March  2021, 41(3): 1387-1413. doi: 10.3934/dcds.2020322

Global large solutions and optimal time-decay estimates to the Korteweg system

1. 

School of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China

2. 

School of Mathematics, South China University of Technology, Guangzhou, 510640, China

Received  December 2019 Revised  May 2020 Published  March 2021 Early access  September 2020

We prove the global solutions to the Korteweg system without smallness condition imposed on the vertical component of the incompressible part of the velocity. The weighted Chemin-Lerner-norm technique which is well-known for the incompressible Navier-Stokes equations is introduced to derive the a priori estimates. As a byproduct, we obtain the optimal time decay rates of the solutions by using the pure energy argument (independent of spectral analysis). In contrast to the compressible Navier-Stokes system, the time-decay estimates are more accurate owing to smoothing effect from the Korteweg tensor.

Citation: Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322
References:
[1]

P. Antonelli, L. E. Hientzsch and S. Spirito, Global existence of finite energy weak solutions to the Quantum Navier-Stokes equations with non-trivial far-field behavior, arXiv: 2001.01652.

[2]

P. Antonelli and P. Marcati, On the finite energy weak solutions to a system in quantum fluid dynamics, Commun. Math. Phys., 287 (2009), 657-686.  doi: 10.1007/s00220-008-0632-0.

[3]

P. Antonelli and P. Marcati, The quantum hydrodynamics system in two space dimensions, Arch. Rational Mech. Anal., 203 (2012), 499-527.  doi: 10.1007/s00205-011-0454-7.

[4]

P. Antonelli and S. Spirito, Global existence of finite energy weak solutions of the quantum Navier-Stokes equations, Arch. Rational Mech. Anal., 255 (2017), 1161-1199.  doi: 10.1007/s00205-017-1124-1.

[5]

P. Antonelli and S. Spirito, On the compactness of weak solutions to the Navier-Stokes-Korteweg equations for capillary fluids, Nonlinear Anal., 187 (2019), 110-124.  doi: 10.1016/j.na.2019.03.020.

[6]

C. Audiard and B. Haspot, Global well-posedness of the Euler Korteweg system for small irrotational data, Commun. Math. Phys., 351 (2017), 201-247.  doi: 10.1007/s00220-017-2843-8.

[7]

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

[8]

D. BreschB. Desjardins and C.-K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Comm. Part. Diffe. Equ., 28 (2003), 843-868.  doi: 10.1081/PDE-120020499.

[9]

D. BreschM. Gisclon and I. Lacroix-Violet, On Navier-Stokes-Korteweg and Euler-Korteweg systems: Application to quantum fluids models, Arch. Rational Mech. Anal., 233 (2019), 975-1025.  doi: 10.1007/s00205-019-01373-w.

[10]

F. Charve and R. Danchin, A global existence result for the compressible Navier-Stokes equations in the critical $L^p$ framework, Arch. Rational Mech. Anal., 198 (2010), 233-271.  doi: 10.1007/s00205-010-0306-x.

[11]

F. Charve, R. Danchin and J. Xu, Gevrey analyticity and decay for the compressible Navier-Stokes system with capillarity, arXiv: 1805.01764.

[12]

J.-Y. Chemin and I. Gallagher, Wellposedness and stability results for the Navier-Stokes equations in ${\mathbb R}^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 599-624.  doi: 10.1016/j.anihpc.2007.05.008.

[13]

J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs no lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), 314–328. doi: 10.1006/jdeq.1995.1131.

[14]

Q. Chen, C. Miao and Z. Zhang, Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math., 63 (2010), 1173–1224. doi: 10.1002/cpa.20325.

[15]

Z.-M. Chen and X. Zhai, Global large solutions and incompressible limit for the compressible Navier-Stokes equations, J. Math. Fluid Mech., 21 (2019), Art. 26, 23 pp. doi: 10.1007/s00021-019-0428-3.

[16]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579–614. doi: 10.1007/s002220000078.

[17]

R. Danchin and B. Desjardins, Existence of solutions for compressible fluid models of Korteweg type, Annales de l'IHP, Analyse nonlinéaire, 18 (2001), 97-133.  doi: 10.1016/S0294-1449(00)00056-1.

[18]

R. Danchin and L. He, The incompressible limit in $L^p$ type critical spaces, Math. Ann., 366 (2016), 1365-1402.  doi: 10.1007/s00208-016-1361-x.

