Global solutions to the non-local Navier-Stokes equations

Joelma Azevedo; Juan Carlos Pozo; Arlúcio Viana

doi:10.3934/dcdsb.2021146

Previous Article
Global well-posedness and exponential stability for the fermion equation in weighted Sobolev spaces
DCDS-B Home
This Issue
Next Article
Optimal control of the SIR epidemic model based on dynamical systems theory

May 2022, 27(5): 2515-2535. doi: 10.3934/dcdsb.2021146

Global solutions to the non-local Navier-Stokes equations

Joelma Azevedo ^1, , Juan Carlos Pozo ^2, and Arlúcio Viana ^3,,

1.	Departamento de Matemática, Universidade de Pernambuco, Nazaré da Mata, Brazil
2.	Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Santiago, Chile
3.	Departamento de Matemática, Universidade Federal de Sergipe, São Cristóvão, Brazil

^* Corresponding author: Arlúcio Viana

Received December 2020 Revised April 2021 Published May 2022 Early access May 2021

This paper is devoted to the study of the global well-posedness for a non-local-in-time Navier-Stokes equation. Our results recover in particular other existing well-posedness results for the Navier-Stokes equations and their time-fractional version. We show the appropriate manner to apply Kato's strategy and this context, with initial conditions in the divergence-free Lebesgue space $ L^\sigma_d(\mathbb{R}^d) $. Temporal decay at $ 0 $ and $ \infty $ are obtained for the solution and its gradient.

Keywords: Nonlocal Navier-Stokes, PDEs in connection with fluid mechanics, well-posedness, long-time behavior, uniqueness.

Mathematics Subject Classification: Primary: 35R09, 35R11, 35Q35, 35Q30; Secondary: 33E12, 76D03, 35B30, 35B40.

Citation: Joelma Azevedo, Juan Carlos Pozo, Arlúcio Viana. Global solutions to the non-local Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2515-2535. doi: 10.3934/dcdsb.2021146

References:

