# American Institute of Mathematical Sciences

October  2018, 23(8): 2967-2988. doi: 10.3934/dcdsb.2017149

## Fractional Navier-Stokes equations

 Institute of Mathematics, University of Silesia in Katowice, 40-007 Katowice, Poland

* Corresponding author:Jan W. Cholewa

Received  December 2016 Revised  February 2017 Published  April 2017

Fund Project: J.C. partially supported by grant MTM2012-31298 from Ministerio de Economia y Competividad, Spain; T.D. partially supported by NCN grant DEC-2012/05/B/ST1/00546, Poland

We consider fractional Navier-Stokes equations in a smooth bound-ed domain $Ω\subset\mathbb{R}^N$, $N≥2$. Following the geometric theory of abstract parabolic problems we give the detailed analysis concerning existence, uniqueness, regularization and continuation properties of the solution. For the original Navier-Stokes problem we construct next global solution of the Leray-Hopf type satisfying also Duhamel's integral formula. Focusing finally on the 3-D model with zero external force we estimate a time after which the latter solution regularizes to strong solution. We also estimate a time such that if a local strong solution exists until that time, then it exists for ever.

Citation: Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149
##### References:
 [1] H. Amann, Linear and Quasilinear Parabolic Problems, Volume Ⅰ, Abstract Linear Theory Birkhaüser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9221-6. Google Scholar [2] J. Arrieta and A. N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, Trans. Amer. Math. Soc., 352 (2000), 285-310. doi: 10.1090/S0002-9947-99-02528-3. Google Scholar [3] J. M. Ball, Strongly continuous semigroups, weak solutions and the variation of constants formula, Proc. Am. Math. Soc., 63 (1977), 370-373. doi: 10.2307/2041821. Google Scholar [4] H. Brezis, Analyse Fonctionelle. Théorie et Applications, Masson, Paris, 1983. Google Scholar [5] L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903. Google Scholar [6] A. N. Carvalho and J. W. Cholewa, Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities, J. Math. Anal. Appl., 310 (2005), 557-578. doi: 10.1016/j.jmaa.2005.02.024. Google Scholar [7] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511526404. Google Scholar [8] J. W. Cholewa, C. Quesada and A. Rodriguez-Bernal, Nonlinear evolution equations in scales of Banach spaces and applications to PDEs, preprint.Google Scholar [9] T. Dlotko, Navier-Stokes equation and its fractional approximations Appl. Math. Optim. (2016). doi: 10.1007/s00245-016-9368-y. Google Scholar [10] T. Dlotko, M. B. Kania and C. Sun, Quasi-geostrophic equation in R2, J. Differential Equations, 259 (2015), 531-561. doi: 10.1016/j.jde.2015.02.022. Google Scholar [11] C. L. Fefferman, Existence and smoothness of the Navier-Stokes equation http://www.claymath.org/sites/default/files/navierstokes.pdfGoogle Scholar [12] C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754. Google Scholar [13] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. Google Scholar [14] H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Rational Mech. Anal., 16 (1964), 269-315. doi: 10.1007/BF00276188. Google Scholar [15] D. Fujiwara and H. Morimoto, An Lr-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo, 24 (1977), 685-700. Google Scholar [16] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-state Problems Springer Monographs in Mathematics, Springer, New York, 2011. doi: 10.1007/978-0-387-09620-9. Google Scholar [17] Y. Giga, Analyticity of the semigroup generated by the Stokes operator in Lr spaces, Math. Z., 178 (1981), 297-329. doi: 10.1007/BF01214869. Google Scholar [18] Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 61 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3. Google Scholar [19] Y. Giga, Domains of fractional powers of the Stokes operator in Lr spaces, Arch. Rational Mech. Anal., 89 (1985), 251-265. doi: 10.1007/BF00276874. Google Scholar [20] Y. Giga and T. Miyakawa, Solutions in Lr of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267-281. doi: 10.1007/BF00276875. Google Scholar [21] B. Guo, D. Huang, Q. Li and C. Sun, Dynamics for a generalized incompressible Navier-Stokes equations in $\mathbb{R}^2$, Adv. Nonlinear Stud., 16 (2016), 249-272. doi: 10.1515/ans-2015-5018. Google Scholar [22] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981. Google Scholar [23] A. Inoue and M. Wakimoto, On existence of solutions of the Navier-Stokes equation