-
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
Global solutions to the non-local Navier-Stokes equations
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 |
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.
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 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]