doi: 10.3934/dcdsb.2021146
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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 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 & Continuous Dynamical Systems - B, 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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