in a time dependent domain, J. Fac. Sci. Univ. Tokyo, Sec. I, 24 (1977), 303-319. Google Scholar [24] T. Kato and H. Fujita, On the nonstationary Navier-Stokes system, Rend. Sem. Math. Univ. Padova, 32 (1962), 243-260. Google Scholar [25] H. Komatsu, Fractional powers of operators, Pacific J. Math., 19 (1966), 285-346. doi: 10.2140/pjm.1966.19.285. Google Scholar [26] O. A. Ladyzhenskaya, On some gaps in two of my papers on the Navier-Stokes equations and the way of closing them, J. Math. Sci., 115 (2003), 2789-2891. Google Scholar [27] O. A. Ladyzhenskaya, On the uniqueness and smoothness of generalized solutions of the Navier-Stokes equations, Zap. Nauchn. Semin. LOMI, (1967), 169-185. Google Scholar [28] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2 Gordon and Breach, Science Publishers, New York-London-Paris, 1969. Google Scholar [29] J. -L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod Gauthier-Villars, Paris, 1969. Google Scholar [30] J. -L. Lions and E. Magenes, Probl}mes aux Limites non Homogénes et Applications, Vol. Ⅰ, Dunod, Paris 1968. Google Scholar [31] G. Łukaszewicz and P. Kalita, Navier-Stokes Equations. An Introduction with Applications Advances in Mechanics and Mathematics 34, Springer International Publishing, 2016. doi: 10.1007/978-3-319-27760-8. Google Scholar [32] C. Martínez Carracedo and M. Sanz Alix, The Theory of Fractional Powers of Operators Elsevier, Amsterdam, 2001. Google Scholar [33] K. Masuda, Weak solutions of Navier-Stokes equations, Tôhoku Math. Journ., 36 (1984), 623-646. doi: 10.2748/tmj/1178228767. Google Scholar [34] J. Mattingly and Y. Sinai, An elementary proof of the existence and uniqueness theorem for the Navier-Stokes equation, Commun. Contemp. Math., 1 (1999), 497-516. doi: 10.1142/S0219199799000183. Google Scholar [35] T. Miyakawa, On the initial value problem for the Navier-Stokes equations in $L^p$ spaces, Hiroshima Math. J., 11 (1981), 9-20. Google Scholar [36] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Springer, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar [37] G. Prodi, Un teorema di unicita per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173-182. doi: 10.1007/BF02410664. Google Scholar [38] F. Ribaud, A remark on the uniqueness problem for the weak solutions of Navier-Stokes equations, Ann. Fac. Sci. Toulouse Math., 11 (2002), 225-238. doi: 10.5802/afst.1024. Google Scholar [39] J. C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0. Google Scholar [40] J. Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Problems (Proc. Sympos. , Madison, Wis. , 1962), (1963), 69{98, Univ. of Wisconsin Press, Madison, Wisconsin. Google Scholar [41] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar [42] P. E. Sobolevskii, On non-stationary equations of hydrodynamics for viscous fluid, Dokl. Akad. Nauk SSSR, 128 (1959), 45-48 (in Russian). Google Scholar [43] P. E. Sobolevskii, On equations of parabolic type in a Banach space, Trudy Moskov. Mat. Obsc. , 10 (1961), 297-350 (in Russian); Equations of parabolic type in a Banach space, Amer. Math. Soc. Transl. , 49 (1966), 1{62. Google Scholar [44] H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8255-2. Google Scholar [45] W. A. Strauss, On continuity of functions with values in various Banach spaces, Pacific J. Math., 19 (1966), 543-551. doi: 10.2140/pjm.1966.19.543. Google Scholar [46] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1979. Google Scholar [47] R. Temam, On the Euler equations of incompressible perfect fluids, J. Functional Analysis, 20 (1975), 32-43. doi: 10.1016/0022-1236(75)90052-X. Google Scholar [48] W. von Wahl, Equations of Navier-Stokes and Abstract Parabolic Equations Vieweg, Braunschweig/Wiesbaden, 1985. doi: 10.1007/978-3-663-13911-9. Google Scholar [49] W. von Wahl, Global solutions to evolution equations of parabolic type, in: Differential Equations in Banach Spaces, Proceedings, 1985 (Eds. A. Favini, E. Obrecht), Springer-Verlag, Berlin, 1223 (1986), 254{266 Google Scholar [50] F. B. Weissler, The Navier-Stokes initial value problem in $L^p$, Arch. Rational Mech. Anal., 74 (1980), 219-230. doi: 10.1007/BF00280539. Google Scholar [51] J. Wu, Generalized MHD equations, J. Differential Equations, 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007. Google Scholar [52] H. Wu and J. Fan, Weak-strong uniqueness for the generalized Navier-Stokes equations, Appl. Math. Lett., 25 (2012), 423-428. doi: 10.1016/j.aml.2011.09.028. Google Scholar