[19]

R. Danchin and P. B. Mucha, Compressible Navier-Stokes system: large solutions and incompressible limit, Adv. Math., 320 (2017), 904-925.  doi: 10.1016/j.aim.2017.09.025.

[20]

R. Danchin and J. Xu, Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical $L^p$ framework, Arch. Rational Mech. Anal., 224 (2017), 53-90.  doi: 10.1007/s00205-016-1067-y.

[21]

D. DonatelliE. Feireisl and P. Marcati, Well/ill posedness for the Euler-Korteweg-Poisson system and related problems, Comm. Part. Diffe. Equ., 40 (2015), 1314-1335.  doi: 10.1080/03605302.2014.972517.

[22]

J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Rational Mech. Anal., 88 (1985), 95-133.  doi: 10.1007/BF00250907.

[23]

E. Feireisl, Dynamics of Viscous Compressible Fluids., Oxford Univ. Press, Oxford, 2004.

[24]

E. Feireisl and A. Novotný, H. Petzeltová, On the global existence of globally defined weak solutions to the Navier-Stokes equations of isentropic compressible fluids, J. Math. Fluid Mech., 3 (2001), 358–392. doi: 10.1007/PL00000976.

[25]

E. Feireisl, Compressible Navier-Stokes equations with a non-monotone pressure law, J. Differential Equations, 184 (2002), 97–108. doi: 10.1006/jdeq.2001.4137.

[26]

E. Feireisl, A. Novotný and Y. Sun, Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids, Indiana Univ. Math. J., 60 (2011), 611–631. doi: 10.1512/iumj.2011.60.4406.

[27]

A. N. Gorban and I. V. Karlin, Beyond Navier-Stokes equations: Capillarity of ideal gas, Contemporary physics, 58 (2017), 70-90.  doi: 10.1080/00107514.2016.1256123.

[28]

G. Gui and P. Zhang, Stability to the global solutions of 3-D Navier-Stokes equations, Adv. Math., 225 (2010), 1248-1284.  doi: 10.1016/j.aim.2010.03.022.

[29]

B. Haspot, Existence of global weak solution for compressible fluid models of Korteweg type, J. Math. Fluid Mech., 13 (2011), 223-249.  doi: 10.1007/s00021-009-0013-2.

[30]

B. Haspot, Well-posedness in critical spaces for the system of compressible Navier-Stokes in larger spaces, J. Differential Equations, 251 (2011), 2262-2295.  doi: 10.1016/j.jde.2011.06.013.

[31]

B. Haspot, Existence of global strong solution for Korteweg system with large infinite energy initial data, J. Math. Anal. Appl., 438 (2016), 395-443.  doi: 10.1016/j.jmaa.2016.01.047.

[32]

B. Haspot, Global strong solution for the Korteweg system with quantum pressure in dimension $N \geq 2$, Math. Ann., 367 (2017), 667-700.  doi: 10.1007/s00208-016-1391-4.

[33]

H. Hattori and D. Li, The existence of global solutions to a fluid dynamic model for materials for Korteweg type, J. Partial Differential Equations, 9 (1996), 323-342. 

[34]

L. HeJ. Huang and C. Wang, Global stability of large solutions to the 3D compressible Navier-Stokes equations, Arch. Rational Mech. Anal., 234 (2019), 1167-1222.  doi: 10.1007/s00205-019-01410-8.

[35]

M. Heida and J. Málek, On compressible Korteweg fluid-like materials, Internat. J. Engrg. Sci., 48 (2010), 1313-1324.  doi: 10.1016/j.ijengsci.2010.06.031.

[36]

A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045.  doi: 10.1137/090776068.

[37]

M. Kawashita, On global solution of Cauchy problems for compressible Navier-Stokes equation, Nonlinear Anal., 48 (2002), 1087-1105.  doi: 10.1016/S0362-546X(00)00238-8.

[38]

D. J. Korteweg, Sur la forme que prennent les équations du mouvement des fluides si l'on tient compte des forces capillaires par des variations de densité, Arch. Néer. Sci. Exactes Sér., 6 (1901), 1-24. 

[39]

M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type, Annales de l'IHP, Analyse nonlinéaire, 25 (2008), 679-696.  doi: 10.1016/j.anihpc.2007.03.005.

[40]

H.-K. Li and T. Zhang, Large time behavior of isentropic compressible Navier-Stokes system in ${\mathbb R}^3$, Math. Methods Appl. Sci., 34 (2011), 670-682.  doi: 10.1002/mma.1391.

[41]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  doi: 10.1215/kjm/1250522322.

