar

[37]

J. ZhaoQ. Liu and S. Cui, Regularizing and decay rate estimates for solutions to the Cauchy problem of the Debye-Hückel system, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 1-18.  doi: 10.1007/s00030-011-0115-4.  Google Scholar

show all references

References:
[1]

D. R. Adams, A note on Riesz potentials, Duke Math. J., 42 (1975), 765-778.  doi: 10.1215/S0012-7094-75-04265-9.  Google Scholar

[2]

H. BaeA. 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.  Google Scholar

[3]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, vol. 343, Springer-Verlag, Berlin, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[4]

M. Z. Bazant, K. Thornton and A. Ajdari, Diffuse-charge dynamics in electrochemical systems, Phys. Rev. E, 70 (2004), 021506. doi: 10.1103/PhysRevE.70.021506.  Google Scholar

[5]

M. Cannone, Y. Meyer and F. Planchon, Solutions auto-similaires des équations de Navier-Stokes(French), Séminaire sur les équations aux Dérivées Partielles, 1993–1994.  Google Scholar

[6]

M. Cannone and G. Wu, Global well-posedness for Navier-Stokes equations in critical Fourier-Herz spaces, Nonlinear Anal., 75 (2012) 3754–3760. doi: 10.1016/j.na.2012.01.029.  Google Scholar

[7]

C. DengJ. Zhao and S. Cui, Well-posedness of a dissipative nonlinear electrohydrodynamic system in modulation spaces, Nonlinear Anal., 73 (2010), 2088-2100.  doi: 10.1016/j.na.2010.05.037.  Google Scholar

[8]

C. DengJ. 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.  Google Scholar

[9]

C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369. doi: 10.1016/0022-1236(89)90015-3.  Google Scholar

[10]

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.  Google Scholar

[11]

C. Huang and B. Wang, Analyticity for the (generalized) Navier-Stokes equations with rough initial data, arXiv: 1310.2141. Google Scholar

[12]

T. Iwabuchi and R. Takada, Global well-posedness and ill-posedness for the Navier-Stokes equations with the Coriolis force in function spaces of Besov type, J. Funct. Anal., 267 (2014), 1321-1337.  doi: 10.1016/j.jfa.2014.05.022.  Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

T. Kato, Strong solutions of the Navier-Stokes equation in Morrey spaces, Bol. Soc. Brasil. Mat.(N.S.), 22 (1992), 127–155. doi: 10.1007/BF01232939.  Google Scholar

[16]

H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937.  Google Scholar

[17]

P. Konieczny and T. Yoneda, On dispersive effect of the Coriolis force for the stationary Navier-Stokes equations, J. Differential Equations, 250 (2011), 3859-3873.  doi: 10.1016/j.jde.2011.01.003.  Google Scholar

[18]

H. Kozono and M. Yamazaki, Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data, Comm. Partial Differential Equations, 19 (1994), 959-1014.  doi: 10.1080/03605309408821042.  Google Scholar

[19]

Z. Lei and F. Lin, Global mild solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 64 (2011), 1297-1304. doi: 10.1002/cpa.20361.  Google Scholar

[20] P. G. Lemarié-Rieusset, The Navier-Stokes Problem in the 21st Century, CRC Press, Boca Raton, FL, 2016.  doi: 10.1201/b19556.  Google Scholar
[21]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[22]

Q. Liu and S. Cui, Regularizing rate estimates for mild solutions of the incompressible magneto-hydrodynamic system, Commun. Pure Appl. Anal., 11 (2012), 1643–1660. doi: 10.3934/cpaa.2012.11.1643.  Google Scholar

[23]

Q. LiuJ. Zhao and S. Cui, Existence and regularizing rate estimates of solutions to a generalized magneto-hydrodynamic system in pseudomeasure spaces, Ann. Mat. Pura Appl., 191 (2012), 293-309.  doi: 10.1007/s10231-010-0184-8.  Google Scholar

[24]

F. Li, Quasineutral limit of the electro-diffusion model arising in electrohydrodynamics, J. Differential Equations, 246 (2009), 3620-3641.  doi: 10.1016/j.jde.2009.01.027.  Google Scholar

[25]

A. L. Mazzucato, Besov-Morrey spaces function space theory and applications to non-linear PDE, Trans. Amer. Math. Soc., 355 (2003), 1297-1364.  doi: 10.1090/S0002-9947-02-03214-2.  Google Scholar

[26]

J. Newman and K. Thomas-Alyea, Electrochemical Systems(3rd Edition), J. Wiley, Hoboken, 2004. Google Scholar

[27]

M. Oliver and E. S. Titi, Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations in $\mathbb{R}^n$, J. Funct. Anal., 172 (2000), 1-18.  doi: 10.1006/jfan.1999.3550.  Google Scholar

[28]

R. J. Ryham, An energetic variational approach to mathematical modeling of charged fluids: Charge phases, simulation and well posedness (Doctoral dissertation), The Pennsylvania State University, 2006, 83pp.  Google Scholar

[29]

M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system, Math. Models Methods Appl. Sci., 19 (2009), 993-1014.  doi: 10.1142/S0218202509003693.  Google Scholar

[30]

J. Sun, M. Yang and S. Cui, Existence and analyticity of mild solutions for the 3D rotating Navier-Stokes equations, Ann. Mat. Pura Appl., 196 (2017), no. 4, 1203–1229. doi: 10.1007/s10231-016-0613-4.  Google Scholar