show all references

##### References:
 [1] H. Amann, Linear and Quasilinear Parabolic Problems, Volume Ⅰ, Abstract Linear Theory Birkhaüser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9221-6. Google Scholar [2] J. Arrieta and A. N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, Trans. Amer. Math. Soc., 352 (2000), 285-310. doi: 10.1090/S0002-9947-99-02528-3. Google Scholar [3] J. M. Ball, Strongly continuous semigroups, weak solutions and the variation of constants formula, Proc. Am. Math. Soc., 63 (1977), 370-373. doi: 10.2307/2041821. Google Scholar [4] H. Brezis, Analyse Fonctionelle. Théorie et Applications, Masson, Paris, 1983. Google Scholar [5] L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903. Google Scholar [6] A. N. Carvalho and J. W. Cholewa, Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities, J. Math. Anal. Appl., 310 (2005), 557-578. doi: 10.1016/j.jmaa.2005.02.024. Google Scholar [7] J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511526404. Google Scholar [8] J. W. Cholewa, C. Quesada and A. Rodriguez-Bernal, Nonlinear evolution equations in scales of Banach spaces and applications to PDEs, preprint.Google Scholar [9] T. Dlotko, Navier-Stokes equation and its fractional approximations Appl. Math. Optim. (2016). doi: 10.1007/s00245-016-9368-y. Google Scholar [10] T. Dlotko, M. B. Kania and C. Sun, Quasi-geostrophic equation in R2, J. Differential Equations, 259 (2015), 531-561. doi: 10.1016/j.jde.2015.02.022. Google Scholar [11] C. L. Fefferman, Existence and smoothness of the Navier-Stokes equation http://www.claymath.org/sites/default/files/navierstokes.pdfGoogle Scholar [12] C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754. Google Scholar [13] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. Google Scholar [14] H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Rational Mech. Anal., 16 (1964), 269-315. doi: 10.1007/BF00276188. Google Scholar [15] D. Fujiwara and H. Morimoto, An Lr-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo, 24 (1977), 685-700. Google Scholar [16] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-state Problems Springer Monographs in Mathematics, Springer, New York, 2011. doi: 10.1007/978-0-387-09620-9. Google Scholar [17] Y. Giga, Analyticity of the semigroup generated by the Stokes operator in Lr spaces, Math. Z., 178 (1981), 297-329. doi: 10.1007/BF01214869. Google Scholar [18] Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 61 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3. Google Scholar [19] Y. Giga, Domains of fractional powers of the Stokes operator in Lr spaces, Arch. Rational Mech. Anal., 89 (1985), 251-265. doi: 10.1007/BF00276874. Google Scholar [20] Y. Giga and T. Miyakawa, Solutions in Lr of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267-281. doi: 10.1007/BF00276875. Google Scholar [21] B. Guo, D. Huang, Q. Li and C. Sun, Dynamics for a generalized incompressible Navier-Stokes equations in $\mathbb{R}^2$, Adv. Nonlinear Stud., 16 (2016), 249-272. doi: 10.1515/ans-2015-5018. Google Scholar [22] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981. Google Scholar [23] A. Inoue and M. Wakimoto, On existence of solutions of the Navier-Stokes equation in a time dependent domain, J. Fac. Sci. Univ. Tokyo, Sec. I, 24 (1977), 303-319. Google Scholar [24] T. Kato and H. Fujita, On the nonstationary Navier-Stokes system, Rend. Sem. Math. Univ. Padova, 32 (1962), 243-260. Google Scholar [25] H. Komatsu, Fractional powers of operators, Pacific J. Math., 19 (1966), 285-346. doi: 10.2140/pjm.1966.19.285. Google Scholar [26] O. A. Ladyzhenskaya, On some gaps in two of my papers on the Navier-Stokes equations and the way of closing them, J. Math. Sci., 115 (2003), 2789-2891. Google Scholar [27] O. A. Ladyzhenskaya, On the uniqueness and smoothness of generalized solutions of the Navier-Stokes