[42]

M. Murata and Y. Shibata, The global well-posedness for the compressible fluid model of Korteweg type, arXiv: 1908.07224.

[43]

M. Okita, Optimal decay rate for strong solutions in critical spaces to the compressible Navier-Stokes equations, J. Differential Equations, 257 (2014), 3850-3867.  doi: 10.1016/j.jde.2014.07.011.

[44]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces, Comm. Math. Phys., 307 (2011), 713-759.  doi: 10.1007/s00220-011-1350-6.

[45]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible inhomogeneous Navier- Stokes system, J. Funct. Anal., 262 (2012), 3556-3584.  doi: 10.1016/j.jfa.2012.01.022.

[46]

G. Ponce, Global existence of small solution to a class of nonlinear evolution equations, Nonlinear Anal. TMA., 9 (1985), 339-418.  doi: 10.1016/0362-546X(85)90001-X.

[47]

K. Takayuki and T. Kazuyuki, Global existence and time decay estimate of solutions to the compressible Navier-Stokes-Korteweg system under critical condition, Asympt. Anal., (2020), Publishing. doi: 10.3233/ASY-201600.

[48]

J. F. Van der Waals, Thermodynamische Theorie der Kapillarität unter Voraussetzung stetiger Dichteänderung, Phys. Chem., 13 (1894), 657-725.  doi: 10.1515/zpch-1894-1338.

[49]

K. Watanabe, Global existence of the Navier-Stokes-Korteweg equations with a non-decreasing pressure in $L^p$-framework, arXiv: 1907.07752.

[50]

Z. Xin and J. Xu, Optimal decay for the compressible Navier-Stokes equations without additional smallness assumptions, arXiv: 1812.11714v2.

[51]

H. Xu, Y. Li and F. Chen, Global solution to the incompressible inhomogeneous Navier-Stokes equations with some large initial data, J. Math. Fluid Mech., 19 (2017), 315–328. doi: 10.1007/s00021-016-0282-5.

[52]

X. ZhaiY. Li and F. Zhou, Global large solutions to the three dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal., 52 (2020), 1806-1843.  doi: 10.1137/19M1265843.

[53]

S. Zhang, A class of global large solutions to the compressible Navier-Stokes-Korteweg system in critical Besov spaces, J. Evol. Equ., (2020). doi: 10.1007/s00028-020-00565-2.

[54]

T. Zhang, Global wellposedness problem for the 3-D incompressible anisotropic Navier-Stokes equations in an anisotropic space, Comm. Math. Phys., 287 (2009), 211-224.  doi: 10.1007/s00220-008-0631-1.

show all references

References:
[1]

P. Antonelli, L. E. Hientzsch and S. Spirito, Global existence of finite energy weak solutions to the Quantum Navier-Stokes equations with non-trivial far-field behavior, arXiv: 2001.01652.

[2]

P. Antonelli and P. Marcati, On the finite energy weak solutions to a system in quantum fluid dynamics, Commun. Math. Phys., 287 (2009), 657-686.  doi: 10.1007/s00220-008-0632-0.

[3]

P. Antonelli and P. Marcati, The quantum hydrodynamics system in two space dimensions, Arch. Rational Mech. Anal., 203 (2012), 499-527.  doi: 10.1007/s00205-011-0454-7.

[4]

P. Antonelli and S. Spirito, Global existence of finite energy weak solutions of the quantum Navier-Stokes equations, Arch. Rational Mech. Anal., 255 (2017), 1161-1199.  doi: 10.1007/s00205-017-1124-1.

[5]

P. Antonelli and S. Spirito, On the compactness of weak solutions to the Navier-Stokes-Korteweg equations for capillary fluids, Nonlinear Anal., 187 (2019), 110-124.  doi: 10.1016/j.na.2019.03.020.

[6]

C. Audiard and B. Haspot, Global well-posedness of the Euler Korteweg system for small irrotational data, Commun. Math. Phys., 351 (2017), 201-247.  doi: 10.1007/s00220-017-2843-8.

[7]

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

[8]

D. BreschB. Desjardins and C.-K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Comm. Part. Diffe. Equ., 28 (2003), 843-868.  doi: 10.1081/PDE-120020499.

[9]

D. BreschM. Gisclon and I. Lacroix-Violet, On Navier-Stokes-Korteweg and Euler-Korteweg systems: Application to quantum fluids models, Arch. Rational Mech. Anal., 233 (2019), 975-1025.  doi: 10.1007/s00205-019-01373-w.

