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

Xiaoping Zhai; Yongsheng Li

doi:10.3934/dcds.2020322

Article Contents

2021, Volume 41, Issue 3: 1387-1413. Doi: 10.3934/dcds.2020322

This issue Previous Article Sharp regularity for degenerate obstacle type problems: A geometric approach Next Article A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system

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

Xiaoping Zhai^1, and
Yongsheng Li^2,

1.
School of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China
2.
School of Mathematics, South China University of Technology, Guangzhou, 510640, China

Received: November 30, 2019

Revised: April 30, 2020

Early access: September 2020

Published: March 2021

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Abstract

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.

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Mathematics Subject Classification: 35Q35, 76N10.

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[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. 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.
[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.
[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. 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.