equations, Zap. Nauchn. Semin. LOMI, (1967), 169-185. Google Scholar [28] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2 Gordon and Breach, Science Publishers, New York-London-Paris, 1969. Google Scholar [29] J. -L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod Gauthier-Villars, Paris, 1969. Google Scholar [30] J. -L. Lions and E. Magenes, Probl}mes aux Limites non Homogénes et Applications, Vol. Ⅰ, Dunod, Paris 1968. Google Scholar [31] G. Łukaszewicz and P. Kalita, Navier-Stokes Equations. An Introduction with Applications Advances in Mechanics and Mathematics 34, Springer International Publishing, 2016. doi: 10.1007/978-3-319-27760-8. Google Scholar [32] C. Martínez Carracedo and M. Sanz Alix, The Theory of Fractional Powers of Operators Elsevier, Amsterdam, 2001. Google Scholar [33] K. Masuda, Weak solutions of Navier-Stokes equations, Tôhoku Math. Journ., 36 (1984), 623-646. doi: 10.2748/tmj/1178228767. Google Scholar [34] J. Mattingly and Y. Sinai, An elementary proof of the existence and uniqueness theorem for the Navier-Stokes equation, Commun. Contemp. Math., 1 (1999), 497-516. doi: 10.1142/S0219199799000183. Google Scholar [35] T. Miyakawa, On the initial value problem for the Navier-Stokes equations in $L^p$ spaces, Hiroshima Math. J., 11 (1981), 9-20. Google Scholar [36] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Springer, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar [37] G. Prodi, Un teorema di unicita per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173-182. doi: 10.1007/BF02410664. Google Scholar [38] F. Ribaud, A remark on the uniqueness problem for the weak solutions of Navier-Stokes equations, Ann. Fac. Sci. Toulouse Math., 11 (2002), 225-238. doi: 10.5802/afst.1024. Google Scholar [39] J. C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0. Google Scholar [40] J. Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Problems (Proc. Sympos. , Madison, Wis. , 1962), (1963), 69{98, Univ. of Wisconsin Press, Madison, Wisconsin. Google Scholar [41] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar [42] P. E. Sobolevskii, On non-stationary equations of hydrodynamics for viscous fluid, Dokl. Akad. Nauk SSSR, 128 (1959), 45-48 (in Russian). Google Scholar [43] P. E. Sobolevskii, On equations of parabolic type in a Banach space, Trudy Moskov. Mat. Obsc. , 10 (1961), 297-350 (in Russian); Equations of parabolic type in a Banach space, Amer. Math. Soc. Transl. , 49 (1966), 1{62. Google Scholar [44] H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8255-2. Google Scholar [45] W. A. Strauss, On continuity of functions with values in various Banach spaces, Pacific J. Math., 19 (1966), 543-551. doi: 10.2140/pjm.1966.19.543. Google Scholar [46] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1979. Google Scholar [47] R. Temam, On the Euler equations of incompressible perfect fluids, J. Functional Analysis, 20 (1975), 32-43. doi: 10.1016/0022-1236(75)90052-X. Google Scholar [48] W. von Wahl, Equations of Navier-Stokes and Abstract Parabolic Equations Vieweg, Braunschweig/Wiesbaden, 1985. doi: 10.1007/978-3-663-13911-9. Google Scholar [49] W. von Wahl, Global solutions to evolution equations of parabolic type, in: Differential Equations in Banach Spaces, Proceedings, 1985 (Eds. A. Favini, E. Obrecht), Springer-Verlag, Berlin, 1223 (1986), 254{266 Google Scholar [50] F. B. Weissler, The Navier-Stokes initial value problem in $L^p$, Arch. Rational Mech. Anal., 74 (1980), 219-230. doi: 10.1007/BF00280539. Google Scholar [51] J. Wu, Generalized MHD equations, J. Differential Equations, 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007. Google Scholar [52] H. Wu and J. Fan, Weak-strong uniqueness for the generalized Navier-Stokes equations, Appl. Math. Lett., 25 (2012), 423-428. doi: 10.1016/j.aml.2011.09.028. Google Scholar
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