[31]

M. E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Differential Equations, 17 (1992), 1407–1456. doi: 10.1080/03605309208820892.  Google Scholar

[32]

M. Yang, Z. Fu and J. Sun, Existence and Gevrey regularity for a two-species chemotaxis system in homogeneous Besov spaces, Sci. China Math., 60 (2017), 1837-1856. doi: 10.1007/s11425-016-0490-y.  Google Scholar

[33]

M. Yang and J. Sun, Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces, Commun. Pure Appl. Anal., 16 (2017), 1617-1639.  doi: 10.3934/cpaa.2017078.  Google Scholar

[34]

M. YangZ. Fu and J. Sun, Existence and large time behavior to coupled chemotaxis-fluid equations in Besov-Morrey spaces, J. Differential Equations, 266 (2019), 5867-5894.  doi: 10.1016/j.jde.2018.10.050.  Google Scholar

[35]

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), 093101. doi: 10.1063/1.3484184.  Google Scholar

[36]

J. ZhaoC. 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.  Google Scholar

[37]

J. ZhaoQ. Liu and S. Cui, Regularizing and decay rate estimates for solutions to the Cauchy problem of the Debye-Hückel system, NoDEA Nonlinear Differential Equations Appl., 19 (2012), 1-18.  doi: 10.1007/s00030-011-0115-4.  Google Scholar

[1]

Minghua Yang, Jinyi Sun. Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1617-1639. doi: 10.3934/cpaa.2017078

[2]

Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319

[3]

Chia-Yu Hsieh. Stability of radial solutions of the Poisson-Nernst-Planck system in annular domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2657-2681. doi: 10.3934/dcdsb.2018269

[4]

Qunyi Bie, Qiru Wang, Zheng-An Yao. On the well-posedness of the inviscid Boussinesq equations in the Besov-Morrey spaces. Kinetic & Related Models, 2015, 8 (3) : 395-411. doi: 10.3934/krm.2015.8.395

[5]

Victor A. Kovtunenko, Anna V. Zubkova. Mathematical modeling of a discontinuous solution of the generalized Poisson-Nernst-Planck problem in a two-phase medium. Kinetic & Related Models, 2018, 11 (1) : 119-135. doi: 10.3934/krm.2018007

[6]

Peter Constantin, Gregory Seregin. Global regularity of solutions of coupled Navier-Stokes equations and nonlinear Fokker Planck equations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1185-1196. doi: 10.3934/dcds.2010.26.1185

[7]

Jonathan Zinsl. Exponential convergence to equilibrium in a Poisson-Nernst-Planck-type system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2915-2930. doi: 10.3934/dcds.2016.36.2915

[8]

Hammadi Abidi, Taoufik Hmidi, Sahbi Keraani. On the global regularity of axisymmetric Navier-Stokes-Boussinesq system. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 737-756. doi: 10.3934/dcds.2011.29.737

[9]

Nan Chen, Cheng Wang, Steven Wise. Global-in-time Gevrey regularity solution for a class of bistable gradient flows. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1689-1711. doi: 10.3934/dcdsb.2016018

[10]

Minghua Yang, Zunwei Fu, Jinyi Sun. Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3427-3460. doi: 10.3934/dcdsb.2018284

[11]

L. Bedin, Mark Thompson. Existence theory for a Poisson-Nernst-Planck model of electrophoresis. Communications on Pure & Applied Analysis, 2013, 12 (1) : 157-206. doi: 10.3934/cpaa.2013.12.157

[12]

Lan Luo, Hongjun Yu. Global solutions to the relativistic Vlasov-Poisson-Fokker-Planck system. Kinetic & Related Models, 2016, 9 (2) : 393-405. doi: 10.3934/krm.2016.9.393

[13]

Daoyuan Fang, Chenyin Qian. Regularity criterion for 3D Navier-Stokes equations in Besov spaces. Communications on Pure & Applied Analysis, 2014, 13 (2) : 585-603. doi: 10.3934/cpaa.2014.13.585

[14]

Keyan Wang. On global regularity of incompressible Navier-Stokes equations in $\mathbf R^3$. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1067-1072. doi: 10.3934/cpaa.2009.8.1067

[15]

John W. Barrett, Endre Süli. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 371-408. doi: 10.3934/dcdss.2010.3.371

[16]

Mimi Dai, Han Liu. Low modes regularity criterion for a chemotaxis-Navier-Stokes system. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2713-2735. doi: 10.3934/cpaa.2020118

[17]

Atanas Stefanov. On the Lipschitzness of the solution map for the 2 D Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1471-1490. doi: 10.3934/dcds.2010.26.1471

[18]

Yuhua Zhu. A local sensitivity and regularity analysis for the Vlasov-Poisson-Fokker-Planck system with multi-dimensional uncertainty and the spectral convergence of the stochastic Galerkin method. Networks & Heterogeneous Media, 2019, 14 (4) : 677-707. doi: 10.3934/nhm.2019027

[19]

Francesca Crispo, Paolo Maremonti. A remark on the partial regularity of a suitable weak solution to the Navier-Stokes Cauchy problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1283-1294. doi: 10.3934/dcds.2017053

[20]

Jiří Neustupa. A note on local interior regularity of a suitable weak solution to the Navier--Stokes problem. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1391-1400. doi: 10.3934/dcdss.2013.6.1391

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (29)
  • HTML views (71)
  • Cited by (0)

Other articles
by authors

[Back to Top]