[10]

F. Charve and R. Danchin, A global existence result for the compressible Navier-Stokes equations in the critical $L^p$ framework, Arch. Rational Mech. Anal., 198 (2010), 233-271.  doi: 10.1007/s00205-010-0306-x.

[11]

F. Charve, R. Danchin and J. Xu, Gevrey analyticity and decay for the compressible Navier-Stokes system with capillarity, arXiv: 1805.01764.

[12]

J.-Y. Chemin and I. Gallagher, Wellposedness and stability results for the Navier-Stokes equations in ${\mathbb R}^3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 599-624.  doi: 10.1016/j.anihpc.2007.05.008.

[13]

J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs no lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), 314–328. doi: 10.1006/jdeq.1995.1131.

[14]

Q. Chen, C. Miao and Z. Zhang, Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math., 63 (2010), 1173–1224. doi: 10.1002/cpa.20325.

[15]

Z.-M. Chen and X. Zhai, Global large solutions and incompressible limit for the compressible Navier-Stokes equations, J. Math. Fluid Mech., 21 (2019), Art. 26, 23 pp. doi: 10.1007/s00021-019-0428-3.

[16]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579–614. doi: 10.1007/s002220000078.

[17]

R. Danchin and B. Desjardins, Existence of solutions for compressible fluid models of Korteweg type, Annales de l'IHP, Analyse nonlinéaire, 18 (2001), 97-133.  doi: 10.1016/S0294-1449(00)00056-1.

[18]

R. Danchin and L. He, The incompressible limit in $L^p$ type critical spaces, Math. Ann., 366 (2016), 1365-1402.  doi: 10.1007/s00208-016-1361-x.

[19]

R. Danchin and P. B. Mucha, Compressible Navier-Stokes system: large solutions and incompressible limit, Adv. Math., 320 (2017), 904-925.  doi: 10.1016/j.aim.2017.09.025.

[20]

R. Danchin and J. Xu, Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical $L^p$ framework, Arch. Rational Mech. Anal., 224 (2017), 53-90.  doi: 10.1007/s00205-016-1067-y.

[21]

D. DonatelliE. Feireisl and P. Marcati, Well/ill posedness for the Euler-Korteweg-Poisson system and related problems, Comm. Part. Diffe. Equ., 40 (2015), 1314-1335.  doi: 10.1080/03605302.2014.972517.

[22]

J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Rational Mech. Anal., 88 (1985), 95-133.  doi: 10.1007/BF00250907.

[23]

E. Feireisl, Dynamics of Viscous Compressible Fluids., Oxford Univ. Press, Oxford, 2004.

[24]

E. Feireisl and A. Novotný, H. Petzeltová, On the global existence of globally defined weak solutions to the Navier-Stokes equations of isentropic compressible fluids, J. Math. Fluid Mech., 3 (2001), 358–392. doi: 10.1007/PL00000976.

[25]

E. Feireisl, Compressible Navier-Stokes equations with a non-monotone pressure law, J. Differential Equations, 184 (2002), 97–108. doi: 10.1006/jdeq.2001.4137.

[26]

E. Feireisl, A. Novotný and Y. Sun, Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids, Indiana Univ. Math. J., 60 (2011), 611–631. doi: 10.1512/iumj.2011.60.4406.

[27]

A. N. Gorban and I. V. Karlin, Beyond Navier-Stokes equations: Capillarity of ideal gas, Contemporary physics, 58 (2017), 70-90.  doi: 10.1080/00107514.2016.1256123.

[28]

G. Gui and P. Zhang, Stability to the global solutions of 3-D Navier-Stokes equations, Adv. Math., 225 (2010), 1248-1284.  doi: 10.1016/j.aim.2010.03.022.

[29]

B. Haspot, Existence of global weak solution for compressible fluid models of Korteweg type, J. Math. Fluid Mech., 13 (2011), 223-249.  doi: 10.1007/s00021-009-0013-2.

[30]

B. Haspot, Well-posedness in critical spaces for the system of compressible Navier-Stokes in larger spaces, J. Differential Equations, 251 (2011), 2262-2295.  doi: 10.1016/j.jde.2011.06.013.

[31]

B. Haspot, Existence of global strong solution for Korteweg system with large infinite energy initial data, J. Math. Anal. Appl., 438 (2016), 395-443.  doi: 10.1016/j.jmaa.2016.01.047.

