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Sharp regularity for degenerate obstacle type problems: A geometric approach
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 |
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.
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. Google Scholar |
[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. Bresch, B. 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. Bresch, M. 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. Google Scholar |
[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. Donatelli, E. 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. He, J. 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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[50] |
Z. Xin and J. Xu, Optimal decay for the compressible Navier-Stokes equations without additional smallness assumptions, arXiv: 1812.11714v2. Google Scholar |
[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. Zhai, Y. 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. Google Scholar |
[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. Bresch, B. 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. Bresch, M. 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. Google Scholar |
[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. Donatelli, E. 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. He, J. 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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[50] |
Z. Xin and J. Xu, Optimal decay for the compressible Navier-Stokes equations without additional smallness assumptions, arXiv: 1812.11714v2. Google Scholar |
[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. Zhai, Y. 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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