[1]	H. Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (1984), 593-676.
[2]	G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials with Memory, Springer, New York, 2012. doi: 10.1007/978-1-4614-1692-0.
[3]	V. Barbu and S. S. Sritharan, Navier-Stokes equation with hereditary viscosity, Z. Angew. Math. Phys., 54 (2003), 449-461. doi: 10.1007/s00033-003-1087-y.
[4]	M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoamericana, 13 (1997), 515-541. doi: 10.4171/RMI/229.
[5]	R. Carlone, A. Fiorenza and L. Tentarelli, The action of Volterra integral operators with highly singular kernels on Hölder continuous, Lebesgue and Sobolev functions, J. Funct. Anal., 273 (2017), 1258-1294. doi: 10.1016/j.jfa.2017.04.013.
[6]	Ph. Clément and J. A. Nohel, Abstract linear and nonlinear Volterra equations preserving positivity, SIAM J. Math. Anal., 10 (1979), 365-388. doi: 10.1137/0510035.
[7]	Ph. Clément and J. A. Nohel, Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM J. Math. Anal., 12 (1981), 514-535. doi: 10.1137/0512045.
[8]	P. M. de Carvalho-Neto and G. Planas, Mild solutions to the time fractional Navier-Stokes equations in $\Bbb{R}^N$, J. Differential Equations, 259 (2015), 2948-2980. doi: 10.1016/j.jde.2015.04.008.
[9]	Z. Z. Ganji, D. D. Ganji, D. Ammar and M. Rostamian, Analytical solution of time-fractional Navier-Stokes equation in polar coordinate by homotopy perturbation method, Numer. Methods Partial Differential Equations, 26 (2010), 117-124. doi: 10.1002/num.20420.
[10]	L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004.
[11]	T. Kato, Strong $L^{p}$-solutions of the Navier-Stokes equation in $\Bbb{R}^{m}$, with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182.
[12]	J. Kemppainen, J. Siljander, V. Vergara and R. Zacher, Decay estimates for time-fractional and other non-local in time subdiffusion equations in $\Bbb{R}^d$, Math. Ann., 366 (2016), 941-979. doi: 10.1007/s00208-015-1356-z.
[13]	A. N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl., 340 (2008), 252-281. doi: 10.1016/j.jmaa.2007.08.024.
[14]	T. Kodama and T. Koide, Memory effects and transport coefficients for non-Newtonian fluids, J. Phys. G: Nucl. Part. Phys., 36 (2009), 6 pp. doi: 10.1088/0954-3899/36/6/064063.
[15]	Q. Li, Y. Chen, Y. Huang and Y. Wang, Two-grid methods for semilinear time fractional reaction diffusion equations by expanded mixed finite element method, Appl. Numer. Math., 157 (2020), 38-54. doi: 10.1016/j.apnum.2020.05.024.
[16]	R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 77 pp. doi: 10.1016/S0370-1573(00)00070-3.
[17]	S. Momani and Z. Odibat, Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Appl. Math. Comput., 177 (2006), 488-494. doi: 10.1016/j.amc.2005.11.025.
[18]	L. Peng, Y. Zhou, B. Ahmad and A. Alsaedi, The Cauchy problem for fractional Navier-Stokes equations in Sobolev spaces, Chaos, Solitons Fractals, 102 (2017), 218-228. doi: 10.1016/j.chaos.2017.02.011.
[19]	J. C. Pozo and V. Vergara, Fundamental solutions and decay of fully non-local problems, Discrete Contin. Dyn. Syst., 39 (2019), 639-666. doi: 10.3934/dcds.2019026.
[20]	J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, Vol. 87, Birkhäuser, Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.
[21]	Y. Wang and T. Liang, Mild solutions to the time fractional Navier-Stokes delay differential inclusions, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3713-3740. doi: 10.3934/dcdsb.2018312.
[22]	L. Xu, T. Shen, X. Yang and J. Liang, Analysis of time fractional and space nonlocal stochastic incompressible Navier-Stokes equation driven by white noise, Comput. Math. Appl., 78 (2019), 1669-1680. doi: 10.1016/j.camwa.2018.12.022.
[23]	J. Xu, Z. Zhang and T. Caraballo, Mild solutions to time fractional stochastic 2d-stokes equations with bounded and unbounded delay, J. Dyn. Diff. Equat., (2019). doi: 10.1007/s10884-019-09809-3.
[24]	P. Xu, C. Zeng and J. Huang, Well-posedness of the time-space fractional stochastic Navier-Stokes equations driven by fractional Brownian motion, Math. Model. Nat. Phenom., 13 (2018), Paper No. 11, 18 pp. doi: 10.1051/mmnp/2018003.
[25]	J. Zhang and J. Wang, Numerical analysis for Navier-Stokes equations with time fractional derivatives, Appl. Math. Comput., 336 (2018), 481-489. doi: 10.1016/j.amc.2018.04.036.
[26]	R. Zheng and X. Jiang, Spectral methods for the time-fractional Navier-Stokes equation, Appl. Math. Lett., 91 (2019), 194-200. doi: 10.1016/j.aml.2018.12.018.
[27]	Y. Zhou and L. Peng, On the time-fractional Navier-Stokes equations, Comput. Math. Appl., 73 (2017), 874-891. doi: 10.1016/j.camwa.2016.03.026.
[28]	Y. Zhou and L. Peng, Weak solutions of the time-fractional Navier-Stokes equations and optimal control, Comput. Math. Appl., 73 (2017), 1016-1027. doi: 10.1016/j.camwa.2016.07.007.
[29]	Y. Zhou, L. Peng and Y. Huang, Existence and Hölder continuity of solutions for time-fractional Navier-Stokes equations, Math. Methods Appl. Sci., 41 (2018), 7830-7838. doi: 10.1002/mma.5245.
[30]	L. Peng, A. Debbouche and Y. Zhou, Existence and approximation of solutions for time-fractional Navier-Stokes equations, Math. Methods Appl. Sci., 41 (2018), 8973-8984. doi: 10.1002/mma.4779.
[31]	Y. Zhou, L. Peng, B. Ahmad, Ba shir and A. Alsaedi, Energy methods for fractional Navier-Stokes equations, Chaos, Solitons Fractals, 102 (2017), 78-85. doi: 10.1016/j.chaos.2017.03.053.
[32]	G. Zou, G. Lv and J.-L. Wu, Stochastic Navier-Stokes equations with Caputo derivative driven by fractional noises, J. Math. Anal. Appl., 461 (2018), 595-609. doi: 10.1016/j.jmaa.2018.01.027.

show all references

References:

[1]	H. Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (1984), 593-676.
[2]	G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials with Memory, Springer, New York, 2012. doi: 10.1007/978-1-4614-1692-0.
[3]	V. Barbu and S. S. Sritharan, Navier-Stokes equation with hereditary viscosity, Z. Angew. Math. Phys., 54 (2003), 449-461. doi: 10.1007/s00033-003-1087-y.
[4]	M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoamericana, 13 (1997), 515-541. doi: 10.4171/RMI/229.
[5]	R. Carlone, A. Fiorenza and L. Tentarelli, The action of Volterra integral operators with highly singular kernels on Hölder continuous, Lebesgue and Sobolev functions, J. Funct. Anal., 273 (2017), 1258-1294. doi: 10.1016/j.jfa.2017.04.013.
[6]	Ph. Clément and J. A. Nohel, Abstract linear and nonlinear Volterra equations preserving positivity, SIAM J. Math. Anal., 10 (1979), 365-388. doi: 10.1137/0510035.
[7]	Ph. Clément and J. A. Nohel, Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM J. Math. Anal., 12 (1981), 514-535. doi: 10.1137/0512045.
[8]	P. M. de Carvalho-Neto and G. Planas, Mild solutions to the time fractional Navier-Stokes equations in $\Bbb{R}^N$, J. Differential Equations, 259 (2015), 2948-2980. doi: 10.1016/j.jde.2015.04.008.
[9]	Z. Z. Ganji, D. D. Ganji, D. Ammar and M. Rostamian, Analytical solution of time-fractional Navier-Stokes equation in polar coordinate by homotopy perturbation method, Numer. Methods Partial Differential Equations, 26 (2010), 117-124. doi: 10.1002/num.20420.
[10]	L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004.
[11]	T. Kato, Strong $L^{p}$-solutions of the Navier-Stokes equation in $\Bbb{R}^{m}$, with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182.
[12]	J. Kemppainen, J. Siljander, V. Vergara and R. Zacher, Decay estimates for time-fractional and other non-local in time subdiffusion equations in $\Bbb{R}^d$, Math. Ann., 366 (2016), 941-979. doi: 10.1007/s00208-015-1356-z.
[13]	A. N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl., 340 (2008), 252-281. doi: 10.1016/j.jmaa.2007.08.024.
[14]	T. Kodama and T. Koide, Memory effects and transport coefficients for non-Newtonian fluids, J. Phys. G: Nucl. Part. Phys., 36 (2009), 6 pp. doi: 10.1088/0954-3899/36/6/064063.
[15]	Q. Li, Y. Chen, Y. Huang and Y. Wang, Two-grid methods for semilinear time fractional reaction diffusion equations by expanded mixed finite element method, Appl. Numer. Math., 157 (2020), 38-54. doi: 10.1016/j.apnum.2020.05.024.
[16]	R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 77 pp. doi: 10.1016/S0370-1573(00)00070-3.
[17]	S. Momani and Z. Odibat, Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Appl. Math. Comput., 177 (2006), 488-494. doi: 10.1016/j.amc.2005.11.025.
[18]	L. Peng, Y. Zhou, B. Ahmad and A. Alsaedi, The Cauchy problem for fractional Navier-Stokes equations in Sobolev spaces, Chaos, Solitons Fractals, 102 (2017), 218-228. doi: 10.1016/j.chaos.2017.02.011.
[19]	J. C. Pozo and V. Vergara, Fundamental solutions and decay of fully non-local problems, Discrete Contin. Dyn. Syst., 39 (2019), 639-666. doi: 10.3934/dcds.2019026.
[20]	J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, Vol. 87, Birkhäuser, Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.
[21]	Y. Wang and T. Liang, Mild solutions to the time fractional Navier-Stokes delay differential inclusions, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3713-3740. doi: 10.3934/dcdsb.2018312.
[22]	L. Xu, T. Shen, X. Yang and J. Liang, Analysis of time fractional and space nonlocal stochastic incompressible Navier-Stokes equation driven by white noise, Comput. Math. Appl., 78 (2019), 1669-1680. doi: 10.1016/j.camwa.2018.12.022.
[23]	J. Xu, Z. Zhang and T. Caraballo, Mild solutions to time fractional stochastic 2d-stokes equations with bounded and unbounded delay, J. Dyn. Diff. Equat., (2019). doi: 10.1007/s10884-019-09809-3.
[24]	P. Xu, C. Zeng and J. Huang, Well-posedness of the time-space fractional stochastic Navier-Stokes equations driven by fractional Brownian motion, Math. Model. Nat. Phenom., 13 (2018), Paper No. 11, 18 pp. doi: 10.1051/mmnp/2018003.
[25]	J. Zhang and J. Wang, Numerical analysis for Navier-Stokes equations with time fractional derivatives, Appl. Math. Comput., 336 (2018), 481-489. doi: 10.1016/j.amc.2018.04.036.
[26]	R. Zheng and X. Jiang, Spectral methods for the time-fractional Navier-Stokes equation, Appl. Math. Lett., 91 (2019), 194-200. doi: 10.1016/j.aml.2018.12.018.
[27]	Y. Zhou and L. Peng, On the time-fractional Navier-Stokes equations, Comput. Math. Appl., 73 (2017), 874-891. doi: 10.1016/j.camwa.2016.03.026.
[28]	Y. Zhou and L. Peng, Weak solutions of the time-fractional Navier-Stokes equations and optimal control, Comput. Math. Appl., 73 (2017), 1016-1027. doi: 10.1016/j.camwa.2016.07.007.
[29]	Y. Zhou, L. Peng and Y. Huang, Existence and Hölder continuity of solutions for time-fractional Navier-Stokes equations, Math. Methods Appl. Sci., 41 (2018), 7830-7838. doi: 10.1002/mma.5245.
[30]	L. Peng, A. Debbouche and Y. Zhou, Existence and approximation of solutions for time-fractional Navier-Stokes equations, Math. Methods Appl. Sci., 41 (2018), 8973-8984. doi: 10.1002/mma.4779.
[31]	Y. Zhou, L. Peng, B. Ahmad, Ba shir and A. Alsaedi, Energy methods for fractional Navier-Stokes equations, Chaos, Solitons Fractals, 102 (2017), 78-85. doi: 10.1016/j.chaos.2017.03.053.
[32]	G. Zou, G. Lv and J.-L. Wu, Stochastic Navier-Stokes equations with Caputo derivative driven by fractional noises, J. Math. Anal. Appl., 461 (2018), 595-609. doi: 10.1016/j.jmaa.2018.01.027.