[32]

B. Haspot, Global strong solution for the Korteweg system with quantum pressure in dimension $N \geq 2$, Math. Ann., 367 (2017), 667-700.  doi: 10.1007/s00208-016-1391-4.

[33]

H. Hattori and D. Li, The existence of global solutions to a fluid dynamic model for materials for Korteweg type, J. Partial Differential Equations, 9 (1996), 323-342. 

[34]

L. HeJ. Huang and C. Wang, Global stability of large solutions to the 3D compressible Navier-Stokes equations, Arch. Rational Mech. Anal., 234 (2019), 1167-1222.  doi: 10.1007/s00205-019-01410-8.

[35]

M. Heida and J. Málek, On compressible Korteweg fluid-like materials, Internat. J. Engrg. Sci., 48 (2010), 1313-1324.  doi: 10.1016/j.ijengsci.2010.06.031.

[36]

A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045.  doi: 10.1137/090776068.

[37]

M. Kawashita, On global solution of Cauchy problems for compressible Navier-Stokes equation, Nonlinear Anal., 48 (2002), 1087-1105.  doi: 10.1016/S0362-546X(00)00238-8.

[38]

D. J. Korteweg, Sur la forme que prennent les équations du mouvement des fluides si l'on tient compte des forces capillaires par des variations de densité, Arch. Néer. Sci. Exactes Sér., 6 (1901), 1-24. 

[39]

M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type, Annales de l'IHP, Analyse nonlinéaire, 25 (2008), 679-696.  doi: 10.1016/j.anihpc.2007.03.005.

[40]

H.-K. Li and T. Zhang, Large time behavior of isentropic compressible Navier-Stokes system in ${\mathbb R}^3$, Math. Methods Appl. Sci., 34 (2011), 670-682.  doi: 10.1002/mma.1391.

[41]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  doi: 10.1215/kjm/1250522322.

[42]

M. Murata and Y. Shibata, The global well-posedness for the compressible fluid model of Korteweg type, arXiv: 1908.07224.

[43]

M. Okita, Optimal decay rate for strong solutions in critical spaces to the compressible Navier-Stokes equations, J. Differential Equations, 257 (2014), 3850-3867.  doi: 10.1016/j.jde.2014.07.011.

[44]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces, Comm. Math. Phys., 307 (2011), 713-759.  doi: 10.1007/s00220-011-1350-6.

[45]

M. Paicu and P. Zhang, Global solutions to the 3-D incompressible inhomogeneous Navier- Stokes system, J. Funct. Anal., 262 (2012), 3556-3584.  doi: 10.1016/j.jfa.2012.01.022.

[46]

G. Ponce, Global existence of small solution to a class of nonlinear evolution equations, Nonlinear Anal. TMA., 9 (1985), 339-418.  doi: 10.1016/0362-546X(85)90001-X.

[47]

K. Takayuki and T. Kazuyuki, Global existence and time decay estimate of solutions to the compressible Navier-Stokes-Korteweg system under critical condition, Asympt. Anal., (2020), Publishing. doi: 10.3233/ASY-201600.

[48]

J. F. Van der Waals, Thermodynamische Theorie der Kapillarität unter Voraussetzung stetiger Dichteänderung, Phys. Chem., 13 (1894), 657-725.  doi: 10.1515/zpch-1894-1338.

[49]

K. Watanabe, Global existence of the Navier-Stokes-Korteweg equations with a non-decreasing pressure in $L^p$-framework, arXiv: 1907.07752.

[50]

Z. Xin and J. Xu, Optimal decay for the compressible Navier-Stokes equations without additional smallness assumptions, arXiv: 1812.11714v2.

[51]

H. Xu, Y. Li and F. Chen, Global solution to the incompressible inhomogeneous Navier-Stokes equations with some large initial data, J. Math. Fluid Mech., 19 (2017), 315–328. doi: 10.1007/s00021-016-0282-5.

[52]

X. ZhaiY. Li and F. Zhou, Global large solutions to the three dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal., 52 (2020), 1806-1843.  doi: 10.1137/19M1265843.

[53]

S. Zhang, A class of global large solutions to the compressible Navier-Stokes-Korteweg system in critical Besov spaces, J. Evol. Equ., (2020). doi: 10.1007/s00028-020-00565-2.

[54]

T. Zhang, Global wellposedness problem for the 3-D incompressible anisotropic Navier-Stokes equations in an anisotropic space, Comm. Math. Phys., 287 (2009), 211-224.  doi: 10.1007/s00220-008-0631-1.

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