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

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

* 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.

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. CarloneA. 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. GanjiD. D. GanjiD. 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. KemppainenJ. SiljanderV. 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. LiY. ChenY. 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. PengY. ZhouB. 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. XuT. ShenX. 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. ZhouL. 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. PengA. 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. ZhouL. PengB. AhmadBa 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. ZouG. 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. CarloneA. 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. GanjiD. D. GanjiD. 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. KemppainenJ. SiljanderV. 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. LiY. ChenY. 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. PengY. ZhouB. 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. XuT. ShenX. 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. ZhouL. 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. PengA. 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. ZhouL. PengB. AhmadBa 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. ZouG. 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.

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