[1]	Andrea Giorgini. On the Swift-Hohenberg equation with slow and fast dynamics: well-posedness and long-time behavior. Communications on Pure and Applied Analysis, 2016, 15 (1) : 219-241. doi: 10.3934/cpaa.2016.15.219
[2]	Igor Chueshov, Irena Lasiecka, Justin Webster. Flow-plate interactions: Well-posedness and long-time behavior. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 925-965. doi: 10.3934/dcdss.2014.7.925
[3]	Giulio Schimperna, Antonio Segatti, Ulisse Stefanelli. Well-posedness and long-time behavior for a class of doubly nonlinear equations. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 15-38. doi: 10.3934/dcds.2007.18.15
[4]	Oscar Jarrín, Manuel Fernando Cortez. On the long-time behavior for a damped Navier-Stokes-Bardina model. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 3661-3707. doi: 10.3934/dcds.2022028
[5]	Daoyuan Fang, Ruizhao Zi. On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3517-3541. doi: 10.3934/dcds.2013.33.3517
[6]	Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations I: Local well-posedness. Evolution Equations and Control Theory, 2012, 1 (1) : 195-215. doi: 10.3934/eect.2012.1.195
[7]	Matthias Hieber, Sylvie Monniaux. Well-posedness results for the Navier-Stokes equations in the rotational framework. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5143-5151. doi: 10.3934/dcds.2013.33.5143
[8]	Jan Prüss, Vicente Vergara, Rico Zacher. Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 625-647. doi: 10.3934/dcds.2010.26.625
[9]	Quanrong Li, Shijin Ding. Global well-posedness of the Navier-Stokes equations with Navier-slip boundary conditions in a strip domain. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3561-3581. doi: 10.3934/cpaa.2021121
[10]	Myeongju Chae, Kyungkeun Kang, Jihoon Lee. Global well-posedness and long time behaviors of chemotaxis-fluid system modeling coral fertilization. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2135-2163. doi: 10.3934/dcds.2020109
[11]	Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148
[12]	Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142
[13]	Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101
[14]	Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure and Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35
[15]	Keyan Wang, Yao Xiao. Local well-posedness for Navier-Stokes equations with a class of ill-prepared initial data. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2987-3011. doi: 10.3934/dcds.2020158
[16]	Weimin Peng, Yi Zhou. Global well-posedness of axisymmetric Navier-Stokes equations with one slow variable. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3845-3856. doi: 10.3934/dcds.2016.36.3845
[17]	Yoshihiro Shibata. Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 315-342. doi: 10.3934/dcdss.2016.9.315
[18]	Jingjing Zhang, Ting Zhang. Local well-posedness of perturbed Navier-Stokes system around Landau solutions. Electronic Research Archive, 2021, 29 (4) : 2719-2739. doi: 10.3934/era.2021010
[19]	Haydi Israel. Well-posedness and long time behavior of an Allen-Cahn type equation. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2811-2827. doi: 10.3934/cpaa.2013.12.2811
[20]	Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3765-3788. doi: 10.3934/dcds.2018163

American Institute of Mathematical Sciences

Global solutions to the non-local Navier-Stokes equations

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Global solutions to the non-local Navier-Stokes equations

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors